Asymptotic Formulas for Macdonald Polynomials

Asymptotic Formulas for Macdonald Polynomials and
the Boundary of the $\boldsymbol{(q,t)}$-Gelfand–Tsetlin Graph

Cesar CUENCA

C. Cuenca

Department of Mathematics, Massachusetts Institute of Technology, USA
<EMAIL_ADDRESS>http://math.mit.edu/~cuenca/

Received April 21, 2017, in final form December 09, 2017; Published online
January 02, 2018

We introduce Macdonald characters and use algebraic properties of Macdonald
polynomials to study them. As a result, we produce several formulas for
Macdonald characters, which are generalizations of those obtained by Gorin and
Panova in [Ann. Probab. 43 (2015), 3052–3132], and are expected to provide
tools for the study of statistical mechanical models, representation theory
and random matrices. As first application of our formulas, we characterize the
boundary of the $(q,t)$-deformation of the Gelfand–Tsetlin graph when
$t=q^{\theta}$ and $\theta$ is a positive integer.

Branching graph; Macdonald polynomials; Gelfand–Tsetlin graph

33D52; 33D90; 60B15; 60C05

##  1 Introduction

Macdonald polynomials are remarkable two-parameter $q$, $t$ generalizations of
Schur polynomials. They were first introduced by Ian G. Macdonald in [31]; the
canonical reference is his classical book [32]. The Macdonald polynomials are
very interesting objects for representation theory and integrable systems, due
to their connections with quantum groups, e.g., [20, 33], double affine Hecke
algebras, e.g., [13, 28], etc. More recently, and releveant for us, Macdonald
polynomials have been heavily used to study probabilistic models arising in
mathematical physics and random matrix theory. The important work [4] of
Borodin and Corwin showed how to use the algebraic properties of these
polynomials to obtain analytic formulas that allow asymptotic analysis of the
so-called Macdonald processes. Remarkably, by specialization or degeneration
of the parameters $q$, $t$ defining Macdonald processes, the paper [4] yields
tools that can be used to analyze interacting particle systems [5], beta
Jacobi corners processes [6], probabilistic models from asymptotic
representation theory [3], among others; see the survey [9] and references
therein.

It should be noted that the special case $t=q$ of Macdonald processes are
known as Schur processes and they have the special property of being
determinantal point processes, thus allowing much more control over their
asymptotics. Schur processes were introduced by Okounkov and Reshetikhin, as
generalizations of the classical Plancherel measures, several years before the
work of Borodin and Corwin [35, 38]. The Schur processes, though a very
special case of Macdonald processes, produced various applications to
statistical models of plane partitions and random matrices, see for example
[27, 39]. However, most of the physical models that were studied with the
Macdonald processes do not have a determinantal structure, and therefore they
could not have been analyzed solely by means of the Schur processes machinery.

The conclusion from the story of Schur and Macdonald processes is that
studying the more complicated object can allow one to tackle more complicated
questions, despite losing some integrability (such as the determinantal
structure in the case of Schur processes). We follow this philosophy in our
work, by introducing and studying Macdonald characters, two-parameter $q$, $t$
generalizations of normalized characters of unitary groups. The normalized
characters of the unitary groups are expressible in terms of Schur
polynomials, reason why we will call them Schur characters, whereas the
generalization we present involves Macdonald polynomials.

Our main results are asymptotic formulas for Macdonald characters, which are
generalizations of those for Schur characters, proved in [24] by different
methods. As it is expected, the asymptotic formulas for Schur characters are
simpler and they involve certain determinantal structure, whereas the formulas
for Macdonald polynomials are more complicated and the determinantal structure
is no longer present. However the advantage of our work on Macdonald
polynomials, much like the advantage of Macdonald processes over Schur
processes, is that we are able to access a number of asymptotic questions that
are more general than those given in [24]. The tools we obtain in this paper
are therefore very exciting, given that the formulas for Schur characters have
already produced several applications to stochastic discrete particle systems,
lozenge and domino tiling models, and asymptotic representation theory [10,
11, 12, 22, 24, 44].

The paper [15] is this article’s companion, in which the author studies Jack
characters, a natural degeneration of Macdonald characters and obtains their
asymptotics in the Vershik–Kerov limit regime. The approach to the study of
asymptotics of Jack characters is different from the approach we use here to
study the asymptotics of Macdonald characters; in particular, it relies
heavily on the Pieri integral formula, see [15] for further details. The tools
from this paper and [15] afford us a very strong control over the asymptotics
of Macdonald characters, Jack characters and Bessel functions [36, 43], if the
number of variables remains fixed and the rank tends to infinity. As another
application of the developed toolbox, we have studied a Jack–Gibbs model of
lozenge tilings in the spirit of [7, 25]. The author was able to prove the
weak convergence of statistics of the Jack–Gibbs lozenge tilings model near
the edge of the boundary to the well-known Gaussian beta ensemble, see, e.g.,
Forrester [1, Chapter 20] and references therein. This result, and its
rational limit concerning corner processes of Gaussian matrix ensembles, will
appear in a forthcoming publication.

We proceed with a more detailed description of the results of the present
paper.

###  1.1 Description of the formulas

The main object of study in this paper are the Macdonald characters, which we
define as follows. For integers $1\leq m\leq N$, a Macdonald character of rank
$N$ and $m$ variables is a polynomial, with coefficients in $\mathbb{C}(q,t)$,
of the form

$\displaystyle
P_{\lambda}(x_{1},\dots,x_{m};N,q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{P_{\lambda}\big{(}x_{1},\dots,x_{m},1,t,\dots,t^{N-m-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}},$

where $P_{\lambda}(x_{1},\dots,x_{N};q,t)$ is the Macdonald polynomial of $N$
variables parametrized by the signature
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N})\in\mathbb{Z}^{N}$.
Macdonald characters, under the specialization $t=q$, turn into $q$-Schur
characters, which have appeared previously in [21, 24]. The reason behind the
use of the word “character” is that $q$-Schur characters turn into normalized
characters of the irreducible rational representations of unitary groups,
after the degeneration $q\rightarrow 1$.

We make one further comment about terminology. Macdonald characters, as
defined here, are two-parameter $q$, $t$ degenerations of normalized and
irreducible characters of unitary groups. One could also consider a two-
parameter degeneration of the characters of the symmetric groups, in the
spirit of Lassalle’s work [29], where a one-parameter degeneration of
symmetric group characters was considered. Thus a better name for our object
would be Macdonald unitary character. For convenience, we simply will use the
name Macdonald character. We remark that Macdonald symmetric group characters
have not been considered yet, to the author’s best knowledge. However, there
have been many articles studying the structural theory and asymptotics on Jack
symmetric group characters, notably several recent works by Maciej Dołȩga,
Valentin Féray and Piotr Śniady, see, e.g., [16, 17, 18, 46].

The main theorems of this paper fall into two categories:

  1. (A)

Integral representations for Macdonald characters of one variable and
arbitrary rank $N$.

The initial idea that led to the integral representations in this paper is due
to Andrei Okounkov, see [24, remark following Theorem 3.6]. An example of the
integral formulas we prove is the following theorem. Observe that the
integrand is a simple expression in terms of $q$-Gamma functions and can be
analyzed by well known methods of asymptotic analysis, such as the method of
steepest descent or the saddle-point method [14]. In this paper, we study the
regime in which the signatures grow to infinity, whereas the remaining
parameters are fixed.

######  Theorem 1.1 (consequence of Theorem 3.2).

Assume $q\in(0,1)$ and $\theta>0$. Let $N\in\mathbb{N}$,
$\lambda\in\mathbb{GT}_{N}$ and $x\in\mathbb{C}\setminus\\{0\\}$, $|x|\leq
q^{\theta(1-N)}$. The integral below converges absolutely and the identity
holds

$\displaystyle P_{\lambda}\big{(}x;N,q,q^{\theta}\big{)}=\frac{\ln
q}{1-q}\frac{\big{(}xq^{\theta};q\big{)}_{\infty}}{\big{(}xq^{1+\theta(1-N)};q\big{)}_{\infty}}\frac{\Gamma_{q}(\theta
N)}{2\pi\sqrt{-1}}$
$\displaystyle\hphantom{P_{\lambda}\big{(}x;N,q,q^{\theta}\big{)}=}{}\times\int_{\mathcal{C}^{+}}{\big{(}xq^{\theta(1-N)}\big{)}^{z}\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}{\rm
d}z},$ (1.1)

where $\mathcal{C}^{+}$ is a certain contour described in Theorem 3.2, and
which looks as in Fig. 2. In the formula above, we used the $q$-Pochhammer
symbol $(z;q)_{\infty}$ and the $q$-Gamma function $\Gamma_{q}(z)$; see
Appendix A.

  1. (B)

Formulas expressing Macdonald characters of $m$ variables (and rank $N$) in
terms of Macdonald characters of one variable (and rank $N$).

These formulas will involve certain $q$-difference operators. Formulas of this
kind will be called multiplicative formulas.111The reason for the name is that
analogous formulas for Schur characters were used to prove statements of the
form
$\lim\limits_{N\rightarrow\infty}{F_{\lambda(N)}(x_{1},\dots,x_{m})}=\prod\limits_{i=1}^{m}{\lim\limits_{N\rightarrow\infty}{F_{\lambda(N)}(x_{i})}}$,
where the functions $F$ are certain normalizations of Schur characters, see,
e.g., [24, Corollaries 3.10 and 3.12] One of the simplest multiplicative
formulas we prove is the one below that expresses a Macdonald character of two
variables in terms of those of one variable; the general formula is given
below in Theorem 4.1.

######  Theorem 1.2 (reformulation of Corollary 4.2).

Let $\theta\in\mathbb{N}$, $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}_{N}$. Then

$\displaystyle
P_{\lambda}\big{(}x_{1},x_{2};N,q,q^{\theta}\big{)}=\frac{q^{-(N-\frac{3}{2})\theta^{2}+\frac{1}{2}\theta}(1-q)^{\theta}}{\prod\limits_{i=1}^{\theta}{(1-q^{\theta
N-i})}}\frac{1}{\prod\limits_{i=1}^{\theta(N-1)-1}{(x_{1}-q^{i-\theta})(x_{2}-q^{i-\theta})}}$
$\displaystyle\qquad{}\times\left(\frac{1}{x_{1}-x_{2}}\circ(D_{q,x_{2}}-D_{q,x_{1}})\right)^{\theta}\left\\{\prod_{i=1}^{2}{\left(P_{\lambda}\big{(}x_{i};N,q,q^{\theta}\big{)}\prod_{j=1}^{\theta
N-1}{\big{(}x_{i}-q^{j-\theta}\big{)}}\right)}\right\\},$

where $D_{q,x_{i}}$, $i=1,2$, are the linear operators in
$\mathbb{C}(q)[x_{1},x_{2}]$ acting on monomials by
$D_{q,x_{i}}(x_{1}^{m_{1}}x_{2}^{m_{2}})$
$=\frac{1-q^{m_{i}}}{1-q}(x_{1}^{m_{1}}x_{2}^{m_{2}})$, $i=1,2$.

Observe that the multiplicative formula above requires $\theta\in\mathbb{N}$.
It is somewhat surprising that all the identities we prove in this paper, even
the integral representations, behave better for $\theta\in\mathbb{N}$.

###  1.2 The boundary of the $\boldsymbol{(q,t)}$-Gelfand–Tsetlin graph

As an application of our formulas, we characterize the space of central
probability measures in the path-space of the $(q,t)$-Gelfand–Tsetlin graph,
when $t=q^{\theta}$ and $\theta$ is a positive integer. To state our result,
we first introduce a few notions. Assume that $q,t\in(0,1)$ are generic real
parameters for the moment.

The Gelfand–Tsetlin graph, or simply GT graph, is an undirected graph whose
vertices are the signatures of all lengths
$\mathbb{GT}=\bigsqcup\limits_{N\geq 0}{\mathbb{GT}_{N}}$; we also include the
empty signature $\varnothing$ as the only element of $\mathbb{GT}_{0}$ for
convenience. The set of edges is determined by the interlacing constraints,
namely the edges in the GT graph can only join signatures whose lengths differ
by $1$ and $\mu\in\mathbb{GT}_{N}$ is joined to $\lambda\in\mathbb{GT}_{N+1}$
if and only if

$\displaystyle\lambda_{N+1}\leq\mu_{N}\leq\lambda_{N}\leq\dots\leq\lambda_{2}\leq\mu_{1}\leq\lambda_{1}.$

If the above inequalities are satisfied, we write $\mu\prec\lambda$. If
$\mu\in\mathbb{GT}_{N}$ and $\lambda\in\mathbb{GT}_{N+1}$ are joined by an
edge, then we consider the expression $\Lambda^{N+1}_{N}(\lambda,\mu)$ given
by

$\displaystyle\Lambda^{N+1}_{N}(\lambda,\mu)=\psi_{\lambda/\mu}(q,t)\frac{P_{\mu}\big{(}t^{N},\dots,t^{2},t;q,t\big{)}}{P_{\lambda}\big{(}t^{N},\dots,t,1;q,t\big{)}},$

where $\psi_{\lambda/\mu}(q,t)$ is given in the branching rule for Macdonald
polynomials, see Theorem 2.5 below. If $\mu\in\mathbb{GT}_{N}$ is not joined
to $\lambda\in\mathbb{GT}_{N+1}$, set $\Lambda^{N+1}_{N}(\lambda,\mu)=0$. One
can easily show, see, e.g., Theorem 2.3 below,

$\displaystyle\Lambda^{N+1}_{N}(\lambda,\mu)\geq
0,\qquad\forall\,\lambda\in\mathbb{GT}_{N+1},\quad\mu\in\mathbb{GT}_{N},$
$\displaystyle\sum_{\mu\in\mathbb{GT}_{N}}{\Lambda^{N+1}_{N}(\lambda,\mu)}=1,\qquad\forall\,\lambda\in\mathbb{GT}_{N+1}.$

Thus, for any $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}_{N+1}$,
$\Lambda^{N+1}_{N}(\lambda,\cdot)$ is a probability measure on
$\mathbb{GT}_{N}$. For this reason, we will call the expressions
$\Lambda^{N+1}_{N}(\lambda,\mu)$ cotransition probabilities.

Next we define the path-space $\mathcal{T}$ of the GT graph as the set of
infinite paths in the GT graph that begin at $\varnothing\in\mathbb{GT}_{0}$:
$\mathcal{T}=\\{\tau=(\varnothing=\tau^{(0)}\prec\tau^{(1)}\prec\tau^{(2)}\prec\cdots)\colon\tau^{(n)}\in\mathbb{GT}_{n}\
\forall\,n\in\mathbb{Z}_{\geq 0}\\}$. Each finite path of the form
$\phi=\big{(}\varnothing=\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(n)}\big{)}$
defines a cylinder set
$S_{\phi}=\big{\\{}\tau\in\mathcal{T}\colon\tau^{(1)}=\phi^{(1)},\dots,\tau^{(n)}=\phi^{(n)}\big{\\}}\subset\mathcal{T}$.
We equip $\mathcal{T}$ with the $\sigma$-algebra generated by the cylinder
sets $S_{\phi}$, over all finite paths $\phi$. Equivalently, the
$\sigma$-algebra of $\mathcal{T}$ is its Borel $\sigma$-algebra if we equip
$\mathcal{T}$ with the topology it inherits as a subspace of the product
$\prod\limits_{n\geq 0}{\mathbb{GT}_{n}}$. Each probability measure $M$ on
$\mathcal{T}$ admits a pushforward to a probability measure on
$\mathbb{GT}_{m}$ via the obvious projection map

$\displaystyle\operatorname{Proj}_{m}\colon\ \mathcal{T}\subset\prod_{n\geq
0}{\mathbb{GT}_{n}}\longrightarrow\mathbb{GT}_{N},$
$\displaystyle\hphantom{\operatorname{Proj}_{m}\colon}{}\
\tau=\big{(}\tau^{(0)}\prec\tau^{(1)}\prec\tau^{(2)}\prec\cdots\big{)}\mapsto\tau^{(N)}.$

We say that a probability measure $M$ on $\mathcal{T}$ is a $(q,t)$-central
measure if

$\displaystyle
M\big{(}S\big{(}\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(N-1)}\prec\phi^{(N)}\big{)}\big{)}$
$\displaystyle\qquad{}=\Lambda^{N}_{N-1}\big{(}\phi^{(N)},\phi^{(N-1)}\big{)}\cdots\Lambda^{1}_{0}\big{(}\phi^{(1)},\phi^{(0)}\big{)}M_{N}\big{(}\phi^{(N)}\big{)},$

for all $N\geq 0$, all finite paths $\phi^{(0)}\prec\cdots\prec\phi^{(N)}$,
and for some probability measures $M_{N}$ on $\mathbb{GT}_{N}$. It then
automatically follows that $M_{N}=(\operatorname{Proj}_{N})_{*}M$ are the
pushforwards of $M$; moreover, they satisfy the coherence relations

$\displaystyle
M_{N}(\mu)=\sum_{\lambda\in\mathbb{GT}_{N+1}}{M_{N+1}(\lambda)\Lambda^{N+1}_{N}(\lambda,\mu)},\qquad\forall\,N\geq
0,\quad\forall\,\mu\in\mathbb{GT}_{N}.$

We denote by $M_{\rm prob}(\mathcal{T})$ the set of $(q,t)$-central
(probability) measures on $\mathcal{T}$; it is clearly a convex subset of the
Banach space of all finite and signed measures on $\mathcal{T}$. Let us denote
by $\Omega_{q,t}=\textrm{Ex}(M_{\rm prob}(\mathcal{T}))$ the set of its
extreme points. From a general theorem, we can deduce that
$\Omega_{q,t}\subset M_{\rm prob}(\mathcal{T})$ is a Borel subset. We call
$\Omega_{q,t}$, with its inherited topology, the boundary of the
$(q,t)$-Gelfand–Tsetlin graph. The theorem stated below, which is our main
application, completely characterizes the topological space $\Omega_{q,t}$.
Before stating it, let us make a couple of relevant definitions.

Consider the set of weakly increasing integers
$\mathcal{N}=\\{\nu=(\nu_{1}\leq\nu_{2}\leq\cdots)\colon\nu_{1},\nu_{2},\dots\in\mathbb{Z}\\}$
and equip it with the topology inherited from the product
$\mathbb{Z}^{\infty}=\mathbb{Z}\times\mathbb{Z}\times\cdots$ of countably many
discrete spaces. For each $k\in\mathbb{Z}$, we can define the automorphism
$A_{k}$ of $\mathcal{N}$ by $\nu\mapsto
A_{k}\nu=(\nu_{1}+k\leq\nu_{2}+k\leq\cdots)$. Clearly $A_{k}$ has inverse
$A_{-k}$. There is a similar automorphism of $\mathbb{GT}$, given by
$\lambda\mapsto A_{k}\lambda=(\lambda_{1}+k\geq\lambda_{2}+k\geq\cdots)$,
$\varnothing\mapsto A_{k}\varnothing=\varnothing$, which restricts to
automorphisms $\mathbb{GT}_{m}\rightarrow\mathbb{GT}_{m}$, for each
$m\in\mathbb{Z}_{\geq 0}$.

For any $k\in\mathbb{Z}$, one can easily show that $\mu\in\mathbb{GT}_{m}$
interlaces with $\lambda\in\mathbb{GT}_{m+1}$ iff $A_{k}\mu\in\mathbb{GT}_{m}$
interlaces with $A_{k}\lambda\in\mathbb{GT}_{m+1}$, that is, $\mu\prec\lambda$
iff $A_{k}\mu\prec A_{k}\lambda$. This allows us to define automorphisms
$A_{k}$ of $\mathcal{T}$ by

$\displaystyle A_{k}\colon\ \mathcal{T}\longrightarrow\mathcal{T},$
$\displaystyle\hphantom{A_{k}\colon}{}\
\tau=\big{(}\varnothing\prec\tau^{(1)}\prec\tau^{(2)}\prec\cdots\big{)}\mapsto
A_{k}\tau=\big{(}\varnothing\prec A_{k}\tau^{(1)}\prec
A_{k}\tau^{(2)}\prec\cdots\big{)}.$

One can similarly obtain maps $A_{k}$ on the set of finite paths of length $n$
by $\phi=(\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(n)})\mapsto
A_{k}\phi=(A_{k}\phi^{(0)}\prec A_{k}\phi^{(1)}\prec\cdots\prec
A_{k}\phi^{(n)})$. Consequently we can also define automorphisms on cylinder
sets by $A_{k}S_{\phi}=S_{A_{k}\phi}$, for all finite paths
$\phi=(\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(n)})$, in the natural
way.

We named several maps above by the same letter $A_{k}$, but there should be no
risk of confusion.

For our main theorem, we make the assumption $\theta\in\mathbb{N}$,
$t=q^{\theta}$. We believe the theorem can be generalized for any $\theta>0$,
but we do not have a proof at the moment.

######  Theorem 1.3.

Assume $q\in(0,1)$, $\theta\in\mathbb{N}$ and set $t=q^{\theta}$.

  1. $1.$

There exists a homeomorphism
$\mathbf{N}\colon\mathcal{N}\rightarrow\Omega_{q,t}$ sending each
$\nu\in\mathcal{N}$ to the $(q,t)$-central probability measure
$M^{\nu}\in\Omega_{q,t}$ determined by the relations

$\displaystyle\sum_{\lambda\in\mathbb{GT}_{m}}{M_{m}^{\nu}(\lambda)\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{m-1};q,t\big{)}}}=\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t\big{)},$
$\displaystyle\forall\,m\in\mathbb{N},\qquad\forall\,(x_{1},\dots,x_{m})\in\mathbb{T}^{m}.$
(1.2)

In (1.2), we denoted by $\\{M^{\nu}_{m}\\}_{m\geq 1}$ the corresponding
sequence of pushforwards of $M^{\nu}$ under the projection maps
$\operatorname{Proj}_{m}\colon\mathcal{T}\rightarrow\mathbb{GT}_{m}$. The left
side in (1.2) is absolutely convergent on $\mathbb{T}^{m}$,
$\mathbb{T}=\\{z\in\mathbb{C}\colon|z|=1\\}$, and the functions $\Phi^{\nu}$
in the right side are defined in (5.2) and (5.8). The probability measure
$M^{\nu}$ is determined uniquely by the relations (1.2).

  2. $2.$

For each $k\in\mathbb{Z}$, the probability measures $M^{\nu}$ and
$M^{A_{k}\nu}$ are related by

$\displaystyle M^{A_{k}\nu}(S_{A_{k}\phi})=M^{\nu}(S_{\phi}),\quad\textrm{for
all finite
paths}\quad\phi=\big{(}\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(n)}\big{)}.$
(1.3)

Moreover the $(q,t)$-coherent sequences $\\{M^{\nu}_{m}\\}_{m\geq 0}$ and
$\\{M^{A_{k}\nu}_{m}\\}_{m\geq 0}$ are related by

$\displaystyle
M_{m}^{A_{k}\nu}(A_{k}\lambda)=M^{\nu}_{m}(\lambda),\qquad\forall\,m\geq
0,\qquad\lambda\in\mathbb{GT}_{m}.$ (1.4)

Another main result of this article is Theorem 7.9, where we characterize the
Martin boundary of the $(q,t)$-Gelfand–Tseltin graph for $t=q^{\theta}$ and
$\theta\in\mathbb{N}$. In fact, we first prove that the Martin boundary is
homeomorphic to $\mathcal{N}$ and then show that the minimal boundary
$\Omega_{q,t}$ coincides with the Martin boundary. See Sections 7.1 and 7.2
for the definition and characterization of the Martin boundary.

###  1.3 Comments on Theorem 1.3 and connections to existing literature

Our first comment is that Theorem 1.3 is a generalization of the main theorem
in the article of Vadim Gorin [21], which is the special case $\theta=1$ of
our theorem, and characterizes the boundary of the $q$-Gelfand–Tsetlin graph.
Some ideas in the proofs are the same, especially the overall scheme of using
the ergodic method of Vershik–Kerov, see [47], but we need many new arguments
as well. For example, [21] makes heavy use of the shifted Macdonald
polynomials, in particular the binomial formula for shifted Macdonald
polynomials at $t=q$, [34], while we do not use them at all. Moreover, in
order to prove that the boundary of the $q$-Gelfand–Tsetlin graph is
homeomorphic to $\mathcal{N}$, [21] made use of the following closed formula
for the shifted-Schur generating function of $M_{N}^{\nu,\theta=1}$, in the
case that $\nu_{1}\geq 0$:

$\displaystyle\sum_{\lambda\in\mathbb{GT}^{+}_{N}}{M_{N}^{\nu,\theta=1}(\lambda)\frac{s_{\lambda}^{*}\big{(}q^{N-1}x_{1},\dots,q^{N-1}x_{N};q^{-1}\big{)}}{s_{\lambda}^{*}\big{(}0,\dots,0;q^{-1}\big{)}}}=H^{\nu}(x_{1})\cdots
H^{\nu}(x_{N}),$ (1.5)

where

$\displaystyle
H^{\nu}(x)=\frac{\prod\limits_{i=0}^{\infty}{(1-q^{i}t)}}{\prod\limits_{j=1}^{\infty}{(1-q^{\nu_{j}+j-1}t)}}.$

In the formula above, $s_{\lambda}^{*}(x_{1},\dots,x_{N};q)$ is the shifted
Macdonald polynomial at $t=q$. In addition to the usefulness of the closed
formula (1.5) above, the multiplicative structure is surprising. It would be
interesting to find a closed formula for the shifted Macdonald generating
function of the measures $M_{N}^{\nu}$, for general $\theta\in\mathbb{N}$, and
find out if the multiplicative structure still holds in this generality.

It is shown in [21] that their main statement is equivalent to the
characterization of certain Gibbs measures on lozenge tilings. A conjectural
characterization of positive $q$-Toeplitz matrices is also given in that
paper. Finally, it is mentioned that the asymptotics of $q$-Schur functions is
related to quantum traces and the representation theory of
$U_{\epsilon}(\mathfrak{gl}_{\infty})$. It would be interesting to extend some
of these statements to the Macdonald case, especially to connect the
asymptotics of Macdonald characters to the representation theory of inductive
limits of quantum groups.

Several other “boundary problems” have appeared in the literature in various
contexts. For instance, in the limiting case $t=q\rightarrow 1$, the problem
of characterizing the boundary $\Omega_{q,t}$ becomes equivalent to
characterizing the space of extreme characters of the infinite-dimensional
unitary group ${\rm U}(\infty)=\lim\limits_{\rightarrow}{{\rm U}(N)}$. The
answer also characterizes totally positive Toeplitz matrices [19, 47, 48].
Also in the degenerate case $t=q^{2}$ or $t=q^{1/2}$ and $q\rightarrow 1$, the
boundary problem becomes equivalent to characterizing the space of extreme
spherical functions of the infinite-dimensional Gelfand pairs $({\rm
U}(2\infty),{\rm Sp}(\infty))$ and $({\rm U}(\infty),{\rm O}(\infty))$,
respectively. This question, and in fact a more general one-parameter
“Jack”-degeneration, was solved in [37]. A similar degenerate question in the
setting of random matrix theory was studied in [42]. Some of the tools in this
paper can be degenerated easily to these scenarios and they may provide an
alternative approach to their proof as well; for example, the special case of
our toolbox in the case $t=q\rightarrow 1$ was used in [24] to study the
corresponding boundary problem, and in [15] we also study refine the
asymptotic result that is needed to solve the boundary problem of [37].

Another similar boundary problem in a somewhat different direction is the
following. Assume we consider the Young graph instead of the Gelfand–Tsetlin
graph, e.g., see [8]. Assume also that the cotransition probabilities coming
from the branching rule of Macdonald polynomials are replaced by the
cotransition probabilities coming from the Pieri-rule. In this setting, the
boundary problem has not been solved yet, but it is expected that the answer
is given by Kerov’s conjecture, which characterizes Macdonald-positive
specializations, see [4, Section 2].

Finally, it was brought to my attention, after I completed the results of this
paper, that Grigori Olshanski has obtained a characterization of the extreme
set of $(q,t)$-central measures in the extended Gelfand–Tsetlin graph for more
general parameters $q$, $t$ by different methods. His work follows the setting
of the paper [23] of Gorin–Olshanski, which is some sort of analytic
continuation to our proposed boundary problem. Interestingly, new features
arise, e.g., two copies of the space $\mathcal{N}$ characterize the boundary
in his context, one can define and work with suitable analogues of
$zw$-measures, etc. Another related work in the $t=q$ case is his recent
article [41].

###  1.4 Organization of the paper

The present work is organized as follows. In Section 2, we briefly recall some
important algebraic properties of Macdonald polynomials that will be used to
obtain our main results. We prove integral representations for Macdonald
characters of one variable in Section 3. Next, in Section 4, we obtain
multiplicative formulas for Macdonald characters of a given number of
variables $m\in\mathbb{N}$ in terms of those of one variable. By making use of
our formulas, in Section 5 we obtain asymptotics of Macdonald characters as
the signatures grow to infinity in a specific limit regime. In Sections 6 and
7, we define and characterize the boundary of the $(q,t)$-Gelfand–Tsetlin
graph in the case that $\theta\in\mathbb{N}$ and $t=q^{\theta}$. The
asymptotic statements of Section 5 play the key role in the characterization
of the boundary.

In Appendix A, we have bundled the necessary language and results of
$q$-theory that are used throughout the paper. In Appendix B, we make some
computations with expressions that appear in the multiplicative formulas for
Macdonald polynomials.

##  2 Symmetric Laurent polynomials

A canonical reference for symmetric polynomials is [32]. We choose to give a
brief overview of the tools that we need from [32], in order to fix
terminology and to introduce lesser known objects, such as signatures and
Macdonald Laurent polynomials.

###  2.1 Partitions, signatures and symmetric Laurent polynomials

A partition is a finite sequence of weakly decreasing nonnegative integers
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k})$,
$\lambda_{i}\in\mathbb{Z}_{\geq 0}$ $\forall\,i$. We identify partitions that
differ by trailing zeroes; for example, $(4,2,2,0,0)$ and $(4,2,2)$ are the
same partition. We define the size of $\lambda$ to be the sum
$|\lambda|\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\lambda_{1}+\dots+\lambda_{k}$, and its length $\ell(\lambda)$ to
be the number of strictly positive elements of it. The dominance order for
partitions is a partial order given by letting $\mu\leq\lambda$ if
$|\mu|=|\lambda|$ and
$\mu_{1}+\cdots+\mu_{i}\leq\lambda_{1}+\cdots+\lambda_{i}$ for all $i$. As
usual, we let $\mu<\lambda$ if $\mu\leq\lambda$ and $\mu\neq\lambda$.

Partitions can be graphically represented by their Young diagrams. The Young
diagram of partition $\lambda$ is the array of boxes with coordinates $(i,j)$
with $1\leq j\leq\lambda_{i}$, $1\leq i\leq\ell(\lambda)$, where the
coordinates are in matrix notation (row labels increase from top to bottom and
column labels increase from left to right), see Fig. 1.

centertableaux &

s

Figure 1: Young diagram for the partition $\lambda=(5,4,4,2)$. Square
$s=(3,3)$ has arm length, arm colength, leg length and leg colength given by
$a(s)=1$, $a^{\prime}(s)=2$, $l(s)=0$, $l^{\prime}(s)=2$.

A signature is a sequence of weakly decreasing integers
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{k})$,
$\lambda_{i}\in\mathbb{Z}$ $\forall\,i$. A positive signature is a signature
whose elements are all nonnegative. The length of a signature, or positive
signature, is the number $k$ of elements of it. Positive signatures which
differ by trailing zeroes are not identified, in contrast to partitions; for
example, $(4,2,2,0,0)$ and $(4,2,2)$ are different positive signatures, the
first of length $5$ and the second of length $3$. We shall denote
$\mathbb{GT}_{N}$ (resp. $\mathbb{GT}^{+}_{N}$) the set of signatures (resp.
positive signatures) of length $N$. Evidently $\mathbb{GT}^{+}_{N}$ can be
identified with the set of all partitions of length $\leq N$. Under this
identification, we are allowed to talk about the Young diagram of a positive
signature $\lambda\in\mathbb{GT}^{+}_{N}$, its size, the dominance order, and
other attributes that are typically associated to partitions. Note, however,
that length is defined differently for partitions and for positive signatures.

Let us now switch to notions pertaining to symmetric (Laurent) polynomials.
Fix a positive integer $N$. Consider the field $F=\mathbb{C}(q,t)$ and recall
the algebra $\Lambda_{F}[x_{1},\dots,x_{N}]$ of symmetric polynomials on the
variables $x_{1},\dots,x_{N}$ with coefficients in $F$. For any
$m\in\mathbb{Z}_{\geq 0}$, recall also the subalgebra
$\Lambda_{F}^{m}[x_{1},\dots,x_{N}]$ of symmetric polynomials on
$x_{1},\dots,x_{N}$ that are homogeneous of degree $m$; then

$\displaystyle\Lambda_{F}[x_{1},\dots,x_{N}]=\bigoplus_{m\geq
0}{\Lambda_{F}^{m}[x_{1},\dots,x_{N}]}.$

We also denote by $\Lambda_{F}[x_{1}^{\pm},\dots,x_{N}^{\pm}]$ the algebra of
symmetric (with respect to the transpositions $x_{i}\leftrightarrow x_{i+1}$
for $i=1,\dots,N-1$) Laurent polynomials in the variables $x_{1},\dots,x_{N}$.

The connection between partitions/signatures and symmetric polynomials comes
from the observation that
$\dim_{F}\left(\Lambda_{F}^{m}[x_{1},\dots,x_{N}]\right)$ is the number of
partitions of size $m$ and length $\leq N$, or equivalently the number of
positive signatures of size $m$ and length $N$. A basis for the space
$\Lambda_{F}^{m}[x_{1},\dots,x_{N}]$ is given by the monomial symmetric
polynomials $m_{\lambda}(x_{1},\dots,x_{N})$, with $|\lambda|=m$,
$\ell(\lambda)\leq N$, defined by

$\displaystyle
m_{\lambda}(x_{1},\dots,x_{N})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{\mu\in S_{N}\cdot\lambda}{x_{1}^{\mu_{1}}\cdots
x_{N}^{\mu_{N}}},$

where $S_{N}\cdot\lambda$ is the orbit of $\lambda$ under the permutation
action of $S_{N}$, and the sum runs over distinct elements $\mu$ of that
orbit. It is implied that
$\\{m_{\lambda}(x_{1},\dots,x_{N})\colon\ell(\lambda)\leq N\\}$ is a basis of
$\Lambda_{F}[x_{1},\dots,x_{N}]$.

###  2.2 Macdonald polynomials and Macdonald characters

######  Proposition/Definition 2.1 ([32, Chapter VI, Sections 3, 4, 9]).

The Macdonald polynomials $P_{\lambda}(x_{1},\dots,x_{N};q,t)$, for partitions
$\lambda$ with $\ell(\lambda)\leq N$, are the unique elements of
$\Lambda_{F}[x_{1},\dots,x_{N}]$ satisfying the following two properties

  * •

Triangular decomposition: $\displaystyle
P_{\lambda}(x_{1},\dots,x_{N};q,t)=m_{\lambda}+\sum\limits_{\mu\colon\mu<\lambda}{c_{\lambda,\mu}m_{\mu}}$,
for some $c_{\lambda,\mu}\in F$, and the sum is over partitions $\mu$ with
$\ell(\mu)\leq N$, and $\mu<\lambda$ in the dominance order.

  * •

Orthogonality relation: Let $[\cdot]_{0}\colon
F[x_{1}^{\pm},\dots,x_{N}^{\pm}]\rightarrow F$ be the constant term map

$\displaystyle\left[\sum_{\lambda=(\lambda_{1}\geq\cdots\geq\lambda_{N})\in\mathbb{Z}^{N}}{a_{\lambda}x_{1}^{\lambda_{1}}\cdots
x_{N}^{\lambda_{N}}}\right]_{0}=a_{(0,\dots,0)}.$

The Macdonald polynomials are orthogonal with respect to the inner product
$(\cdot,\cdot)_{q,t}$ on $\Lambda_{F}[x_{1}^{\pm},\dots,x_{N}^{\pm}]$ given by
$\displaystyle(f,g)_{q,t}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}[f(x_{1},\dots,x_{N})g\big{(}x_{1}^{-1},\dots,x_{N}^{-1}\big{)}\Delta_{q,t}]_{0}$,
where

$\Delta_{q,t}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod_{1\leq i\neq j\leq
N}\prod_{k=0}^{\infty}{\frac{1-q^{k}x_{i}x_{j}^{-1}}{1-q^{k}tx_{i}x_{j}^{-1}}}.$

Note that $P_{\varnothing}(q,t)=1$. If $N<\ell(\lambda)$, we set
$P_{\lambda}(x_{1},\dots,x_{N};q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}0$ for convenience.

When we are talking about Macdonald polynomials and some of their properties
which hold regardless of the number $N$ of variables, as long as $N$ is large
enough, we simply write $P_{\lambda}(q,t)$ instead of
$P_{\lambda}(x_{1},\dots,x_{N};q,t)$.

From the triangular decomposition of Macdonald polynomials, $P_{\lambda}(q,t)$
is a homogeneous polynomial of degree $|\lambda|$. Moreover
$\\{P_{\lambda}(x_{1},\dots,x_{N})\colon\ell(\lambda)\leq N\\}$ is a basis of
$\Lambda_{F}[x_{1},\dots,x_{N}]$. Finally, we have the following index
stability property:

$\displaystyle
P_{(\lambda_{1}+1,\dots,\lambda_{N}+1)}(x_{1},\dots,x_{N};q,t)=(x_{1}\cdots
x_{N})\cdot P_{\lambda}(x_{1},\dots,x_{N};q,t).$ (2.1)

As pointed out before, the set of partitions of length $\leq N$ is in
bijection with $\mathbb{GT}^{+}_{N}$. Thus we can index the Macdonald
polynomials by positive signatures rather than by partitions: for any
$\lambda\in\mathbb{GT}^{+}_{N}$, we let $P_{\lambda}(x_{1},\dots,x_{N};q,t)$
be the Macdonald polynomial corresponding to the partition associated to
$\lambda$. We can slightly extend the definition above and introduce Macdonald
Laurent polynomials $P_{\lambda}(x_{1},\dots,x_{N};q,t)$ for any
$\lambda\in\mathbb{GT}_{N}$. Let $\lambda\in\mathbb{GT}_{N}$ be arbitrary. If
$\lambda_{N}\geq 0$, then $\lambda\in\mathbb{GT}^{+}_{N}$ and
$P_{\lambda}(x_{1},\dots,x_{N};q,t)$ is already defined. If $\lambda_{N}<0$,
choose $m\in\mathbb{N}$ such that $\lambda_{N}+m\geq 0$ and so
$(\lambda_{1}+m,\dots,\lambda_{N}+m)\in\mathbb{GT}^{+}_{N}$. Then define

$\displaystyle
P_{\lambda}(x_{1},\dots,x_{N};q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}(x_{1}\cdots x_{N})^{-m}\cdot
P_{(\lambda_{1}+m,\dots,\lambda_{N}+m)}(x_{1},\dots,x_{N};q,t).$ (2.2)

By virtue of the index stability property, the Macdonald (Laurent) polynomial
$P_{\lambda}(x_{1},\dots,x_{N};q,t)$ is well-defined and does not depend on
the value of $m$ that we choose. For simplicity, we call
$P_{\lambda}(x_{1},\dots,x_{N};q,t)$ a Macdonald polynomial, whether
$\lambda\in\mathbb{GT}^{+}_{N}$ or not. In a similar fashion, we can define
monomial symmetric polynomials $m_{\lambda}$, for any
$\lambda\in\mathbb{GT}_{N}$.

Recall the definitions of the arm-length, arm-colength, leg-length, leg-
colength $a(s)$, $a^{\prime}(s)$, $l(s)$, $l^{\prime}(s)$ of the square
$s=(i,j)$ of the Young diagram of $\lambda$, given by $a(s)=\lambda_{i}-j$,
$a^{\prime}(s)=j-1$, $l(s)=\lambda_{j}^{\prime}-i$, $l^{\prime}(s)=i-1$; we
note that $\lambda_{j}^{\prime}=|\\{i\colon\lambda_{i}\geq j\\}|$ is the
length of the $j$th part of the conjugate partition $\lambda^{\prime}$, see
Fig. 1.

We use terminology from $q$-analysis, see Appendix A; particularly we use the
definition of $q$-Pochhammer symbols
$(z;q)_{n}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod\limits_{i=0}^{n-1}{(1-zq^{i})}$ and
$(z;q)_{\infty}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod\limits_{i=0}^{\infty}{(1-zq^{i})}$.

For $\lambda\in\bigsqcup\limits_{N\geq 0}{\mathbb{GT}^{+}_{N}}$, define the
dual Macdonald polynomials $Q_{\lambda}(q,t)$ as the following normalization
of Macdonald polynomials

$\displaystyle
Q_{\lambda}(q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}b_{\lambda}(q,t)P_{\lambda}(q,t),\qquad
b_{\lambda}(q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod_{s\in\lambda}{\frac{1-q^{a(s)}t^{l(s)+1}}{1-q^{a(s)+1}t^{l(s)}}}.$
(2.3)

The complete homogeneous symmetric $($Macdonald$)$ polynomials
$g_{0}=1,g_{1},g_{2},\dots$ are the one-row dual Macdonald polynomials:

$\displaystyle
g_{n}(q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}Q_{(n)}(q,t)=\frac{(q;q)_{n}}{(t;q)_{n}}P_{(n)}(q,t).$ (2.4)

For convenience, we also set
$g_{n}(q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}0$, $\forall\,n<0$.

Now we come to several important theorems on Macdonald polynomials, which will
be our main tools.

######  Theorem 2.2 (index-argument symmetry; [32, Chapter VI, Property 6.6]).

Let $N\in\mathbb{N}$, $\lambda,\mu\in\mathbb{GT}^{+}_{N}$, then

$\displaystyle\frac{P_{\lambda}\big{(}q^{\mu_{1}}t^{N-1},q^{\mu_{2}}t^{N-2},\dots,q^{\mu_{N}};q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}=\frac{P_{\mu}\big{(}q^{\lambda_{1}}t^{N-1},q^{\lambda_{2}}t^{N-2},\dots,q^{\lambda_{N}};q,t\big{)}}{P_{\mu}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}.$

######  Theorem 2.3 (evaluation identity; [32, Chapter VI, equations (6.11)
and (6.11′)]).

Let $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}^{+}_{N}$, then

$\displaystyle
P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}=t^{n(\lambda)}\prod_{1\leq
i<j\leq
N}{\frac{\big{(}q^{\lambda_{i}-\lambda_{j}}t^{j-i};q\big{)}_{\infty}(t^{j-i+1};q)_{\infty}}{\big{(}q^{\lambda_{i}-\lambda_{j}}t^{j-i+1};q\big{)}_{\infty}(t^{j-i};q)_{\infty}}}$
$\displaystyle\hphantom{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}{}=t^{n(\lambda)}\prod_{s\in\lambda}{\frac{1-q^{a^{\prime}(s)}t^{N-l^{\prime}(s)}}{1-q^{a(s)}t^{l(s)+1}}},$

where and
$n(\lambda)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\lambda_{2}+2\lambda_{3}+\cdots+(N-1)\lambda_{N}$. The first
equality holds, more generally, for any signature $\lambda\in\mathbb{GT}_{N}$
by virtue of the definition (2.2) of Macdonald Laurent polynomials
$P_{\lambda}(x_{1},\dots,x_{N};q,t)$.

From the second equality of Theorem 2.3 and the definition of dual Macdonald
polynomials, we obtain

######  Corollary 2.4.

Let $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}^{+}_{N}$, then

$\displaystyle
Q_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}=t^{n(\lambda)}\prod_{s\in\lambda}{\frac{1-q^{a^{\prime}(s)}t^{N-l^{\prime}(s)}}{1-q^{a(s)+1}t^{l(s)}}}.$

Since the Macdonald polynomial $P_{\lambda}(x_{1},x_{2},\dots,x_{N};q,t)$ is
symmetric in $x_{1},x_{2},\dots,x_{N}$, it is also a symmetric polynomial on
$x_{2},\dots,x_{N}$; thus it is a linear combination of Macdonald polynomials
$P_{\mu}(x_{2},\dots,x_{N};q,t)$ with coefficients in $F[x_{1}]$. More
precisely, we have the so-called branching rule for Macdonald polynomials:

######  Theorem 2.5 (branching rule; [32, Chapter VI, equation (7.13′),
Example 2(b) on p. 342]).

Let $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}^{+}_{N}$, then

$\displaystyle
P_{\lambda}(x_{1},x_{2},\dots,x_{N};q,t)=\sum_{\mu\in\mathbb{GT}^{+}_{N-1}\colon\mu\prec\lambda}{\psi_{\lambda/\mu}(q,t)x_{1}^{|\lambda|-|\mu|}P_{\mu}(x_{2},\dots,x_{N};q,t)},$

where the branching coefficients are

$\displaystyle\psi_{\lambda/\mu}(q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod_{1\leq i\leq j\leq
N-1}\frac{\big{(}q^{\mu_{i}-\mu_{j}}t^{j-i+1};q\big{)}_{\infty}\big{(}q^{\lambda_{i}-\lambda_{j+1}}t^{j-i+1};q\big{)}_{\infty}}{\big{(}q^{\lambda_{i}-\mu_{j}}t^{j-i+1};q\big{)}_{\infty}\big{(}q^{\mu_{i}-\lambda_{j+1}}t^{j-i+1};q\big{)}_{\infty}}$
$\displaystyle\hphantom{\psi_{\lambda/\mu}(q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod_{1\leq i\leq j\leq
N-1}}{}\times\frac{\big{(}q^{\lambda_{i}-\mu_{j}+1}t^{j-i};q\big{)}_{\infty}\big{(}q^{\mu_{i}-\lambda_{j+1}+1}t^{j-i};q\big{)}_{\infty}}{\big{(}q^{\mu_{i}-\mu_{j}+1}t^{j-i};q\big{)}_{\infty}\big{(}q^{\lambda_{i}-\lambda_{j+1}+1}t^{j-i};q\big{)}_{\infty}},$

and the sum is over positive signatures $\mu\in\mathbb{GT}^{+}_{N-1}$ that
satisfy the interlacing constraint

$\displaystyle\lambda_{N}\leq\mu_{N-1}\leq\lambda_{N-1}\leq\cdots\leq\mu_{1}\leq\lambda_{1},$

which is written succinctly as $\mu\prec\lambda$.

Observe that $\psi_{\lambda/\mu}(q,t)>0$, whenever $q,t\in(0,1)$. By applying
the branching rule several times, we can deduce the following.

######  Corollary 2.6.

The coefficients $c_{\lambda,\mu}=c_{\lambda,\mu}(q,t)$ in the expansion

$\displaystyle P_{\lambda}(q,t)=\sum_{\mu}{c_{\lambda,\mu}m_{\mu}}$

are such that $c_{\lambda,\lambda}=1$ and $c_{\lambda,\mu}\geq 0$, whenever
$q,t\in(0,1)$. Moreover, $c_{\lambda,\mu}=0$ unless $\lambda\geq\mu$.

We come to our final tool on Macdonald polynomials. It is the main theorem of
[30], and is called the Jacobi–Trudi formula for Macdonald polynomials. For
any $n\in\mathbb{N}$, nonnegative integers $\tau_{1},\dots,\tau_{n}$,
variables $u_{1},\dots,u_{n}$, define the rational functions
$C_{\tau_{1},\dots,\tau_{n}}^{(q,t)}(u_{1},\dots,u_{n})$ by

$\displaystyle
C_{\tau_{1},\dots,\tau_{n}}^{(q,t)}(u_{1},\dots,u_{n})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod_{k=1}^{n}{t^{\tau_{k}}\frac{(q/t;q)_{\tau_{k}}}{(q;q)_{\tau_{k}}}\frac{(qu_{k};q)_{\tau_{k}}}{(qtu_{k};q)_{\tau_{k}}}}\prod_{1\leq
i<j\leq
n}{\frac{(qu_{i}/tu_{j};q)_{\tau_{i}}}{(qu_{i}/u_{j};q)_{\tau_{i}}}\frac{(tu_{i}/(q^{\tau_{i}}u_{j});q)_{\tau_{i}}}{(u_{i}/(q^{\tau_{i}}u_{j});q)_{\tau_{i}}}}$
$\displaystyle\qquad{}\times\frac{1}{\Delta(q^{\tau_{1}}u_{1},\dots,q^{\tau_{n}}u_{n})}\det_{1\leq
i,j\leq
n}{\left[(q^{\tau_{i}}u_{i})^{n-j}\\!\left(\\!1-t^{j-1}\frac{1-tq^{\tau_{i}}u_{i}}{1-q^{\tau_{i}}u_{i}}\prod_{k=1}^{n}{\frac{u_{k}-q^{\tau_{i}}u_{i}}{tu_{k}-q^{\tau_{i}}u_{i}}}\right)\\!\right]},\\!\\!\\!$
(2.5)

where
$\Delta(z_{1},\dots,z_{n})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod\limits_{1\leq i<j\leq
n}{(z_{i}-z_{j})}=\det\big{[}z_{i}^{n-j}\big{]}_{i,j=1}^{n}$ is known as the
Vandermonde determinant.

######  Theorem 2.7 (Jacobi–Trudi formula; [30, Theorem 5.1]).

Let $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}^{+}_{N}$, then

$\displaystyle Q_{\lambda}(x_{1},\dots,x_{N};q,t)$
$\displaystyle\qquad{}=\sum_{\tau\in
M^{(N)}}\left\\{\prod_{s=1}^{N-1}{C^{(q,t)}_{\tau_{1,s+1},\dots,\tau_{s,s+1}}\left(u_{i}=q^{\lambda_{i}-\lambda_{s+1}+\sum\limits_{j=s+2}^{N}{(\tau_{i,j}-\tau_{s+1,j})}}t^{s-i}\colon
1\leq i\leq s\right)}\right.$
$\displaystyle\left.\qquad\quad{}\times\prod_{s=1}^{N}{g_{\lambda_{s}+\tau_{s}^{+}-\tau_{s}^{-}}(x_{1},\dots,x_{N};q,t)}\right\\},$

where $M^{(N)}$ is the set of strictly upper-triangular matrices with
nonnegative entries, and for each $1\leq s\leq N$, the integers
$\tau_{s}^{+}$, $\tau_{s}^{-}$, depend only on the indexing matrix $\tau$ and
are defined by

$\displaystyle\tau_{s}^{+}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{i=s+1}^{N}{\tau_{s,i}},\qquad\tau_{s}^{-}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{i=1}^{s-1}{\tau_{i,s}}.$ (2.6)

######  Remark 2.8.

Observe that, even though $M^{(N)}$ is an infinite set, the only nonvanishing
terms in the sum above are those $\tau\in M^{(N)}$ such that
$\lambda_{s}+\tau_{s}^{+}-\tau_{s}^{-}\geq 0$ $\forall\,s=1,\dots,N$. In other
words, the sum is indexed by points of the discrete
$\frac{N(N-1)}{2}$-dimensional simplex with coordinates
$\\{\tau_{i,j}\\}_{1\leq i<j\leq N}$ satisfying

$\displaystyle\tau_{i,j}\geq 0,\qquad\textrm{for all}\quad 1\leq i<j\leq N,$
$\displaystyle\lambda_{n}+\sum_{i=n+1}^{N}{\tau_{n,i}}-\sum_{i=1}^{n-1}{\tau_{i,n}}\geq
0,\qquad\textrm{for all}\quad n=1,\dots,N.$

Let us introduce the last piece of terminology and main object of study in
this paper.

######  Definition 2.9.

For any $m,N\in\mathbb{N}$ with $1\leq m\leq N$, $\lambda\in\mathbb{GT}_{N}$,
define

$\displaystyle
P_{\lambda}(x_{1},\dots,x_{m};N,q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{P_{\lambda}\big{(}x_{1},\dots,x_{m},1,t,\dots,t^{N-m-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}$
(2.7)

and call $P_{\lambda}(x_{1},\dots,x_{m};N,q,t)$ the Macdonald unitary
character of rank $N$, number of variables $m$ and parametrized by $\lambda$.
For simplicity of terminology, we call $P_{\lambda}(x_{1},\dots,x_{m};N,q,t)$
a Macdonald character rather than a Macdonald unitary character. Observe that
if $q,t\in\mathbb{C}$ are such that $|q|,|t|\in(0,1)$, the evaluation identity
for Macdonald polynomials, Theorem 2.3, shows that the denominator of (2.7) is
nonzero.

##  3 Integral formulas for Macdonald characters of one variable

In this section, assume $q$ is a real number in the interval $(0,1)$. There
will also be a parameter $\theta$, typically $\theta>0$, but we also consider
cases when $\theta$ is a complex number with $\Re\theta>0$. In either case,
the parameter $t=q^{\theta}$ satisfies $|t|<1$.

###  3.1 Statements of the theorems

The simplest contour integral representation is the following, which works
only when $t=q^{\theta}$, $\theta\in\mathbb{N}$, and involves a closed contour
around finitely many singularities.

######  Theorem 3.1.

Let $\theta\in\mathbb{N}$, $t=q^{\theta}$, $N\in\mathbb{N}$,
$\lambda\in\mathbb{GT}_{N}$ and
$x\in\mathbb{C}\setminus\big{\\{}0,q,q^{2},\dots,q^{\theta N-1}\big{\\}}$.
Then

$\displaystyle\frac{P_{\lambda}\big{(}x,t,t^{2},\dots,t^{N-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}$
$\displaystyle\qquad{}=\ln(1/q)\prod_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{x-q^{i}}}\frac{1}{2\pi\sqrt{-1}}\oint_{\mathcal{C}_{0}}{\frac{x^{z}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{z-(\lambda_{i}+\theta(N-i)+j)}\big{)}}}{\rm
d}z},$ (3.1)

where $\mathcal{C}_{0}$ is a closed, positively oriented contour enclosing the
real poles $\\{\lambda_{i}+\theta(N-i)+j\colon
i=1,\dots,N,\,j=0,\dots,\theta-1\\}$ of the integrand. For instance, the
rectangular contour with vertices $-M-r\sqrt{-1}$, $-M+r\sqrt{-1}$,
$M+r\sqrt{-1}$ and $M-r\sqrt{-1}$, for any $-\frac{2\pi}{\ln{q}}>r>0$ and any
$M>\max\\{0,-\lambda_{N},\,\lambda_{1}+\theta N-1\\}$, is a suitable contour.

The following two theorems are analytic continuations, in the variable
$\theta$, of Theorem 3.1 above.

######  Theorem 3.2.

Let $\theta>0$, $t=q^{\theta}$, $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}_{N}$
and $x\in\mathbb{C}\setminus\\{0\\}$, $|x|\leq 1$. The integral below
converges absolutely and the equality holds

$\displaystyle\frac{P_{\lambda}\big{(}xt^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}$
$\displaystyle\qquad{}=\frac{\ln
q}{1-q}\frac{\big{(}xt^{N};q\big{)}_{\infty}}{(xq;q)_{\infty}}\frac{\Gamma_{q}(\theta
N)}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}{\rm
d}z}.$ (3.2)

Contour $\mathcal{C}^{+}$ is a positively oriented contour consisting of the
segment $[M+r\sqrt{-1},M-r\sqrt{-1}]$ and the horizontal lines
$[M+r\sqrt{-1},+\infty+r\sqrt{-1})$, $[M-r\sqrt{-1},+\infty-r\sqrt{-1})$, for
some $-\frac{\pi}{2\ln{q}}>r>0$ and $\lambda_{N}>M$, see Fig. 2. Observe that
$\mathcal{C}^{+}$ encloses all real poles of the integrand $($which accumulate
at $+\infty)$ and no other poles.

The reader is referred to Appendix A for a reminder of the definition of the
$q$-Gamma function, its zeroes and poles.

######  Theorem 3.3.

Let $\theta>0$, $t=q^{\theta}$, $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}_{N}$
and $x\in\mathbb{C}$, $|x|\geq 1$. The integral below converges absolutely and
the equality holds

$\displaystyle\frac{P_{\lambda}\big{(}x,t,t^{2},\dots,t^{N-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}$
$\displaystyle\qquad{}=\frac{\ln
q}{q-1}\frac{\big{(}x^{-1}t^{N};q\big{)}_{\infty}}{(x^{-1}q;q)_{\infty}}\frac{\Gamma_{q}(\theta
N)}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{-}}{x^{z}\prod_{i=1}^{N}{\frac{\Gamma_{q}(z-(\lambda_{i}-\theta
i+\theta))}{\Gamma_{q}(z-(\lambda_{i}-\theta i))}}{\rm d}z}.$ (3.3)

Contour $\mathcal{C}^{-}$ is a positively oriented contour consisting of the
segment $[M-r\sqrt{-1},M+r\sqrt{-1}]$ and the horizontal lines
$[M-r\sqrt{-1},-\infty-r\sqrt{-1})$, $[M+r\sqrt{-1},-\infty+r\sqrt{-1})$, for
some $-\frac{\pi}{2\ln{q}}>r>0$ and $M>\lambda_{1}$, see Fig. 3. Observe that
$\mathcal{C}^{-}$ encloses all real poles of the integrand $($which accumulate
at $-\infty)$ and no other poles.

$\Re z$$\Im z$ Figure 2: Contour $\mathcal{C}^{+}$. $\Re z$$\Im z$ Figure 3:
Contour $\mathcal{C}^{-}$.

######  Remark 3.4.

In the formulas above, $x^{z}=\exp(z\ln{x})$. If $x\notin(-\infty,0)$, we can
use the principal branch of the logarithm to define $\ln{x}$, and if
$x\in(-\infty,0)$, then we can define the logarithm in the complex plane cut
along $(-\sqrt{-1}\infty,0]$ such that $\Im\ln{a}=0$ for all $a\in(0,\infty)$.

######  Remark 3.5.

The formulas in Theorems 3.2 and 3.3 probably hold for more general contours
$\mathcal{C}^{+}$, $\mathcal{C}^{-}$, but we do not need more generality for
our purposes.

######  Remark 3.6.

When $\theta=1$, Theorem 3.1 recovers [24, Theorem 3.6].

######  Remark 3.7.

The infinite contours in Theorems 3.2 and 3.3 are needed because there are
infinitely many real poles in the integrand and the contour needs to enclose
all of them. When $\theta\in\mathbb{N}$, the integrands have finitely many
poles, and we can therefore close the contours, obtaining eventually Theorem
3.1. More generally, if $\theta>0$ is such that $\theta N\in\mathbb{N}$, a
similar remark applies. In fact, we can write the product of $q$-Gamma
function ratios appearing in the integrand (3.2) as

$\displaystyle\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}$
$\displaystyle\qquad{}=\frac{\Gamma_{q}(\lambda_{1}+\theta(N-1)-z)}{\Gamma_{q}(\lambda_{2}+\theta(N-1)-z)}\cdots\frac{\Gamma_{q}(\lambda_{N-1}+\theta-z)}{\Gamma_{q}(\lambda_{N}+\theta-z)}\frac{\Gamma_{q}(\lambda_{N}-z)}{\Gamma_{q}(\lambda_{1}+\theta
N-z)},$ (3.4)

and since $\Gamma_{q}(t+1)=\frac{1-q^{t}}{1-q}\Gamma_{q}(t)$, we conclude that
the product above is a rational function in $q^{-z}$ with finitely many real
poles. Thus formula (3.2) is true if we replaced contour $\mathcal{C}^{+}$ by
a closed contour $\mathcal{C}_{0}$ containing all finitely many real poles of
the integrand. Similarly, we can replace $\mathcal{C}^{-}$ by a closed contour
$\mathcal{C}_{0}$ in (3.3).

###  3.2 An example

Before carrying out the proofs of the theorems above in full generality, we
prove some very special cases, by means of the residue theorem and the
$q$-binomial formula. For simplicity, let $|x|<1$ be a complex number, and
consider the empty partition $\lambda=\varnothing$, or equivalently the
$N$-signature $\lambda=\big{(}0^{N}\big{)}=(0,0,\dots,0)$. As remarked in
Section 2.2, we have $P_{(0^{N})}(q,t)=1$, and therefore the left-hand sides
of identities (3.1) and (3.2) are both equal to $1$, when $\lambda=(0^{N})$.
Let us prove that the right-hand sides of (3.1) and (3.2) also equal $1$, for
$\lambda=(0^{N})$.

Let us begin with the case $\theta\notin\mathbb{N}$, i.e., the right-hand side
of (3.2). Since the contour $\mathcal{C}^{-}$ encloses all real poles in the
integrand in its interior, then the right-hand side of (3.2) equals

$\displaystyle\frac{\ln{q}}{1-q}\frac{\big{(}xt^{N};q\big{)}_{\infty}}{(xq;q)_{\infty}}\times\Gamma_{q}(\theta
N)\times\sum_{n=0}^{\infty}{x^{n}\frac{\operatorname{Res}_{z=n}{\Gamma_{q}(-z)}}{\Gamma_{q}(\theta
N-n)}}.$ (3.5)

From the definition of $q$-Gamma functions, see Appendix A, it is evident
that, for any $n\in\mathbb{Z}_{\geq 0}$, we have

$\displaystyle\operatorname{Res}_{z=n}{\Gamma_{q}(-z)}=\frac{(-1)^{n}(1-q)^{n+1}}{\ln{q}}\frac{q^{{n+1\choose
2}}}{(q;q)_{n}}.$

Furthermore, $\Gamma_{q}(t+1)=\frac{1-q^{t}}{1-q}\Gamma_{q}(t)$ gives
$\frac{\Gamma_{q}(\theta N)}{\Gamma_{q}(\theta N-n)}=(1-q)^{-n}(q^{\theta
N-n};q)_{n}$, so (3.5) equals

$\displaystyle\frac{\big{(}xt^{N};q\big{)}_{\infty}}{(xq;q)_{\infty}}\times\sum_{n=0}^{\infty}{\frac{(-1)^{n}q^{{n+1\choose
2}}(q^{\theta N-n};q)_{n}}{(q;q)_{n}}x^{n}}.$

The latter indeed equals $1$ because of the $q$-binomial theorem, Theorem A.3,
applied to $z=xq^{\theta N}=xt^{N}$, $a=q^{1-\theta N}$, and the equality
$q^{\theta Nn}(q^{1-\theta N};q)_{n}=(-1)^{n}q^{{n+1\choose 2}}(q^{\theta
N-n};q)_{n}$ $\forall\,n\geq 0$.

Second, let us consider the case $\theta\in\mathbb{N}$, i.e., the right-hand
side of (3.1). Observe that for $\lambda=(0^{N})$, the integrand in (3.1) can
be rewritten as $x^{z}\cdot\prod\limits_{i=0}^{\theta N-1}{(1-q^{z-i})^{-1}}$,
whose set of poles enclosed in the interior of $\mathcal{C}_{0}$ is
$\\{0,1,2,\dots,\theta N-1\\}$. Since
$\operatorname{Res}_{z=n}{(1-q^{z-n})^{-1}}=-(\ln{q})^{-1}=(\ln{(1/q)})^{-1}$,
similar considerations as above lead us to conclude that the right side of
(3.1) is equal to the finite sum

$\displaystyle\prod_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{x-q^{i}}}\times\sum_{n=0}^{\theta
N-1}{\frac{x^{n}}{\prod\limits_{\begin{subarray}{c}0\leq i\leq\theta N-1\\\
i\neq n\end{subarray}}{(1-q^{n-i})}}}=\prod_{i=1}^{\theta
N-1}{\frac{1-q^{-i}}{1-xq^{-i}}}\times\sum_{n=0}^{\theta
N-1}{\frac{x^{n}}{\big{(}q;q)_{n}(q^{-1};q^{-1}\big{)}_{\theta N-n-1}}}$
$\displaystyle\qquad{}=\frac{1}{\big{(}xq^{-1};q^{-1}\big{)}_{\theta
N-1}}\times\sum_{n=0}^{\theta N-1}{\frac{(-1)^{n}q^{-{n+1\choose
2}}\big{(}q^{-1};q^{-1}\big{)}_{\theta
N-1}}{\big{(}q^{-1};q^{-1}\big{)}_{n}(q^{-1};q^{-1})_{\theta N-n-1}}x^{n}}.$

The latter equals $1$, because of Corollary A.4 of the $q$-binomial formula,
applied to $z=xq^{-1}$ and $M=\theta N-1$.

A simple argument involving the index stability for Macdonald polynomials, see
(2.1), shows that if Theorems 3.1 and 3.2 hold for
$\lambda\in\mathbb{GT}_{N}$, then they hold for
$(\lambda_{1}+n\geq\lambda_{2}+n\geq\dots\geq\lambda_{N}+n)\in\mathbb{GT}_{N}$,
and any $n\in\mathbb{Z}$, cf. Step 5 in Section 3.3 below. Thus the present
example shows how to prove Theorems 3.1 and 3.2 for signatures of the form
$(n,n,\dots,n)\in\mathbb{GT}_{N}$, only by use of the classical $q$-binomial
theorem.

###  3.3 Integral formula when $\boldsymbol{t=q^{\theta}}$,
$\boldsymbol{\theta\in\mathbb{N}}$: Proof of Theorem 3.1

Assume $\theta\in\mathbb{N}$ and $t=q^{\theta}$. The proof of Theorem 3.1 is
broken down into several steps. In the first four steps, we prove the
statement for positive signatures $\lambda\in\mathbb{GT}^{+}_{N}$ (when all
coordinates are nonnegative: $\lambda_{1}\geq\cdots\geq\lambda_{N}\geq 0$),
and in step 5 we extend it for all signatures $\lambda\in\mathbb{GT}_{N}$
(when some coordinates of $\lambda$ could be negative).

Step 1. We derive a contour integral formula for the ratio
$\frac{P_{\lambda}(q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t)}{P_{\lambda}(t^{N-1},\dots,t,1;q,t)}$
of Macdonald polynomials, and any $r\in\mathbb{N}$. The index-argument
symmetry, Theorem 2.2, applied to $\lambda=(\lambda_{1},\dots,\lambda_{N})$
and $\mu=(r)=(r,0^{N-1})$, gives

$\displaystyle\frac{P_{\lambda}\big{(}q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{P_{(r)}\big{(}q^{\lambda_{1}}t^{N-1},\dots,q^{\lambda_{N-1}}t,q^{\lambda_{N}};q,t\big{)}}{P_{(r)}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}.$
(3.6)

The denominator $P_{(r)}(t^{N-1},\dots,t,1;q,t)$ has a simple expression due
to the evaluation identity, Theorem 2.3; it is particularly simple for the row
partition $(r)$:

$\displaystyle
P_{(r)}\big{(}t^{N-1},\dots,t,1;q,t\big{)}=\prod_{j=2}^{N}{\frac{\big{(}q^{r}t^{j-1};q\big{)}_{\infty}(t^{j};q)_{\infty}}{(q^{r}t^{j};q)_{\infty}(t^{j-1};q)_{\infty}}}={\frac{(q^{r}t;q)_{\infty}(t^{N};q)_{\infty}}{(q^{r}t^{N};q)_{\infty}(t;q)_{\infty}}}=\frac{\big{(}t^{N};q\big{)}_{r}}{(t;q)_{r}}.$

Since we also have $P_{(r)}(q,t)=\frac{(q;q)_{r}}{(t;q)_{r}}g_{r}(q,t)$, see
(2.4), identity (3.6) becomes

$\displaystyle\frac{P_{\lambda}\big{(}q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{(q;q)_{r}}{\big{(}t^{N};q\big{)}_{r}}g_{r}\big{(}q^{\lambda_{1}}t^{N-1},\dots,q^{\lambda_{N}};q,t\big{)}.$
(3.7)

The symmetric polynomials $g_{r}(q,t)$, in addition to being essentially one-
row Macdonald polynomials, can be defined in terms of their generating
function as follows, see [32, Chapter VI]:

$\displaystyle\prod_{i=1}^{N}{\frac{(tx_{i}y;q)_{\infty}}{(x_{i}y;q)_{\infty}}}=\sum_{r\geq
0}g_{r}(x_{1},\dots,x_{N};q,t)y^{r}.$ (3.8)

The relation (3.8) holds formally in the ring
$\Lambda_{F}[x_{1},\dots,x_{N}][[y]]$. If we fix nonzero values
$x_{1},\dots,x_{N}\in\mathbb{C}\setminus\\{0\\}$, the identity above is an
equality of real analytic functions in the domain
$\\{y\in\mathbb{C}\colon\max_{1\leq i\leq N}{|x_{i}y|}<1\\}$. Thus we have the
following contour integral representation

$\displaystyle
g_{r}(x_{1},\dots,x_{N};q,t)=\frac{1}{2\pi\sqrt{-1}}\oint_{C}{\frac{1}{y^{r+1}}\prod_{i=1}^{N}{\frac{(tx_{i}y;q)_{\infty}}{(x_{i}y;q)_{\infty}}{\rm
d}y}},$

where $C$ is any circle around the origin and radius smaller than
$(\max_{i}{|x_{i}|})^{-1}$. Let $x_{i}=q^{\lambda_{i}}t^{N-i}$ for
$i=1,2,\dots,N$ in the integral representation of $g_{r}(q,t)$, and replace it
into the right-hand side of (3.7); then

$\displaystyle\frac{P_{\lambda}\big{(}q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{(q;q)_{r}}{\big{(}t^{N};q\big{)}_{r}}\frac{1}{2\pi\sqrt{-1}}\oint_{C}{\frac{1}{y^{r+1}}\prod_{i=1}^{N}{\frac{\big{(}q^{\lambda_{i}}t^{N-i+1}y;q\big{)}_{\infty}}{\big{(}q^{\lambda_{i}}t^{N-i}y;q\big{)}_{\infty}}{\rm
d}y}},$ (3.9)

where $C$ can be taken to be any circle around the origin of radius smaller
than $1$ (we need here that $\lambda\in\mathbb{GT}^{+}_{N}$ implies
$q^{\lambda_{i}}t^{N-i}<1$ $\forall\,i$). For $t=q^{\theta}$,
$\theta\in\mathbb{N}$, we can simplify (3.9) to

$\displaystyle\frac{P_{\lambda}\big{(}q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{(q;q)_{r}}{\big{(}q^{\theta
N};q\big{)}_{r}}\frac{1}{2\pi\sqrt{-1}}\oint_{C}{\frac{y^{-(r+1)}{\rm
d}y}{\prod\limits_{i=1}^{N}{\prod\limits_{j=0}^{\theta-1}{(1-q^{\lambda_{i}+\theta(N-i)+j}y)}}}}.$
(3.10)

Step 2. We obtain a new contour integral representation by modifying (3.10).
The resulting contour integral representation involves an open contour
$\mathcal{C}^{+}$, that looks like that of Fig. 2 (but has a slight difference
from that in Theorem 3.2).

Observe that the absolute value of the integrand in (3.10) is of order
$o(R^{-r-1})=o(R^{-2})$, if $|y|=R$ is large. An application of Cauchy’s
theorem yields that the value of the integral is unchanged if the closed
contour $C$ is deformed into the “keyhole” contour $\mathcal{C}^{\prime}$
shown in Fig. 4. Let us describe the contour $\mathcal{C}^{\prime}$ in words:
it is a positively oriented contour, formed by two lines away from the origin,
of arguments $\pm 3\pi/4$, and the portion of a semicircle of some radius
$0<\delta<1$. Evidently, the straight lines are part of the level lines
$\Im(\ln(y))=\pm 3\pi/4$, while the portion of the semicircle is part of the
level line $\Re(\ln(y))=\delta$ (where $\ln$ is defined in
$\mathbb{C}\setminus(-\infty,0]$).

$\Re z$$\Im z$$O$

Figure 4: Contour $\mathcal{C}^{\prime}$.

Next, we make the change of variables $y=q^{-z}$, or
$z=-\frac{\ln{y}}{\ln{q}}$, where $\ln$ is defined on its principal branch.
Based on the previous observations about the contour $\mathcal{C}^{\prime}$
being composed by level lines, it is clear that the resulting contour for $z$
is a negatively oriented contour formed by one segment and two straight lines,
but we can easily reverse the orientation of the contour at the cost of
switching signs. Let us call the positively oriented contour
$\mathcal{C}^{+}$, see Fig. 2; the integral formula becomes

$\displaystyle\frac{P_{\lambda}\big{(}q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{(q;q)_{r}}{\big{(}q^{\theta
N};q\big{)}_{r}}\frac{1}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{+}}\\!{\frac{q^{(r+1)z}q^{-z}\ln{q}{\rm
d}z}{\prod\limits_{i=1}^{N}{\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{\lambda_{i}+\theta(N-i)+j-z}\big{)}}}}}.\\!\\!\\!\\!$
(3.11)

Note that points in the horizontal lines of contour $\mathcal{C}^{+}$ in
(3.11) have imaginary parts $\pm\frac{3\pi}{4\ln{q}}$, while points in the
vertical segment of $\mathcal{C}^{+}$ have real part $-\ln{\delta}/\ln{q}<0$.
Thus contour $\mathcal{C}^{+}$ encloses exactly all the real poles of the
integrand in (3.11) and no other poles (it also encloses the origin, though
this not important).

Step 3. We make some final modifications to formula (3.11); the resulting
contour integral representation will include a closed contour
$\mathcal{C}_{0}$ as in the statement of the theorem.

Observe that the integrand in (3.11) has finitely many real poles and they are
all enclosed by contour $\mathcal{C}^{+}$; also the integrand is exponentially
small as $|z|\rightarrow\infty$ along the contour $\mathcal{C}^{+}$.
Therefore, as an application of Cauchy’s theorem, we can replace
$\mathcal{C}^{+}$ by a closed contour $\mathcal{C}_{0}$ that encloses all
finitely many real poles of the integrand. Also note that for $r>\theta N$, we
can write $\frac{(q;q)_{r}}{(q^{\theta N};q)_{r}}=\prod\limits_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{1-q^{r+i}}}$. Thus for $r>\theta N$, equation (3.11) can
be rewritten as

$\displaystyle\frac{P_{\lambda}\big{(}q^{r}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}$
$\displaystyle\qquad{}=\frac{\ln{q}}{2\pi\sqrt{-1}}\prod_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{1-q^{r+i}}}\oint_{\mathcal{C}_{0}}{q^{rz}\prod_{i=1}^{N}{\prod_{j=0}^{\theta-1}{\frac{1}{1-q^{\lambda_{i}+\theta(N-i)+j-z}}}{\rm
d}z}}.$ (3.12)

We let $x=q^{r}t^{N}=q^{r+\theta N}$ and replace all instances of $q^{r}$ in
(3.12) above by $x/t^{N}$; then multiply both sides of the identity by
$t^{|\lambda|}$. We claim that the resulting equation is exactly equality
(3.1). In fact, the left-hand side of our equation is

$\displaystyle
t^{|\lambda|}\frac{P_{\lambda}\big{(}x/t,t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{P_{\lambda}\big{(}x,t^{N-1},t^{N-2},\dots,t;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}=\frac{P_{\lambda}\big{(}x,t,t^{2},\dots,t^{N-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}},$

by homogeneity of the Macdonald polynomials. On the other hand, the right-hand
side of our equation is

$\displaystyle t^{|\lambda|}\frac{\ln{q}}{2\pi\sqrt{-1}}\prod_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{1-xq^{i-\theta
N}}}\oint_{\mathcal{C}_{0}}{x^{z}t^{-Nz}\prod_{i=1}^{N}{\prod_{j=0}^{\theta-1}{\frac{1}{1-q^{\lambda_{i}+\theta(N-i)+j-z}}}{\rm
d}z}},$

which can be shown to be equal to the right-hand side of (3.1), by simple
algebraic manipulations.

The conclusion is that we have proved identity (3.1) for all $x=q^{m}$ with
$m\in\mathbb{N}$ large enough (to be precise, $m$ is of the form $r+\theta N$
and $r>\theta N$, so the statement was proved for all integers $m>2\theta N$).

Step 4. Still assuming $\lambda\in\mathbb{GT}^{+}_{N}$, we prove Theorem 3.1
for all $x\in\mathbb{C}\setminus(\\{0\\}\cup\\{q^{i}\colon i=1,2,\dots,\theta
N-1\\})$. Observe that $x\neq 0$ is imposed to make sense of the term
$x^{z}=\exp(z\ln{x})$ and $x\notin\\{q,q^{2},\dots,q^{\theta N-1}\\}$ is
necessary so that the denominator of the right-hand side of (3.1) is nonzero.

We claim that both sides of (3.1) are rational functions of $x$. This would
prove the desired result, given that we have shown in step 3 above that (3.1)
holds for infinitely many points $q^{m}$, where $m$ is large enough. The left-
hand side of (3.1) is obviously a polynomial on $x$. In the right-hand side of
(3.1), we have the product $\prod\limits_{i=1}^{\theta N-1}{(x-q^{i})^{-1}}$
which is a rational function. We only need to check that the contour integral
is also a rational function on $x$. In fact, this follows from the residue
theorem and the fact that there are finitely many poles in the interior of
$\mathcal{C}_{0}$, all of these being simple and integral. The fact that the
poles considered above are simple and integral can be easily checked and is
equivalent to fact that all the values $\lambda_{i}+\theta(N-i)+j$, for $1\leq
i\leq N$, $0\leq j\leq\theta-1$, are pairwise distinct integers.

Step 5. We extend equality (3.1) to all signatures
$\lambda\in\mathbb{GT}_{N}$.

Let $\lambda\in\mathbb{GT}_{N}$ be arbitrary and we aim to prove (3.1). If
$\lambda\in\mathbb{GT}^{+}_{N}$, the result is already proved in the first
four steps above. Otherwise, choose $m\in\mathbb{N}$ such that
$\lambda_{N}+m\geq 0$, and so
$\widetilde{\lambda}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\lambda+\big{(}m^{N}\big{)}=(\lambda_{1}+m,\lambda_{2}+m,\dots,\lambda_{N}+m)\in\mathbb{GT}^{+}_{N}$.

By the index stability (2.1), $(x_{1}\cdots
x_{N})^{m}P_{\lambda}(x_{1},\dots,x_{N};q,t)=P_{\widetilde{\lambda}}(x_{1},\dots,x_{N};q,t)$.
Thus multiplying the left-hand side of the equality (3.1) by
$x^{m}=\big{(}x\cdot t\cdot t^{2}\cdots t^{N-1}\big{)}^{m}(1\cdot t\cdot
t^{2}\cdots t^{N-1})^{-m}$ gives

$\displaystyle\frac{\big{(}x\cdot t\cdot t^{2}\cdots t^{N-1}\big{)}^{m}\cdot
P_{\lambda}\big{(}x,t,t^{2},\dots,t^{N-1};q,t\big{)}}{\big{(}1\cdot t\cdot
t^{2}\cdots t^{N-1}\big{)}^{m}\cdot
P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}=\frac{P_{\widetilde{\lambda}}\big{(}x,t,t^{2},\dots,t^{N-1};q,t\big{)}}{P_{\widetilde{\lambda}}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}.$

If we multiply the right-hand side of equality (3.1) by $x^{m}$, and also make
the change of variables $z\mapsto z-m$ in the contour integral, we obtain

$\displaystyle\ln(1/q)\prod_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{x-q^{i}}}\frac{1}{2\pi\sqrt{-1}}\oint_{\mathcal{C}_{0}}{\frac{x^{z}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{z-(\widetilde{\lambda}_{i}+\theta(N-i)+j)}\big{)}}}{\rm
d}z}.$

Since we have proved equality (3.1) for the positive signature
$\widetilde{\lambda}$ already, then (3.1) also holds for $\lambda$, since we
have multiplied both sides by $x^{m}$ and obtained equal expressions.

###  3.4 Integral formula for general $\boldsymbol{q}$, $\boldsymbol{t}$ I:
Outline of proof of Theorems 3.2 and 3.3

Theorems 3.2 and 3.3 are analytic continuations of Theorem 3.1, with respect
to the variable $\theta$. We shall need the weak version of Carlson’s lemma
below, which is proved in [2, Theorem 2.8.1] (in this reference, the statement
is given for functions that are analytic and bounded on
$\\{z\in\mathbb{C}\colon\Re z\geq 0\\}$, but a simple change of variables
$z\mapsto z+M$, for some large positive integer $M\in\mathbb{N}$ leads us to
the version below).

######  Lemma 3.8 (Carlson’s lemma).

Let $f(z)$ be a holomorphic function on the right half plane
$\\{z\in\mathbb{C}\colon\Re z>0\\}$, such that $f(z)$ is uniformly bounded on
the domain $\\{z\in\mathbb{C}\colon\Re z\geq M\\}$, for some $M>0$. If
$f(n)=0$ for all $n\in\mathbb{N}$, then $f$ is identically zero.

We outline the steps of the proof of Theorem 3.2 below, and we shall carry out
the detailed proof of each step in the next section. Theorem 3.3 can be proved
analogously and we leave the details to the reader.

  * •

Step 0. Prove that the right-hand side of (3.2) is well-defined, i.e., the
contour integral is absolutely convergent. We prove the absolute convergence
for all $|x|\leq 1$, $x\neq 0$, and $\theta>0$.

  * •

Step 1. Reduce the general statement to the case $|x|<1$, $x\neq 0$, and
$\lambda\in\mathbb{GT}^{+}_{N}$. Assume the latter conditions are in place for
the remaining steps in the proof.

  * •

Step 2. Prove that the equation (3.2) of Theorem 3.2 holds when $t=q^{\theta}$
and $\theta\in\mathbb{N}$.

  * •

Step 3. Prove that both sides of equation (3.2) are holomorphic functions of
$\theta$ on the right half plane $\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$.

  * •

Step 4. Prove that, for some (sufficiently large) $M>0$, both sides of
equation (3.2) are uniformly bounded functions of $\theta$ on
$\\{\theta\in\mathbb{C}\colon\Re\theta\geq M\\}$.

Then we can conclude Theorem 3.2 as follows. From steps 2–4 and Lemma 3.8
above, identity (3.2) is proved for all $x\in\mathbb{C}$ with $|x|<1$, $x\neq
0$, $\lambda\in\mathbb{GT}^{+}_{N}$, $\theta\in\\{z\in\mathbb{C}\colon\Re
z>0\\}$, and in particular for all $\theta>0$. Thus step 1 shows the theorem
holds for the more general case $x\in\mathbb{C}\setminus\\{0\\}$, $|x|\leq 1$,
and $\lambda\in\mathbb{GT}_{N}$.

###  3.5 Integral formula for general $\boldsymbol{q}$, $\boldsymbol{t}$ II:
Proof of Theorem 3.2

In this section, we prove Theorem 3.2; as we already mentioned, Theorem 3.3
can be proved in an analogous way, and we leave the details to the reader.
From the last paragraph of the previous section, Theorem 3.2 will be proved if
we verify steps 0–4 stated there.

Proof of Step 0. Fix $\theta>0$, $x\in\mathbb{C}\setminus\\{0\\}$, $|x|\leq
1$, and let

$\displaystyle
F_{q}(z;\theta)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}x^{z}\prod\limits_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}.$

From the definition of the $q$-Gamma function, $F_{q}(z;\theta)$ is a
holomorphic function in a neighborhood of the contour $\mathcal{C}^{+}$, thus
$\int_{\mathcal{C}^{+}}{F_{q}(z;\theta){\rm d}z}$ is a well-posed integral. To
prove its absolute convergence, it suffices to show that
$|F_{q}(z;\theta)|\leq c_{1}\cdot q^{c_{2}\Re z}$, for some $c_{1},c_{2}>0$
and all $z\in\mathcal{C}^{+}$ with $\Re z$ large enough.

Because $|x|\leq 1$ and all $z$ in the contour $\mathcal{C}^{+}$ have bounded
imaginary parts, we have $|x^{z}|=\exp(\Re z\ln{|x|}-\Im z\arg(x))$,
$z\in\mathcal{C}^{+}$, is upper-bounded by a constant. Thus it will suffice to
show

$\displaystyle\left|\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}\right|\leq
c_{1}\cdot q^{c_{2}\Re z},\qquad i=1,2,\dots,N,$

for some $c_{1},c_{2}>0$ and all $z\in\mathcal{C}^{+}$ with $\Re z$ large
enough. Since each $\lambda_{i}+\theta(N-i)$ is real and $|\Im z|$ is constant
for $z\in\mathcal{C}^{+}$, when $\Re z$ is large enough, the statement reduces
to showing

$\displaystyle\left|\frac{\Gamma_{q}(-z)}{\Gamma_{q}(\theta-z)}\right|\leq
c_{1}\cdot q^{c_{2}\Re z},$ (3.13)

for some $c_{1},c_{2}>0$ and all $z\in\mathcal{C}^{+}$ with $\Re z$ large
enough.

Let $\pi/2>d>0$ be the value such that $|\Im z|$, for any point
$z\in\mathcal{C}^{+}$ with $\Re z$ large enough, equals $-d/\ln{q}$ (such $d$
exists by our assumptions on the contour $\mathcal{C}^{+}$). There exists a
real number $a>0$ large enough so that the following inequality holds

$\displaystyle q^{a}\big{(}1+q^{\theta}\big{)}\leq 2\cos d.$ (3.14)

Now we consider only points $z\in\mathcal{C}^{+}$ with $\Re z$ large enough so
that $|\Im z|=-d/\ln{q}$ and $\Re z>\theta+a+1$; let $M=M(z)=\lfloor\Re
z-\theta-a\rfloor\in\mathbb{N}$. From the definition of $a$ in (3.14), the
restriction on the values of $\Re z$, and $q\in(0,1)$, one can easily obtain

$\displaystyle\big{|}1-q^{z-\theta-j}\big{|}\leq\big{|}1-q^{z-j}\big{|},\qquad\forall\,j=0,1,\dots,M.$
(3.15)

As a consequence of inequality (3.15) and the definition of the $q$-Gamma
function, we have

$\displaystyle\left|\frac{\Gamma_{q}(-z)}{\Gamma_{q}(\theta-z)}\right|=\left|(1-q)^{\theta}\frac{\big{(}q^{-z+\theta};q\big{)}_{\infty}}{(q^{-z};q)_{\infty}}\right|$
$\displaystyle\qquad{}=\left|(1-q)^{\theta}\frac{\big{(}1-q^{\theta-z}\big{)}\cdots\big{(}1-q^{\theta-z+M}\big{)}}{(1-q^{-z})\cdots\big{(}1-q^{-z+M}\big{)}}\frac{\big{(}q^{-z+\theta+M+1};q\big{)}_{\infty}}{\big{(}q^{-z+M+1};q\big{)}_{\infty}}\right|$
$\displaystyle\qquad{}=\left|(1-q)^{\theta}\frac{\big{(}1-q^{z-\theta}\big{)}\cdots\big{(}1-q^{z-\theta-M}\big{)}}{(1-q^{z})\cdots\big{(}1-q^{z-M}\big{)}}q^{\theta(M+1)}\frac{\big{(}q^{\theta-z+M+1};q\big{)}_{\infty}}{\big{(}q^{-z+M+1};q\big{)}_{\infty}}\right|$
$\displaystyle\qquad{}\leq(1-q)^{\theta}\cdot
q^{\theta(M+1)}\left|\frac{\big{(}q^{\theta-z+M+1};q\big{)}_{\infty}}{\big{(}q^{-z+M+1};q\big{)}_{\infty}}\right|.$

Since $M=\lfloor\Re z-\theta-a\rfloor$, the previous inequality almost shows
(3.13). We are left to show that the absolute value of
$(q^{\theta-z+M+1};q)_{\infty}/(q^{-z+M+1};q)_{\infty}$ is upper bounded by a
constant independent of $z\in\mathcal{C}^{+}$ as long as $\Re z$ is large
enough. In fact, we have $\Re z-\theta-a\geq M>\Re z-\theta-a-1$, so
$|q^{\theta-z+M+1}|,|q^{-z+M+1}|\leq q^{-\theta-a}$ and $|\Im
q^{\theta-z+M+1}|,|\Im q^{-z+M+1}|\geq q^{\theta+M+1-\Re z}\sin d\geq
q^{1-a}\sin d>0$. Because the $q$-Pochhammer symbol $(x;q)_{\infty}$ is
holomorphic on $x\in\mathbb{C}$, in particular continuous, the absolute value
$|(x;q)_{\infty}|$ attains a maximum $c_{\max}\geq 0$ and a minimum value
$c_{\min}\geq 0$ on the compact subset $\\{x\in\mathbb{C}\colon|x|\leq
q^{-\theta-a},|\Im x|\geq q^{1-a}\sin d\\}$, and moreover $c_{\min}>0$ because
all the roots of $(x;q)_{\infty}=0$ are real. We have thus proved
$|(q^{\theta-z+M+1};q)_{\infty}|/|(q^{-z+M+1};q)_{\infty}|\leq
c_{\max}/c_{\min}<\infty$, as desired.

Proof of Step 1. Assume we proved identity (3.2) when
$x\in\mathbb{C}\setminus\\{0\\}$, $|x|<1$. Then an easy application of the
dominated convergence theorem (we know the integral converges absolutely when
$x=1$ because of step 1) shows the equation would also hold for all
$x\in\mathbb{C}\setminus\\{0\\}$, $|x|\leq 1$.

Also, observe that step 5 of the proof of Theorem 3.1 can be repeated almost
word-by-word to extend the theorem to all $\lambda\in\mathbb{GT}_{N}$,
assuming that it was proved for all $\lambda\in\mathbb{GT}^{+}_{N}$. Therefore
we will assume for convenience in the next steps that $|x|<1$, $x\neq 0$,
$\lambda\in\mathbb{GT}^{+}_{N}$, and prove the theorem only in that case,
without loss of generality.

Proof of Step 2. In this step, we consider the case $\theta\in\mathbb{N}$.
Since we are also assuming $|x|<1$, $x\neq 0$, then clearly
$xt^{N}\notin\big{\\{}0,q,q^{2},\dots,q^{\theta N-1}\big{\\}}$ and therefore
equality (3.1) holds if $x$ was replaced with $xt^{N}$. After multiplying both
sides of the resulting equation by $t^{-|\lambda|}$, we claim that we arrive
at the desired (3.2) with $\mathcal{C}_{0}$ in place of $\mathcal{C}^{+}$. In
fact, the left-hand side of our equation is

$\displaystyle
t^{-|\lambda|}\frac{P_{\lambda}\big{(}xt^{N},t,t^{2},\dots,t^{N-1};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}$
$\displaystyle\qquad{}=\frac{P_{\lambda}\big{(}xt^{N-1},1,t,\dots,t^{N-2};q,t\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N-1};q,t\big{)}}=\frac{P_{\lambda}\big{(}xt^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}$

thanks to the homogeneity and symmetry of Macdonald polynomials. On the other
hand, the right-hand side of our equation is

$\displaystyle t^{-|\lambda|}\ln(1/q)\prod_{i=1}^{\theta
N-1}{\frac{1-q^{i}}{xq^{\theta
N}-q^{i}}}\frac{1}{2\pi\sqrt{-1}}\oint_{\mathcal{C}_{0}}{\frac{x^{z}t^{Nz}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{z-(\lambda_{i}+\theta(N-i)+j)}\big{)}}}{\rm
d}z},$

which can be shown equal to the right-hand side of (3.2) (with
$\mathcal{C}^{+}$ replaced by $\mathcal{C}_{0}$) after simple algebraic
manipulations. Finally observe that contour $\mathcal{C}_{0}$ can be replaced
by $\mathcal{C}^{+}$ by an application of Cauchy’s theorem.

Proof of Step 3. Let us begin by proving holomorphicity of the left-hand side
of (3.2) with respect to the variable $\theta$; observe that $\theta$ only
appears inside the variable $t=q^{\theta}$. The Macdonald polynomials
$P_{\lambda}(x_{1},\dots,x_{N};q,t)$ are holomorphic functions of $\theta$ on
$\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$ because all the branching
coefficients $\psi_{\mu/\nu}(q,t)$ are holomorphic on this domain. Then
$P_{\lambda}\big{(}xt^{N-1},t^{N-2},\dots,t,1;q,t\big{)}$ and
$P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}$ are also holomorphic.
It follows that the ratio of these two quantities is holomorphic if we proved
that the denominator $P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}$
never vanishes for $\Re\theta>0$, or equivalently for $|t|<1$; this is evident
from the evaluation identity for Macdonald polynomials, Theorem 2.3.

Next we prove holomorphicity of the right-hand side of (3.2) as a function of
$\theta$ in the domain $\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$. Clearly
$\big{(}xq^{\theta N};q\big{)}_{\infty}\Gamma_{q}(\theta N)$ is holomorphic in
the given domain of $\theta$, but it is less clear that the integral
$\int_{\mathcal{C}^{+}}{F_{q}(z,\theta){\rm d}z}$ is holomorphic in the right
half-plane, where

$\displaystyle
F_{q}(z;\theta)=x^{z}\prod\limits_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}.$

First we claim that $F_{q}(z;\theta)$ is holomorphic on
$U\times\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$, for some neighborhood $U$
of $\mathcal{C}^{+}$. Indeed, the factor $x^{z}$ is clearly entire on $z$, and
does not depend on $\theta$. We can write the product of ratios of $q$-Gamma
functions in the definition of $F_{q}(z;\theta)$, as we did in Remark 3.7, see
(3.4). For $1\leq i\leq N-1$, we have

$\displaystyle\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i+1}+\theta(N-i)-z)}=\prod_{n=\lambda_{i+1}}^{\lambda_{i}-1}{[n+\theta(N-i)-z]_{q}}=\prod_{n=\lambda_{i+1}}^{\lambda_{i}-1}{\frac{1-q^{n+\theta(N-i)-z}}{1-q}},$

which is clearly holomorphic on $(z,\theta)\in\mathbb{C}^{2}$. Finally the
remaining factor can be written as

$\displaystyle\frac{\Gamma_{q}(\lambda_{N}-z)}{\Gamma_{q}(\lambda_{1}+\theta
N-z)}=(1-q)^{\lambda_{1}+\theta
N-\lambda_{N}}\frac{\big{(}q^{\lambda_{1}+\theta
N-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{N}-z};q\big{)}_{\infty}}.$

It is clear that $(1-q)^{\lambda_{1}+\theta
N-\lambda_{N}}\big{(}q^{\lambda_{1}+\theta N-z};q\big{)}_{\infty}$ is
holomorphic on $(z,\theta)\in\mathbb{C}^{2}$. And also there is a neighborhood
$U$ of $\mathcal{C}^{+}$ on which the function
$\big{(}q^{\lambda_{N}-z};q\big{)}_{\infty}^{-1}$ of $z$ is holomorphic on
$U$.

Secondly, we claim that $\int_{\mathcal{C}^{+}}{F_{q}(z;\theta){\rm d}z}$ is
absolutely convergent and moreover
$\int_{\mathcal{C}^{+}}{|F_{q}(z;\theta)|{\rm d}z}$ is uniformly bounded on
compact subsets of $\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$; this will be
a consequence of the stronger statement

######  Claim 3.9.

Consider any compact subset
$K\subset\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$. There exists a constant
$M_{1}>0$, depending on $K$, such that

$\displaystyle|F_{q}(z;\theta)|<M_{1}|x^{z}|,\qquad\forall\,z\in\mathcal{C}^{+},\quad\theta\in
K.$ (3.16)

Let us first conclude the proof of step 3 from the claim above.

Since $|x|<1$ we have that $|x^{z}|$ decreases exponentially as
$|z|\rightarrow\infty$, $z\in\mathcal{C}^{+}$. Then inequality (3.16) shows
that $\int_{\mathcal{C}^{+}}{F_{q}(z;\theta){\rm d}z}$ is absolutely
convergent and for all $\theta$ belonging to the compact subset
$K\subset\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$, we have the bound
$\int_{\mathcal{C}^{+}}{|F_{q}(z;\theta)|{\rm d}z}<C(K)$, for some constant
$C(K)>0$ that depends on $K$.

Let $T$ be any triangular contour belonging to
$\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$. Then

$\displaystyle\int_{T}{\int_{\mathcal{C}^{+}}{|F_{q}(z;\theta)|{\rm d}z}{\rm
d}\theta}\leq C(T)\int_{T}{|{\rm d}\theta|}<\infty.$

By Fubini’s and Cauchy’s theorems, we have

$\displaystyle\int_{T}{\int_{\mathcal{C}^{+}}{F_{q}(z;\theta){\rm d}z}{\rm
d}\theta}=\int_{\mathcal{C}^{+}}\int_{T}{{F_{q}(z;\theta){\rm d}\theta}{\rm
d}z}=0.$

Morera’s theorem implies that $\int_{\mathcal{C}^{+}}{F_{q}(z;\theta){\rm
d}z}$ is holomorphic on $\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$,
concluding step 3.

###### Proof of Claim 3.9.

Since we have shown before that $F_{q}(z;\theta)$ is holomorphic on
$U\times\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$, then it suffices to prove
inequality (3.16) for all $z\in\mathcal{C}^{+}$, $\Re z>M_{2}$ and all
$\theta\in K$, where $M_{2}$ is an arbitrarily large positive constant.

We can express the product of $q$-Gamma function ratios in the definition of
$F_{q}(z;\theta)$ as we did in (3.4). The last ratio is

$\displaystyle\frac{\Gamma_{q}(\lambda_{N}-z)}{\Gamma_{q}(\lambda_{1}+\theta
N-z)}=\frac{[-z+\lambda_{N}-1]_{q}\cdots[-z+1]_{q}[-z]_{q}}{[-z+\lambda_{1}+\theta
N-1]_{q}\cdots[-z+\theta N+1]_{q}[-z+\theta
N]_{q}}\frac{\Gamma_{q}(-z)}{\Gamma_{q}(\theta N-z)},$

because we are assuming $\lambda\in\mathbb{GT}^{+}_{N}$ (and so
$\lambda_{1}\geq\lambda_{N}\geq 0$). Plugging the equality above into (3.4),
we obtain

$\displaystyle\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}$
$\displaystyle\qquad{}=\prod_{i,j}{[\theta(N-i)+j-z]_{q}}\frac{[\lambda_{N}-z-1]_{q}\cdots[-z]_{q}}{[\lambda_{1}+\theta
N-z-1]_{q}\cdots[\theta N-z]_{q}}$ (3.17)
$\displaystyle\qquad\quad{}\times\frac{\Gamma_{q}(-z)}{\Gamma_{q}(\theta
N-z)},$ (3.18)

where the product in (3.17) is over $1\leq i\leq N-1$ and $\lambda_{i+1}\leq
j<\lambda_{i}$. The term (3.17) is of the form $P(q^{-z})/Q(q^{-z})$, where
$P,Q\in\mathbb{C}(q,q^{\theta})[x]$ are both polynomials of degree
$\lambda_{1}$. It follows that there exist constants
$M_{1}^{\prime},M_{2}^{\prime}>0$ such that $\Re z>M_{2}^{\prime}$ implies
$|P(q^{-z})|/|Q(q^{-z})|<M_{1}^{\prime}$.

Thus we are left to deal with (3.18), which by definition of the $q$-Gamma
function equals $(1-q)^{\theta N}\frac{(q^{\theta
N-z};q)_{\infty}}{(q^{-z};q)_{\infty}}$. For any $\theta$ with $\Re\theta>0$,
we have $|(1-q)^{\theta}|<1$. Thus we only need to prove the existence of
$M_{1}^{\prime\prime},M_{2}^{\prime\prime}>0$ such that $z\in\mathcal{C}^{+}$,
$\Re z>M_{2}^{\prime\prime}$, implies

$\displaystyle\left|\frac{\big{(}q^{\theta
N-z};q\big{)}_{\infty}}{(q^{-z};q)_{\infty}}\right|<M_{1}^{\prime\prime},\qquad\textrm{for
all}\quad\theta\in K.$ (3.19)

There exists $a>0$ large enough such that

$\displaystyle q^{-a}\big{(}1-q^{N\inf_{\theta\in K}{\Re\theta}}\big{)}>2.$
(3.20)

Note that such $a$ exists because
$K\subset\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$ is compact and so
$\inf_{\theta\in K}{\Re\theta}>0$. Now consider only $z\in\mathcal{C}^{+}$
with $\Re z>a+1$, and let
$M=M(z)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\lfloor\Re z-a\rfloor\in\mathbb{N}$. From the triangle
inequality, $|1-q^{\theta N-z+j}|\leq 1+q^{N\Re\theta-\Re z+j}$ and
$|1-q^{-z+j}|\geq q^{-\Re z+j}-1$. Along with condition (3.20), we deduce that
$|1-q^{-z+j}|\geq|1-q^{\theta N-z+j}|$, for all $0\leq j\leq M$. Since we can
write

$\displaystyle\frac{\big{(}q^{\theta
N-z};q\big{)}_{\infty}}{(q^{-z};q)_{\infty}}=\frac{\big{(}1-q^{\theta
N-z}\big{)}\cdots\big{(}1-q^{\theta
N-z+M}\big{)}}{(1-q^{-z})\cdots\big{(}1-q^{-z+M}\big{)}}\cdot\frac{\big{(}q^{\theta
N-z+M+1};q\big{)}_{\infty}}{\big{(}q^{-z+M+1};q\big{)}_{\infty}},$

it follows that

$\displaystyle\left|\frac{\big{(}q^{\theta
N-z};q\big{)}_{\infty}}{(q^{-z};q)_{\infty}}\right|\leq\left|\frac{\big{(}q^{\theta
N-z+M+1};q\big{)}_{\infty}}{\big{(}q^{-z+M+1};q\big{)}_{\infty}}\right|.$
(3.21)

Thus we only need to show that the right-hand side of (3.21) is bounded by a
constant, for all $z\in\mathcal{C}^{+}$ with $\Re z$ large enough.

Since $M=\lfloor\Re z-a\rfloor>\Re z-a-1$, we have $|q^{-z+M+1}|=q^{M+1-\Re
z}\leq q^{-a}$ and $|q^{\theta N-z+M+1}|\leq q^{N\Re\theta-a}<q^{-a}$.
Moreover, for $z\in\mathcal{C}^{+}$ with $\Re z$ large enough, $|\Im
q^{-z+M+1}|\geq q^{-a+1}|\sin(\ln{q}\cdot\Im
z)|\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}m_{1}(z)$ and since $|\Im z|$ is a constant between $0$ and
$-\frac{\pi}{2\ln{q}}$ for $z\in\mathcal{C}^{+}$ with $\Re z$ large enough,
then $m_{1}(z)=m_{1}>0$ is a strictly positive constant independent of
$z\in\mathcal{C}^{+}$ as long as $\Re z$ is large enough. Since the function
$(x;q)_{\infty}$ is continuous on $x\in\mathbb{C}$, we have
$c_{\max}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sup\limits_{|x|\leq q^{-a}}{|(x;q)_{\infty}|}<\infty$, and
$c_{\min}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\inf_{|x|\leq q^{-a},|\Im x|\geq
m_{1}}{|(x;q)_{\infty}|}\in(0,\infty)$. Thus the right-hand side of (3.21) is
upper bounded by the constant $c_{\max}/c_{\min}<\infty$. ∎

Proof of Step 4. We prove a stronger statement than step 4. Let $M>0$ be any
positive number. We show that both sides of (3.2) are uniformly bounded on
$\\{\theta\in\mathbb{C}\colon\Re\theta\geq M\\}$. Let us begin with the left-
hand side of (3.2). Observe that $\theta$ appears in the left side only within
the variable $t=q^{\theta}$ and $|t|=q^{\Re\theta}\leq q^{M}$. Name
$\epsilon=q^{M}\in(0,1)$; we have to prove that there exists a constant $C>0$
such that

$\displaystyle\sup_{|t|\leq\epsilon}{\left|\frac{P_{\lambda}\big{(}xt^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}\right|}\leq
C.$

Thanks to the branching rule for Macdonald polynomials, Theorem 2.5, and the
assumptions $|x|\leq 1$, $x\neq 0$, $\lambda\in\mathbb{GT}^{+}_{N}$, we have

$\displaystyle\left|\frac{P_{\lambda}\big{(}xt^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}\right|\leq\sum_{\mu\prec\lambda}{|\psi_{\lambda/\mu}(q,t)||t|^{(N-1)(|\lambda|-|\mu|)}\left|\frac{P_{\mu}\big{(}t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}\right|}.$

Given $\lambda\in\mathbb{GT}^{+}_{N}$, there are finitely many
$\mu\in\mathbb{GT}^{+}_{N-1}$ with $\mu\prec\lambda$. Thus it suffices to
prove that there exist constants $C_{1},C_{2}>0$ such that

$\displaystyle|\psi_{\lambda/\mu}(q,t)|\leq
C_{1},\qquad\left|t^{(N-1)(|\lambda|-|\mu|)}\frac{P_{\mu}\big{(}t^{N-2},\dots,t,1;q,t\big{)}}{P_{\mu}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}\right|\leq
C_{2},$

where $C_{1}$, $C_{2}$ do not depend on $t$, though they may depend on $\mu$.

The branching coefficient $|\psi_{\lambda/\mu}(q,t)|$, due to the expression
in Theorem 2.5, is a finite product of terms of the form
$\frac{1-q^{a}t^{b}}{1-q^{c}t^{d}}$, with $a,b,c,d\in\mathbb{Z}_{\geq 0}$ and
$(c,d)\neq(0,0)$. We have

$\displaystyle\left|\frac{1-q^{a}t^{b}}{1-q^{c}t^{d}}\right|\leq
2\big{(}1-q^{c}\epsilon^{d}\big{)}^{-1}$

for all $|t|\leq\epsilon$ and $2(1-q^{c}\epsilon^{d})^{-1}$ does not depend on
$t$, so the boundedness of $|\psi_{\lambda/\mu}(q,t)|$ follows.

Due to the evaluation identity for Macdonald polynomials, Theorem 2.3, we have

$\displaystyle
t^{(N-1)(|\lambda|-|\mu|)}\frac{P_{\mu}\big{(}t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=t^{(N-1)(|\lambda|-|\mu|)+n(\mu)-n(\lambda)}$
$\displaystyle\qquad{}\times\prod_{1\leq i<j\leq
N-1}{\frac{(q^{\mu_{i}-\mu_{j}}t^{j-i};q)_{\infty}(t^{j-i+1};q)_{\infty}}{(t^{j-i};q)_{\infty}(q^{\mu_{i}-\mu_{j}}t^{j-i+1};q)_{\infty}}}\prod_{1\leq
i<j\leq
N}{\frac{(t^{j-i};q)_{\infty}(q^{\lambda_{i}-\lambda_{j}}t^{j-i+1};q)_{\infty}}{(q^{\lambda_{i}-\lambda_{j}}t^{j-i};q)_{\infty}(t^{j-i+1};q)_{\infty}}},$

where $n(\lambda)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}(N-1)\lambda_{N}+(N-2)\lambda_{N-1}+\dots+2\lambda_{3}+\lambda_{2}$
and similarly for $n(\mu)$.

The last two terms above are products of a finite number of fractions
$\frac{1-q^{a}t^{b}}{1-q^{c}t^{d}}$, with $a,b,c,d\in\mathbb{Z}_{\geq 0}$,
$(c,d)\neq(0,0)$, and as we saw above it is implied that the absolute value of
the last two terms above are upper bounded by a constant independent of $t$
(as long as $|t|\leq\epsilon$). Thus our only goal is to show there is an
upper bound for $t^{(N-1)(|\lambda|-|\mu|)+n(\mu)-n(\lambda)}$; this fact
follows if the exponent is nonnegative. In fact, we have

$\displaystyle(N-1)(|\lambda|-|\mu|)-(n(\lambda)-n(\mu))$
$\displaystyle\qquad{}=(N-1)(|\lambda|-|\mu|)-((N-1)\lambda_{N}+(N-2)(\lambda_{N-1}-\mu_{N-1})+\dots+(\lambda_{2}-\mu_{2}))$
$\displaystyle\qquad{}\geq(N-1)(|\lambda|-|\mu|)-(N-1)(\lambda_{N}+\lambda_{N-1}-\mu_{N-1}+\dots+\lambda_{2}-\mu_{2})$
$\displaystyle\qquad{}=(N-1)(\lambda_{1}-\mu_{1})\geq 0.$

Let us proceed to prove uniform boundedness of the right-hand side of (3.2) on
$\\{\theta\in\mathbb{C}\colon\Re\theta\geq M\\}$. First of all, the triangular
inequality gives

$\displaystyle\big{|}\big{(}xt^{N};q\big{)}_{\infty}\big{|}\leq\prod_{i=0}^{\infty}{\big{(}1+|x|q^{N\Re\theta+i}\big{)}}\leq\prod_{i=0}^{\infty}{\big{(}1+|x|q^{i}\big{)}},$
$\displaystyle|(xq;q)_{\infty}|=\prod_{i=1}^{\infty}{\big{|}\big{(}1-xq^{i}\big{)}\big{|}}\geq\prod_{i=1}^{\infty}{\big{(}1-|x|q^{i}\big{)}},$

so the factor $\big{(}xt^{N};q\big{)}_{\infty}/(xq;q)_{\infty}$ in (3.2) has
an upper-bounded absolute value. We are left to deal with

$\displaystyle\Gamma_{q}(\theta
N)\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}{\rm
d}z}$ $\displaystyle\qquad{}=\int_{\mathcal{C}^{+}}{x^{z}\Gamma_{q}(\theta
N)\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}{\rm
d}z}$ (3.22)

and prove its absolute value is uniformly bounded on
$\\{\theta\in\mathbb{C}\colon\Re\theta\geq M\\}$.

For any $M_{2}>0$, the contribution of the portion
$\mathcal{C}^{+}\cap\\{z\in\mathbb{C}\colon\Re z\leq M_{2}\\}$ of the contour
is bounded by a constant. In fact, $\Re\theta\geq M$ implies
$|t|=|q^{\theta}|=q^{\Re\theta}\leq q^{M}$ and $\theta$ appears in the
integrand of (3.2) only as part of the exponent of some $q$, thus the
integrand can be written as a function of $z$ and $t$ (with $q\in(0,1)$
fixed). Thus for $(z,t)$ in the compact subset
$(\mathcal{C}^{+}\cap\\{z\in\mathbb{C}\colon\Re z\leq
M_{2}\\})\times\big{[}0,q^{M}\big{]}$, the integrand in (3.22) attains a
maximum value $L<\infty$, and the contribution of the integral in the portion
$\mathcal{C}^{+}\cap\\{z\in\mathbb{C}\colon\Re z\leq M_{2}\\}$ of the contour
is upper bounded by $L$ times the length of that finite portion.

Since $|x|<1$, the term $x^{z}$ decreases exponentially as
$|z|\rightarrow\infty$, $z\in\mathcal{C}^{+}$. Thus to deal with the infinite
portion of the integral $\mathcal{C}^{+}\cap\\{z\in\mathbb{C}\colon\Re z\geq
M_{2}\\}$, it is enough to show that

$\displaystyle\Gamma_{q}(\theta
N)\prod_{i=1}^{N}{\frac{\Gamma_{q}(\lambda_{i}+\theta(N-i)-z)}{\Gamma_{q}(\lambda_{i}+\theta(N-i+1)-z)}}=(1-q)\frac{(q;q)_{\infty}}{(q^{\theta
N};q)_{\infty}}\prod_{i=1}^{N}{\frac{\big{(}q^{\lambda_{i}+\theta(N-i+1)-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{i}+\theta(N-i)-z};q\big{)}_{\infty}}}$

has bounded absolute value for all $z\in\mathcal{C}^{+}$, $\Re z>M_{2}$ and
$\theta\in\mathbb{C}$, $\Re\theta\geq M$, for a suitable constant $M_{2}>0$.
Clearly $\big{|}\big{(}q^{\theta
N};q\big{)}\big{|}^{-1}\leq\big{(}q^{N\Re\theta};q\big{)}^{-1}\leq\big{(}q^{NM};q\big{)}^{-1}$,
thus we only need a bound on the absolute value of

$\displaystyle\prod_{i=1}^{N}{\frac{\big{(}q^{\lambda_{i}+\theta(N-i+1)-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{i}+\theta(N-i)-z};q\big{)}_{\infty}}},$
(3.23)

uniformly over all $z\in\mathcal{C}^{+}$, $\Re z>M_{2}$, and
$\theta\in\mathbb{C}$, $\Re\theta\geq M$. We can bound the absolute value of
(3.23) by

$\displaystyle\prod_{i=1}^{N-1}{\left|\frac{\big{(}q^{\lambda_{i+1}+\theta(N-i)-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{i}+\theta(N-i)-z};q\big{)}_{\infty}}\right|}\times\left|\frac{\big{(}q^{\lambda_{1}+\theta
N-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{N}-z};q\big{)}_{\infty}}\right|$
$\displaystyle\qquad{}=\prod_{\begin{subarray}{c}1\leq i\leq N-1\\\
\lambda_{i+1}\leq
j<\lambda_{i}\end{subarray}}{\big{|}1-q^{\theta(N-i)+j-z}\big{|}}\times\frac{1}{\prod\limits_{\lambda_{N}\leq
j<\lambda_{1}}{|1-q^{j-z}|}}\times\left|\frac{\big{(}q^{\lambda_{1}+\theta
N-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{1}-z};q\big{)}_{\infty}}\right|$
$\displaystyle\qquad{}\leq\prod_{\begin{subarray}{c}1\leq i\leq N-1\\\
\lambda_{i+1}\leq
j<\lambda_{i}\end{subarray}}{\big{(}1+q^{(N-i)\Re\theta+j-\Re
z}\big{)}}\times\frac{1}{\prod\limits_{\lambda_{N}\leq
j<\lambda_{1}}{\big{(}q^{j-\Re
z}-1\big{)}}}\times\left|\frac{\big{(}q^{\lambda_{1}+\theta
N-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{1}-z};q\big{)}_{\infty}}\right|$
$\displaystyle\qquad{}\leq\prod_{\lambda_{N}\leq
j<\lambda_{1}}{\frac{1+q^{j-\Re z}}{q^{j-\Re
z}-1}}\times\left|\frac{\big{(}q^{\lambda_{1}+\theta
N-z};q\big{)}_{\infty}}{\big{(}q^{\lambda_{1}-z};q\big{)}_{\infty}}\right|.$

If $\Re z$ is large enough, the product $\prod\limits_{\lambda_{N}\leq
j<\lambda_{1}}{\frac{1+q^{j-\Re z}}{q^{j-\Re z}-1}}$ is clearly upper bounded
by a constant. We still have to bound $\left|\frac{(q^{\lambda_{1}+\theta
N-z};q)_{\infty}}{(q^{\lambda_{1}-z};q)_{\infty}}\right|$. Since $\lambda_{1}$
is real and $\Im z$ is constant for $z\in\mathcal{C}^{+}$, $\Re z$ large
enough, it suffices to prove the following statement: there exist constants
$M_{1},M_{2}>0$ such that $z\in\mathcal{C}^{+}$, $\Re z>M_{2}$, and
$\Re\theta\geq M$ imply

$\displaystyle\left|\frac{\big{(}q^{\theta
N-z};q\big{)}_{\infty}}{(q^{-z};q)_{\infty}}\right|<M_{1}.$

This statement was proved above in step 3, see (3.19). In that case, $\theta$
varied over a compact subset
$K\subset\\{\theta\in\mathbb{C}\colon\Re\theta>0\\}$, but in this case
$\theta$ varies over a closed infinite domain of the form
$\\{\theta\in\mathbb{C}\colon\Re\theta\geq M\\}$. However, the expression
$\frac{(q^{\theta N-z};q)_{\infty}}{(q^{-z};q)_{\infty}}$ depends on $\theta$
only by means of $q^{\theta}$, so the proof of (3.19) above can be repeated
word-by-word, since we only used $|t|\leq q^{\inf_{\theta\in K}{\Re\theta}}<1$
in that proof.

##  4 Multiplicative formulas for Macdonald characters

We now come to the multiplicative formulas. All of our results require
parameter $\theta$ to be a positive integer. In this section, $q$ is typically
a variable (but of course, we can specialize $q$ to a complex number later).

###  4.1 Statement of the multiplicative theorem and some consequences

We need some non-standard terminology on $q$-difference operators. The
$q$-shift operators $\\{T_{q,x_{i}}\colon i=1,\dots,m\\}$ are linear operators
on $\mathbb{C}(q)[x_{1},\dots,x_{m}]$ that act as

$\displaystyle(T_{q,x_{i}}f)(x_{1},\dots,x_{m})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}f(x_{1},\dots,x_{i-1},qx_{i},x_{i+1},\dots,x_{m}).$

The $q$-degree operators $\\{D_{q,x_{i}}\\}_{i=1,\dots,m}$ are linear
operators on $\mathbb{C}(q)[x_{1},\dots,x_{m}]$ defined by

$\displaystyle
D_{q,x_{i}}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{T_{q,x_{i}}-1}{q-1}\qquad\text{or}$
$\displaystyle(D_{q,x_{i}}f)(x_{1},\dots,x_{m})=\frac{f(x_{1},\dots,x_{i-1},qx_{i},x_{i+1},\dots,x_{m})-f(x_{1},\dots,x_{m})}{q-1}.$

The $q$-difference operators that appear in the multiplicative formulas for
Macdonald polynomials are finite sums of terms

$\displaystyle
c_{i_{1},\dots,i_{m}}(x_{1},\dots,x_{m};q)T_{q,x_{1}}^{i_{1}}\cdots
T_{q,x_{m}}^{i_{m}}\qquad\text{or}\qquad
c_{i_{1},\dots,i_{m}}(x_{1},\dots,x_{m};q)D_{q,x_{1}}^{i_{1}}\cdots
D_{q,x_{m}}^{i_{m}},$

where $(i_{1},\dots,i_{m})$ vary over a finite subset of $\mathbb{Z}_{\geq
0}^{m}$, and $c_{i_{1},\dots,i_{m}}(x_{1},\dots,x_{m};q)$ are rational
functions in the variables $q,x_{1},\dots,x_{m}$. Thus the operators that we
consider are linear operators
$\mathbb{C}(q)[x_{1},\dots,x_{m}]\longrightarrow\mathbb{C}(q,x_{1},\dots,x_{m})$
that act on polynomials and yield rational functions.

Recall the setting of Jacobi Trudi’s formula for Macdonald polynomials,
Theorem 2.7. The expressions
$C_{\tau_{1},\dots,\tau_{n}}^{(q,q^{\theta})}(u_{1},\dots,u_{n})$ were defined
in (2.5); they are rational functions in $q,u_{1},\dots,u_{n}$ whose
denominators are products of linear factors. We define $M_{\theta}^{(m)}$ as
the set of strictly upper-triangular $m\times m$ matrices whose entries belong
to $\\{0,1,\dots,\theta\\}$ (the cardinality of $M_{\theta}^{(m)}$ is
$(\theta+1){m\choose 2}$). For any strictly upper-triangular $m\times m$
matrix $\tau$ and $1\leq i\leq m$, let
$\tau_{i}^{+}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum\limits_{j=i+1}^{m}{\tau_{i,j}}$ (resp.
$\tau_{i}^{-}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum\limits_{j=1}^{i-1}{\tau_{j,i}}$) be the sum of the entries
of $\tau$ to the right of $(i,i)$ (resp. sum of entries of $\tau$ above
$(i,i)$), cf. (2.6). The main theorem of this section is the following.

######  Theorem 4.1.

Let $\theta\in\mathbb{N}$, $t=q^{\theta}$, $N\in\mathbb{N}$,
$\lambda\in\mathbb{GT}_{N}$. Then

$\displaystyle
P_{\lambda}\big{(}x_{1},\dots,x_{m};N,q,q^{\theta}\big{)}=\frac{q^{\theta^{2}{m+1\choose
3}-\left\\{N\theta^{2}-{\theta+1\choose 2}\right\\}{m\choose
2}}\prod\limits_{i=1}^{m}{[\theta(N-i+1)-1]_{q}!}}{\prod\limits_{i=1}^{m}{\prod\limits_{j=1}^{\theta(N-m+1)-1}{(x_{i}-q^{j-\theta})}}}$
$\displaystyle\qquad{}\times\frac{1}{\prod_{\begin{subarray}{c}1\leq i<j\leq
m\\\ 0\leq
k<\theta\end{subarray}}{(x_{i}-q^{k}x_{j})}}\times\mathcal{D}^{(m)}_{q,\theta}\left\\{\prod_{i=1}^{m}{\frac{P_{\lambda}\big{(}x_{i};N,q,q^{\theta}\big{)}\prod\limits_{j=1}^{\theta
N-1}{\big{(}x_{i}-q^{j-\theta}\big{)}}}{[\theta N-1]_{q}!}}\right\\},$ (4.1)

where $\mathcal{D}^{(m)}_{q,\theta}$ is the $q$-difference operator
$\mathbb{C}(q)[x_{1},\dots,x_{m}]\longrightarrow\mathbb{C}(q,x_{1},\dots,x_{m})$
given by

$\displaystyle\mathcal{D}^{(m)}_{q,\theta}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{1}{(q-1)^{\theta{m\choose 2}}}\sum_{\tau\in
M^{(m)}_{\theta}}\left\\{C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m})\prod_{i=1}^{m}{T_{q,x_{i}}^{(i-1)\theta+\tau_{i}^{+}-\tau_{i}^{-}}}\right\\},$
(4.2)

where for any $\tau\in M_{\theta}^{(m)}$, we denoted

$\displaystyle C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m})$
$\displaystyle\qquad{}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\prod_{s=1}^{m-1}{C_{\tau_{1,s+1},\dots,\tau_{s,s+1}}^{(q,q^{\theta})}\left(\\{u_{i}=x_{s+1}^{-1}x_{i}q^{-\theta+\sum\limits_{j=s+2}^{m}{(\tau_{i,j}-\tau_{s+1,j})}}\colon
1\leq i\leq s\\}\right)}.$ (4.3)

The proof of Theorem 4.1 is given in the next subsection. We derive here some
conclusions, namely two special cases when the operator
$\mathcal{D}_{q,\theta}^{(m)}$ has a simple form. The first simple case is
$m=2$.

######  Corollary 4.2.

In the same setting as Theorem 4.1 $($for $m=2)$, we have

$\displaystyle
P_{\lambda}\big{(}x_{1},x_{2};N,q,q^{\theta}\big{)}=\frac{q^{-(N-1)\theta^{2}+{\theta+1\choose
2}}\cdot[\theta(N-1)-1]_{q}!}{\prod\limits_{j=1}^{\theta(N-1)-1}{\big{(}x_{1}-q^{j-\theta}\big{)}\big{(}x_{2}-q^{j-\theta}\big{)}}}$
$\displaystyle\hphantom{P_{\lambda}\big{(}x_{1},x_{2};N,q,q^{\theta}\big{)}=}{}\times\widetilde{\mathcal{D}}_{q,\theta}^{(2)}\left\\{\frac{\prod\limits_{i=1}^{2}{P_{\lambda}\big{(}x_{i};N,q,q^{\theta}\big{)}\prod\limits_{j=1}^{\theta
N-1}{\big{(}x_{i}-q^{j-\theta}\big{)}}}}{[\theta N-1]_{q}!}\right\\}$

where

$\displaystyle\widetilde{\mathcal{D}}_{q,\theta}^{(2)}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\left(\frac{1}{x_{1}-x_{2}}\circ(D_{q,x_{2}}-D_{q,x_{1}})\right)^{\theta}$
$\displaystyle\hphantom{\widetilde{\mathcal{D}}_{q,\theta}^{(2)}}{}=\underbrace{\left(\frac{1}{x_{1}-x_{2}}\circ(D_{q,x_{2}}-D_{q,x_{1}})\right)\circ\cdots\circ\left(\frac{1}{x_{1}-x_{2}}\circ(D_{q,x_{2}}-D_{q,x_{1}})\right)}_{\textrm{composition
of $\theta$ operators}}.$

###### Proof.

For $m=2$, we have $M_{\theta}^{(2)}=\left\\{\left[\begin{smallmatrix}0&n\\\
0&0\end{smallmatrix}\right],\ n\in\\{0,1,\dots,\theta\\}\right\\}$, and thus
the operator $\mathcal{D}_{q,\theta}^{(2)}$ becomes

$\displaystyle\mathcal{D}^{(2)}_{q,\theta}=\frac{1}{(q-1)^{\theta}}\sum_{n=0}^{\theta}\big{\\{}{C_{n}^{(q,q^{\theta})}\big{(}x_{2}^{-1}x_{1}q^{-\theta}\big{)}}T_{q,x_{1}}^{n}T_{q,x_{2}}^{\theta-n}\big{\\}}.$

We let

$\displaystyle
a_{n}^{(\theta)}=\frac{C_{n}^{(q,q^{\theta})}\big{(}x_{2}^{-1}x_{1}q^{-\theta}\big{)}}{\prod\limits_{i=0}^{\theta-1}{(x_{1}-q^{i}x_{2})}},\qquad
0\leq n\leq\theta.$

The statement of the theorem can be easily reduced to prove that the
coefficient of $T_{q,x_{1}}^{n}T_{q,x_{2}}^{\theta-n}$ in the product

$\displaystyle
D_{q,\theta}=\left(\frac{1}{x_{1}-x_{2}}\circ(T_{q,x_{2}}-T_{q,x_{1}})\right)^{\theta}=\sum_{n=0}^{\theta}{b_{n}^{(\theta)}T_{q,x_{1}}^{n}T_{q,x_{2}}^{\theta-n}}$

is equal to $a_{n}^{(\theta)}$, i.e., we prove
$a_{n}^{(\theta)}=b_{n}^{(\theta)}$ for all $\theta,n\in\mathbb{N}$, $0\leq
n\leq\theta$, and we do it by induction on $\theta$. The case $\theta=1$ can
be easily dealt with, using Lemma B.2. Now assume
$a_{n}^{(\theta-1)}=b_{n}^{(\theta-1)}$ for all $0\leq n\leq\theta-1$, and
some $\theta\geq 2$. We prove $a_{n}^{(\theta)}=b_{n}^{(\theta)}$ for all
$0\leq n\leq\theta$. Evidently,
$D_{q,\theta}=\big{(}\frac{1}{x_{1}-x_{2}}\circ(T_{q,x_{2}}-T_{q,x_{1}})\big{)}\circ
D_{q,\theta-1}$ implies

$\displaystyle
b_{n}^{(\theta)}=\frac{1}{x_{1}-x_{2}}\big{(}T_{q,x_{2}}b_{n}^{(\theta-1)}-T_{q,x_{1}}b_{n-1}^{(\theta-1)}\big{)},\qquad
n=1,2,\dots,\theta-1,$ $\displaystyle
b_{0}^{(\theta)}=\frac{1}{x_{1}-x_{2}}T_{q,x_{2}}b_{0}^{(\theta-1)},\qquad
b_{\theta}^{(\theta)}=\frac{1}{x_{2}-x_{1}}T_{q,x_{1}}b_{\theta-1}^{(\theta-1)}.$

From Lemma B.3 in Appendix B, the terms $a_{n}^{(\theta)}$ satisfy

$\displaystyle
a_{n}^{(\theta)}=\frac{1}{x_{1}-x_{2}}\big{(}T_{q,x_{2}}a_{n}^{(\theta-1)}-T_{q,x_{1}}a_{n-1}^{(\theta-1)}\big{)},\qquad
n=1,2,\dots,\theta-1,$ $\displaystyle
a_{0}^{(\theta)}=\frac{1}{\prod\limits_{i=0}^{\theta-1}{(x_{1}-q^{i}x_{2})}},\qquad
a_{\theta}^{(\theta)}=\frac{1}{\prod\limits_{i=0}^{\theta-1}{(x_{2}-q^{i}x_{1})}}.$

It is not difficult to conclude from these relations, and the inductive
hypothesis $a_{m}^{(\theta-1)}=b_{m}^{(\theta-1)}$
$\forall\,m=0,1,\dots,\theta-1$, that $a_{n}^{(\theta)}=b_{n}^{(\theta)}$ for
all $0\leq n\leq\theta$, as desired. ∎

When $\theta=1$ (equivalently $t=q$), the result has a compact form as well.
Let us recall that when $t=q$, the Macdonald polynomials become the well known
Schur polynomials
$s_{\lambda}(x_{1},\dots,x_{N})=P_{\lambda}(x_{1},\dots,x_{N};q,q)$. The Schur
(Laurent) polynomials $s_{\lambda}(x_{1},\dots,x_{N})$,
$\lambda\in\mathbb{GT}_{N}$, can also be defined by the simple determinantal
formula

$\displaystyle
s_{\lambda}(x_{1},\dots,x_{N})=\frac{\det\big{[}x_{i}^{N+1-j}\big{]}_{i,j=1}^{N}}{\prod\limits_{1\leq
i<j\leq N}{(x_{i}-x_{j})}}.$

For any $m\in\mathbb{N}$ with $1\leq m\leq N$, we consider

$\displaystyle
s_{\lambda}(x_{1},\dots,x_{m};N,q)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{s_{\lambda}\big{(}x_{1},\dots,x_{m},1,q,\dots,q^{N-1-m}\big{)}}{s_{\lambda}(1,q,\dots,q^{N-1})}$

and call it a $q$-Schur character of rank $N$, number of variables $m$ and
parametrized by $\lambda$; it was defined before in [21]. We recover the
following theorem.

######  Corollary 4.3 ([24, Theorem 3.5]).

Let $N\in\mathbb{N}$, $\lambda\in\mathbb{GT}_{N}$. Then

$\displaystyle s_{\lambda}(x_{1},\dots,x_{m};N,q)=\frac{q^{{m+1\choose
3}-(N-1){m\choose
2}}\prod\limits_{i=1}^{m}{[N-i]_{q}!}}{\prod\limits_{i=1}^{m}{\prod\limits_{j=1}^{N-m}{\big{(}x_{i}-q^{j-1}\big{)}}}}\frac{1}{\Delta(x_{1},\dots,x_{m})}$
$\displaystyle\hphantom{s_{\lambda}(x_{1},\dots,x_{m};N,q)=}{}\times\mathcal{D}_{q}^{(m)}\left\\{\prod\limits_{i=1}^{m}{\frac{s_{\lambda}(x_{i};N,q)\prod\limits_{j=1}^{N-1}{\big{(}x_{i}-q^{j-1}\big{)}}}{[N-1]_{q}!}}\right\\},$

where

$\displaystyle\mathcal{D}_{q}^{(m)}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\det\big{[}D_{q,x_{i}}^{j-1}\big{]}_{i,j=1}^{m}=\prod_{1\leq
i<j\leq m}{(D_{q,x_{j}}-D_{q,x_{i}})}.$

###### Proof.

Letting $\theta=1$ in Theorem 4.1, we see that the equation above holds if
$\mathcal{D}_{q,1}^{(m)}=\mathcal{D}_{q}^{(m)}$. In fact, thanks to Lemma B.2,
we have

$\displaystyle\mathcal{D}^{(m)}_{q,1}=(q-1)^{-{m\choose 2}}\sum_{\tau\in
M^{(m)}_{1}}\prod_{s=1}^{m-1}{(-1)^{\tau_{1,s+1}+\dots+\tau_{s,s+1}}}\prod_{i=1}^{m}{T_{q,x_{i}}^{i-1+\tau_{i}^{+}-\tau_{i}^{-}}}.$

Evidently
$\mathcal{D}_{q}^{(m)}=\prod\limits_{i<j}{(D_{q,x_{j}}-D_{q,x_{i}})}=(q-1)^{-{m\choose
2}}\prod\limits_{i<j}{(T_{q,x_{j}}-T_{q,x_{i}})}$, thus we need to show

$\displaystyle\prod_{1\leq i<j\leq m}{(T_{q,x_{j}}-T_{q,x_{i}})}=\sum_{\tau\in
M^{(m)}_{1}}{(-1)^{|\tau|}\prod_{k=1}^{m}{T_{q,x_{k}}^{k-1+\tau_{k}^{+}-\tau_{k}^{-}}}},$
(4.4)

where we denoted by $|\tau|$ to the sum of all entries of $\tau\in
M_{1}^{(m)}$. The operators $T_{q,x_{1}},\dots,T_{q,x_{m}}$ pairwise commute.
When expanding the left-hand side of (4.4), it is clear that the resulting
terms can be parametrized by matrices in $M_{1}^{(m)}$: the term corresponding
to $\tau\in M_{1}^{(m)}$ is the product of
$(-1)^{k}T_{q,x_{i}}^{k}T_{q,x_{j}}^{1-k}$, where $k\in\\{0,1\\}$ ranges over
the elements of $\tau$ strictly above the main diagonal. Thus the term
corresponding to $\tau$ is

$\displaystyle\prod_{1\leq i<j\leq
m}{(-1)^{\tau_{i,j}}T_{q,x_{i}}^{\tau_{i,j}}T_{q,x_{j}}^{1-\tau_{i,j}}}=(-1)^{\sum\limits_{1\leq
i<j\leq m}{\tau_{i,j}}}\prod_{1\leq i<j\leq
m}{T_{q,x_{i}}^{\tau_{i,j}}T_{q,x_{j}}^{1-\tau_{i,j}}}.$

We are left to show

$\displaystyle\prod_{1\leq i<j\leq
m}{T_{q,x_{i}}^{\tau_{i,j}}T_{q,x_{j}}^{1-\tau_{i,j}}}=\prod_{k=1}^{m}{T_{q,x_{k}}^{k-1+\tau_{k}^{+}-\tau_{k}^{-}}}.$
(4.5)

Both sides of (4.5) are of the form $T_{q,x_{1}}^{p_{1}}\cdots
T_{q,x_{m}}^{p_{m}}$, for some $p_{1},\dots,p_{m}\in\mathbb{Z}_{\geq 0}$, so
we simply need to check the equality between exponents $p_{k}$ of
$T_{q,x_{k}}$ in both sides, for an arbitrary $1\leq k\leq m$. In the left
side, there are $k-1$ factors of the form
$T_{q,x_{i}}^{\tau_{i,k}}T_{q,x_{k}}^{1-\tau_{i,k}}$, $i\leq k-1$, which
overall contribute $k-1-\sum\limits_{i=1}^{k-1}{\tau_{i,k}}=k-1-\tau_{k}^{-}$
to the exponent of $T_{q,x_{k}}$. Moreover there are $m-k$ factors of the form
$T_{q,x_{k}}^{\tau_{k,j}}T_{q,x_{j}}^{1-\tau_{k,j}}$, $k+1\leq i$, which
contribute $\sum\limits_{i=k+1}^{m}{\tau_{k,i}}=\tau_{k}^{+}$ to the exponent
of $T_{q,x_{k}}$. Therefore the power of $T_{q,x_{k}}$ in the left-hand side
of (4.5) is $T_{q,x_{k}}^{k-1+\tau_{k}^{+}-\tau_{k}^{-}}$. Evidently, the
power of $T_{q,x_{k}}$ in the right-hand side of (4.5) if also
$T_{q,x_{k}}^{k-1+\tau_{k}^{+}-\tau_{k}^{-}}$, which finishes the proof. ∎

######  Example 4.4.

We discuss the first nontrivial example of the multiplicative formula for
Macdonald polynomials (an example that is not dealt with in the Corollaries
above): $\theta=2$, $m=3$. The formula in this case is

$\displaystyle
P_{\lambda}\big{(}x_{1},x_{2},x_{3};N,q,q^{2}\big{)}=\frac{q^{25-12N}}{([2N-1]_{q})^{2}([2N-2]_{q})^{2}[2N-3]_{q}[2N-4]_{q}}$
$\displaystyle\hphantom{P_{\lambda}\big{(}x_{1},x_{2},x_{3};N,q,q^{2}\big{)}=}{}\times\frac{1}{\prod\limits_{j=-1}^{2N-7}{(x_{1}-q^{j})(x_{2}-q^{j})(x_{3}-q^{j})}}\frac{1}{\prod\limits_{1\leq
i<j\leq 3}{(x_{i}-x_{j})(x_{i}-qx_{j})}}$
$\displaystyle\hphantom{P_{\lambda}\big{(}x_{1},x_{2},x_{3};N,q,q^{2}\big{)}=}{}\times\widehat{\mathcal{D}}_{q,2}^{(3)}\left\\{\prod_{i=1}^{3}{P_{\lambda}\big{(}x_{i};N,q,q^{2}\big{)}\prod_{j=-1}^{2N-3}{(x_{i}-q^{j})}}\right\\},$

the $q$-difference operator is

$\displaystyle\widehat{\mathcal{D}}_{q,2}^{(3)}=\frac{1}{(q-1)^{6}}\sum_{\begin{subarray}{c}a,b,c\in\mathbb{Z}\\\
0\leq a,b,c\leq
2\end{subarray}}\big{\\{}C^{(q,q^{2})}_{a}\big{(}x_{2}^{-1}x_{1}q^{-2+b-c}\big{)}$
$\displaystyle\hphantom{\widehat{\mathcal{D}}_{q,2}^{(3)}=\frac{1}{(q-1)^{6}}\sum_{\begin{subarray}{c}a,b,c\in\mathbb{Z}\\\
0\leq a,b,c\leq 2\end{subarray}}}{}\times
C^{(q,q^{2})}_{b,c}\big{(}x_{3}^{-1}x_{1}q^{-2},x_{3}^{-1}x_{2}q^{-2}\big{)}T_{q,x_{1}}^{a+b}T_{q,x_{2}}^{2+c-a}T_{q,x_{3}}^{4-b-c}\big{\\}}.$

We can also write the $q$-difference operator in terms of the $q$-degree
operators $\\{D_{q,x_{i}}\colon i=1,2,3\\}$ by using
$\frac{1}{q-1}T_{q,x_{i}}=\frac{1}{q-1}+D_{q,x_{i}}$. Then we can replace the
operator $\widehat{\mathcal{D}}_{q,2}^{(3)}$ with the sum of operators
$\mathcal{D}_{q,2}^{(3,{\rm top})}+\mathcal{A}_{q,2}^{(3)}$, where

$\displaystyle\mathcal{D}_{q,2}^{(3,{\rm
top})}=\sum_{\begin{subarray}{c}a,b,c\in\mathbb{Z}\\\ 0\leq a,b,c\leq
2\end{subarray}}\big{\\{}C^{(q,q^{2})}_{a}\big{(}x_{2}^{-1}x_{1}q^{-2+b-c}\big{)}$
$\displaystyle\hphantom{\mathcal{D}_{q,2}^{(3,{\rm
top})}=\sum_{\begin{subarray}{c}a,b,c\in\mathbb{Z}\\\ 0\leq a,b,c\leq
2\end{subarray}}}{}\times
C^{(q,q^{2})}_{b,c}\big{(}x_{3}^{-1}x_{1}q^{-2},x_{3}^{-1}x_{2}q^{-2}\big{)}D_{q,x_{1}}^{a+b}D_{q,x_{2}}^{2+c-a}D_{q,x_{3}}^{4-b-c}\big{\\}},$
$\displaystyle\mathcal{A}_{q,2}^{(3)}=\sum_{\begin{subarray}{c}i_{1},i_{2},i_{3}\geq
0\\\ i_{1}+i_{2}+i_{3}\leq
5\end{subarray}}{\frac{f_{i_{1},i_{2},i_{3}}(x_{1},x_{2},x_{3};q,2)}{(q-1)^{6-i_{1}-i_{2}-i_{3}}}D_{q,x_{1}}^{i_{1}}D_{q,x_{2}}^{i_{2}}D_{q,x_{3}}^{i_{3}}},$

and $f_{i_{1},i_{2},i_{3}}(x_{1},x_{2},x_{3};q,2)$ are certain rational
functions. With the help of Sage, we found

$\displaystyle
f_{i_{1},i_{2},i_{3}}(x_{1},x_{2},x_{3};q,2)=0,\qquad\textrm{for all}\quad
i_{1}+i_{2}+i_{3}\leq 2.$

There are nontrivial rational functions
$f_{i_{1},i_{2},i_{3}}(x_{1},x_{2},x_{3};q,2)$ as well, for some
$i_{1}+i_{2}+i_{3}\geq 3$, e.g.,

$\displaystyle
f_{4,1,0}(x_{1},x_{2},x_{3};q,2)=-(q-1)\frac{(x_{2}+x_{3})(x_{1}-qx_{3})(x_{1}-qx_{2})}{(qx_{2}-x_{3})(qx_{1}-x_{3})(qx_{1}-x_{2})},$
(4.6) $\displaystyle
f_{2,1,1}(x_{1},x_{2},x_{3};q,2)=-(q-1)^{2}\frac{(q+1)(x_{2}-x_{3})\big{(}x_{1}^{2}+x_{2}x_{3}+2x_{1}x_{2}+2x_{1}x_{3}\big{)}}{(qx_{1}-x_{2})(qx_{1}-x_{3})(qx_{2}-x_{3})}.$
(4.7)

Two important observations are in order. First, from Corollaries 4.2 and 4.3,
one might believe that $\mathcal{D}_{q,\theta}^{(m)}$ is, in general,
homogeneous of degree $\theta{m\choose 2}$ as a functions of the operators
$\\{D_{q,x_{i}}\colon i=1,2,3\\}$. However, this example disproves it. Second,
the terms $f_{i_{1},i_{2},i_{3}}(x_{1},x_{2},x_{3};q,2)$ above make us suspect
that $f_{i_{1},i_{2},i_{3}}(x_{1},x_{2},x_{3};q,2)$ is divisible by
$(q-1)^{6-i_{1}-i_{2}-i_{3}}$ $\forall\,0\leq i_{1}+i_{2}+i_{3}\leq 5$. We
have checked this fact in the computer. In fact, we believe that the analogous
statement for general $m$, $\theta\in\mathbb{N}$ holds true, but the author
could not prove it.

###  4.2 Proof of Theorem 4.1

Fix a positive signature $\lambda\in\mathbb{GT}^{+}_{N}$ and let us prove
equation (4.1); we extend the result for all signatures
$\lambda\in\mathbb{GT}_{N}$ at the end.

Let us consider $m$ positive integers $n_{1}>n_{2}>\cdots>n_{m}>\theta(N+m)$.
By the index-argument symmetry, Theorem 2.2, applied to
$\lambda\in\mathbb{GT}^{+}_{N}$ and $\mu=(n_{1}\geq\cdots\geq n_{m}\geq
0\geq\cdots\geq 0)\in\mathbb{GT}^{+}_{N}$, as well as the definition of the
dual Macdonald polynomials $Q_{\mu}(\cdot;q,t)$, we obtain

$\displaystyle\frac{P_{\lambda}\big{(}q^{n_{1}}t^{N-1},\dots,q^{n_{m}}t^{N-m},t^{N-m-1},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}$
$\displaystyle\qquad{}=\frac{Q_{(n_{1},\dots,n_{m})}\big{(}q^{\lambda_{1}}t^{N-1},q^{\lambda_{2}}t^{N-2},\dots,q^{\lambda_{N}};q,t\big{)}}{Q_{(n_{1},\dots,n_{m})}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}.$
(4.8)

Apply the Jacobi–Trudi formula for Macdonald polynomials, Theorem 2.7, to the
numerator of (4.8), then multiply and divide the term parametrized by $\tau$
by the product

$\displaystyle\prod_{s=1}^{m}{{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}},$

so (4.8) equals

$\displaystyle\sum_{\tau\in
M^{(m)}}\left\\{\prod_{s=1}^{m-1}{C^{(q,t)}_{\tau_{1,s+1},\dots,\tau_{s,s+1}}\Big{(}\Big{\\{}u_{i}=q^{n_{i}-n_{s+1}+\sum\limits_{j=s+2}^{m}{(\tau_{i,j}-\tau_{s+1,j})}}t^{s-i}\colon
1\leq i\leq s\Big{\\}}\Big{)}}\right.$
$\displaystyle\left.{}\qquad{}\times\prod_{s=1}^{m}{\frac{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}q^{\lambda_{1}}t^{N-1},\dots,q^{\lambda_{N}};q,t\big{)}}{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}}\times\frac{\prod\limits_{s=1}^{m}{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}}{Q_{(n_{1},\dots,n_{m})}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}\right\\}.\\!\\!\\!\\!$
(4.9)

(In equation (4.9), we are setting $g_{n}(q,t)=0$ if $n$ is a nonpositive
integer.) Recall that $t=q^{\theta}$, $\theta\in\mathbb{N}$. In view of Lemma
B.1, the only terms in the sum (4.9) with nonzero contributions are those
parametrized by $m\times m$ matrices whose entries belong to the set
$\\{0,1,\dots,\theta\\}$; define $M^{(m)}_{\theta}$ to be the set of such
matrices. Notice that $n_{s}+\tau_{s}^{+}-\tau_{s}^{-}>0$ for all $\tau\in
M_{\theta}^{(m)}$, because of our initial assumption on the values of
$n_{1},\dots,n_{m}$. By another application of the index-argument symmetry
(and of the identity (2.4) above), we have

$\displaystyle\frac{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{+}}\big{(}q^{\lambda_{1}}t^{N-1},\dots,q^{\lambda_{N}};q,t\big{)}}{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}(t^{N-1},\dots,t,1;q,t)}$
$\displaystyle\qquad{}=\frac{P_{\lambda}\big{(}q^{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}(t^{N-1},\dots,t,1;q,t)}\qquad\forall\,1\leq
s\leq m.$ (4.10)

Plugging (4.2) into (4.9), we obtain

$\displaystyle\frac{P_{\lambda}\big{(}q^{n_{1}+\theta(N-1)},\dots,q^{n_{m}+\theta(N-m)},t^{N-m-1},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},t^{N-2},\dots,1;q,t\big{)}}$
(4.11) $\displaystyle\qquad{}=\sum_{\tau\in
M^{(m)}_{\theta}}\left\\{\prod_{s=1}^{m-1}{C^{(q,q^{\theta})}_{\tau_{1,s+1},\dots,\tau_{s,s+1}}\Big{(}\Big{\\{}u_{i}=q^{n_{i}-n_{s+1}+\sum\limits_{j=s+2}^{m}{(\tau_{i,j}-\tau_{s+1,j})}+\theta(s-i)}\colon
1\leq i\leq s\Big{\\}}\Big{)}}\right.$
$\displaystyle\left.\qquad{}\times\\!\frac{\prod\limits_{s=1}^{m}{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}}{Q_{(n_{1},\dots,n_{m})}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}\\!\times\\!\prod_{s=1}^{m}\\!{\frac{P_{\lambda}\big{(}q^{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}+\theta(N-1)},t^{N-2},\dots,t,1;q,t\big{)}}{P_{\lambda}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}}\right\\}\\!.\\!$

Let us make the change of variables

$\displaystyle
z_{s}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}q^{n_{s}+\theta(N-s)},\qquad 1\leq s\leq m,$ (4.12)

and rewrite some terms from (4.11) in these new variables. Clearly the left-
hand side of equality (4.11) is the Macdonald character
$P_{\lambda}(z_{1},\dots,z_{m};N,q,q^{\theta})$. It is also evident that the
variable $u_{i}$ in the term
$C_{\tau_{1,s+1},\dots,\tau_{s,s+1}}^{(q,q^{\theta})}$ can be rewritten as
$u_{i}=z_{s+1}^{-1}z_{i}q^{-\theta+\sum\limits_{j=s+2}^{m}{(\tau_{i,j}-\tau_{s+1,j})}}$,
for $1\leq i\leq s$. Additionally we have
$q^{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}+\theta(N-1)}=T_{q,z_{s}}^{\theta
s-\theta+\tau_{s}^{+}-\tau_{s}^{-}}(z_{s})$, for all $s$, which implies that
the last product in (4.11) can be written as

$\displaystyle\prod\limits_{s=1}^{m}{T_{q,x_{s}}^{\theta(s-1)+\tau_{s}^{+}-\tau_{s}^{-}}}\left(\prod_{s=1}^{m}{P_{\lambda}\big{(}z_{s};N,q,q^{\theta}\big{)}}\right).$

Next, we need to rewrite

$\displaystyle\frac{\prod\limits_{s=1}^{m}{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}}{Q_{(n_{1},\dots,n_{m})}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}$

in terms of $z_{1},\dots,z_{m}$. From Corollary 2.4 of the evaluation identity
for Macdonald polynomials, we obtain

$\displaystyle
g_{p}\big{(}t^{N-1},t^{N-2},\dots,t,1;q,t\big{)}=\prod_{s=(1,j)\in(p)}{\frac{1-q^{a^{\prime}(s)+\theta(N-l^{\prime}(s))}}{1-q^{a(s)+1+\theta
l(s)}}}$ $\displaystyle\qquad{}=\prod_{i=1}^{p}{\frac{1-q^{i-1+\theta
N}}{1-q^{p-i+1}}}=\prod_{j=1}^{\theta
N-1}{\frac{1-q^{p+j}}{1-q^{j}}}=\frac{1}{(1-q)^{\theta
N-1}}\frac{\prod\limits_{j=1}^{\theta N-1}{(1-q^{p+j})}}{[\theta N-1]_{q}!}$
(4.13)

for any $p\in\mathbb{N}$, $p>\theta N$. By similar, but more complicated,
computations we find

$\displaystyle Q_{(p_{1},\dots,p_{m})}\big{(}t^{N-1},\dots,t,1;q,t\big{)}$
$\displaystyle\qquad{}=\frac{1}{(1-q)^{m(\theta N-1)-\theta
m(m-1)/2}}\times\prod_{s=1}^{\theta}{\prod_{1\leq i<j\leq
m}{\frac{q^{p_{j}}-q^{p_{i}+s+\theta(j-i-1)}}{1-q^{p_{i}+s+\theta(j-i-1)}}}}$
$\displaystyle\qquad\quad{}\times\frac{\prod\limits_{j=1}^{\theta(N-m+1)-1}{\big{(}1-q^{p_{m}+j}\big{)}}\cdot\prod\limits_{j=1}^{\theta(N-m+2)-1}{\big{(}1-q^{p_{m-1}+j}\big{)}}\cdots\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{p_{1}+j}\big{)}}}{[\theta(N-m+1)-1]_{q}![\theta(N-m+2)-1]_{q}!\cdots[\theta
N-1]_{q}!},$ (4.14)

for any partition $(p_{1}>p_{2}>\dots>p_{m}>0)$ with $p_{i}>\theta(N-i+1)$ for
all $1\leq i\leq m$. From (4.13) and (4.14), we obtain

$\displaystyle\frac{\prod\limits_{s=1}^{m}{g_{n_{s}+\tau_{s}^{+}-\tau_{s}^{-}}\big{(}t^{N-1},\dots,t,1\big{)}}}{Q_{(n_{1},\dots,n_{m})}\big{(}t^{N-1},\dots,t,1;q,t\big{)}}=\frac{1}{(1-q)^{\theta
m(m-1)/2}}\prod_{i=1}^{m}{\frac{[\theta(N-i+1)-1]_{q}!}{[\theta N-1]_{q}!}}$
(4.15) $\displaystyle{}{}\times\frac{\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{n_{1}+\tau_{1}^{+}-\tau_{1}^{-}+j}\big{)}}}{\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{n_{1}+j}\big{)}}}\cdot\frac{\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{n_{2}+\tau_{2}^{+}-\tau_{2}^{-}+j}\big{)}}}{\prod\limits_{j=1}^{\theta(N-1)-1}{\big{(}1-q^{n_{2}+j}\big{)}}}\cdots\frac{\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{n_{m}+\tau_{m}^{+}-\tau_{m}^{-}+j}\big{)}}}{\prod\limits_{j=1}^{\theta(N-m+1)-1}{\big{(}1-q^{n_{m}+j}\big{)}}}$
(4.16) $\displaystyle{}\times\prod_{s=1}^{\theta}{\prod_{1\leq i<j\leq
m}{\frac{1-q^{n_{i}+s+\theta(j-i-1)}}{q^{n_{j}}-q^{n_{i}+s+\theta(j-i-1)}}}}.$
(4.17)

Observe that we used our assumption $n_{1}>n_{2}>\dots>n_{m}>\theta(N+m)$ to
guarantee that (4.13) and (4.14) are applicable. We have to rewrite both
(4.16) and (4.17) in terms of $z_{s}$. Let us begin with (4.16), which is a
product of $m$ terms; for $1\leq r\leq m$, the $r^{\rm th}$ term is

$\displaystyle\frac{\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{n_{r}+\tau_{r}^{+}-\tau_{r}^{-}+j}\big{)}}}{\prod\limits_{j=1}^{\theta(N-r+1)-1}{\big{(}1-q^{n_{r}+j}\big{)}}}=\frac{\prod\limits_{j=1}^{\theta
N-1}{\big{(}1-q^{-\theta(N-r)}z_{r}q^{\tau_{r}^{+}-\tau_{r}^{-}+j}\big{)}}}{\prod\limits_{j=1}^{\theta(N-r+1)-1}{\big{(}1-q^{-\theta(N-r)}z_{r}q^{j}\big{)}}}$
$\displaystyle\qquad{}=(-1)^{\theta(r-1)}q^{-\sum\limits_{j=\theta(N-r+1)}^{\theta
N-1}{(\theta(N-r)-j)}}\cdot\frac{\prod\limits_{j=1}^{\theta
N-1}{\big{(}z_{r}q^{\tau_{r}^{+}-\tau_{r}^{+}}-q^{\theta(N-r)-j}\big{)}}}{\prod\limits_{j=1}^{\theta(N-r+1)-1}{\big{(}z_{r}-q^{\theta(N-r)-j}\big{)}}}$
(4.18)
$\displaystyle\qquad{}=(-1)^{\theta(r-1)}q^{-\sum\limits_{j=\theta(N-r+1)}^{\theta
N-1}{(\theta(N-r)-j)}}q^{-(\theta r-\theta)(\theta
N-1)}\frac{\prod\limits_{j=1}^{\theta N-1}{\big{(}z_{r}q^{\theta
r-\theta+\tau_{r}^{+}-\tau_{r}^{-}}-q^{\theta(N-1)-j}\big{)}}}{\prod\limits_{j=1}^{\theta(N-r+1)-1}{\big{(}z_{r}-q^{\theta(N-r)-j}\big{)}}}.$

Under the change of indexing $j\mapsto\theta(N-r+1)-j$, the product in the
denominator of (4.18) becomes

$\displaystyle\prod_{j=1}^{\theta(N-r+1)-1}{\big{(}z_{r}-q^{\theta(N-r)-j}\big{)}}=\prod_{j=1}^{\theta(N-r+1)-1}{\big{(}z_{r}-q^{j-\theta}\big{)}},$

whereas the numerator of (4.18) can be expressed as

$\displaystyle\prod_{j=1}^{\theta N-1}{\big{(}z_{r}q^{\theta
r-\theta+\tau_{r}^{+}-\tau_{r}^{-}}-q^{\theta(N-1)-j}\big{)}}=T_{q,z_{r}}^{\theta
r-\theta+\tau_{r}^{+}-\tau_{r}^{-}}\prod_{j=1}^{\theta
N-1}{\big{(}z_{r}-q^{\theta(N-1)-j}\big{)}}.$

Define also

$\displaystyle c(N,m,\theta)=-\sum_{r=1}^{m}{\sum_{j=\theta(N-r+1)}^{\theta
N-1}{(\theta(N-r)-j)}}-\sum_{r=1}^{m}{(\theta r-\theta)(\theta N-1)},$ (4.19)

which is the power of $q$ coming from the terms (4.18), for $1\leq r\leq m$.
Thus (4.16) equals

$\displaystyle\frac{(-1)^{\theta
m(m-1)/2}q^{c(N,m,\theta)}}{\prod\limits_{r=1}^{m}{\prod\limits_{j=1}^{\theta(N-r+1)-1}{\big{(}z_{r}-q^{j-\theta}\big{)}}}}\cdot\prod_{r=1}^{m}{T_{q,z_{r}}^{\theta
r-\theta+\tau_{r}^{+}-\tau_{r}^{-}}}\left(\prod_{j=1}^{\theta
N-1}{\big{(}z_{r}-q^{\theta(N-1)-j}\big{)}}\right).$ (4.20)

On the other hand, (4.17) can be expressed in terms of the variables $z_{i}$
as follows:

$\displaystyle\prod_{r=1}^{\theta}{\prod_{i<j}{\frac{1-q^{n_{i}+r+\theta(j-i-1)}}{q^{n_{j}}-q^{n_{i}+r+\theta(j-i-1)}}}}=\prod_{r=1}^{\theta}{\prod_{i<j}{\frac{q^{n_{i}+r+\theta(j-i-1)}-1}{q^{n_{i}+r+\theta(j-i-1)}-q^{n_{j}}}}}$
$\displaystyle\qquad{}=\prod_{r=1}^{\theta}{\prod_{i<j}{\frac{q^{n_{i}+r+\theta(N-i-1)}-q^{\theta(N-j)}}{q^{n_{i}+r+\theta(N-i-1)}-q^{n_{j}+\theta(N-j)}}}}=\prod_{r=1}^{\theta}{\prod_{i<j}{\frac{z_{i}q^{r-\theta}-q^{\theta(N-j)}}{z_{i}q^{r-\theta}-z_{j}}}}$
$\displaystyle\qquad{}=\prod_{r=1}^{\theta}{\prod_{i<j}{\frac{z_{i}-q^{\theta(N-j+1)-r}}{z_{i}-z_{j}q^{\theta-r}}}}.$
(4.21)

The denominator of (4.21) is $\prod\limits_{\begin{subarray}{c}1\leq i<j\leq
m\\\ 0\leq k<\theta\end{subarray}}{(z_{i}-q^{k}z_{j})}$, and the numerator is
$\prod\limits_{r=1}^{m}{\prod\limits_{j=\theta(N-m+1)}^{\theta(N-i+1)-1}{(z_{r}-q^{j-\theta})}}$.
Then (4.17) equals

$\displaystyle\frac{\prod\limits_{r=1}^{m}{\prod\limits_{j=\theta(N-m+1)}^{\theta(N-r+1)-1}{\big{(}z_{r}-q^{j-\theta}\big{)}}}}{\prod\limits_{\begin{subarray}{c}1\leq
i<j\leq m\\\ 0\leq k<\theta\end{subarray}}{(z_{i}-q^{k}z_{j})}}.$ (4.22)

Therefore after the change of variables (4.12), and using (4.15), (4.16),
(4.17), (4.20), (4.22) and the identities after (4.12), equation (4.11) yields
the desired (4.1) for any $x_{s}=q^{n_{s}+\theta(N-s)}$, $1\leq s\leq m$, such
that $n_{1}>\dots>n_{m}>\theta(N+m)$, provided the equality

$\displaystyle
c(N,m,\theta)=\theta^{2}\dbinom{m+1}{3}-\left\\{N\theta^{2}-\dbinom{\theta+1}{2}\right\\}\dbinom{m}{2}$

holds, where $c(N,m,\theta)$ is defined in (4.19). This is an easy exercise.

We have just proved that the statement of Theorem 4.1 holds for all
$x_{s}=q^{n_{s}+\theta(N-s)}$, for all $n_{1}>n_{2}>\dots>n_{m}>\theta(N+m)$.
Since both sides of the equality (4.1) are evidently rational functions in
$x_{1},\dots,x_{m}$, an easy algebro-geometric argument shows the equality
holds for all $x_{1},\dots,x_{m}\in\mathbb{C}$, as desired.

We still have to extend the theorem to all signatures. Let us prove (4.1) for
an arbitrary $\lambda\in\mathbb{GT}_{N}$. If $\lambda\in\mathbb{GT}^{+}_{N}$,
then we are done. Otherwise, choose any $p\in\mathbb{N}$ large enough so that
$\widetilde{\lambda}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}(\lambda_{1}+p,\lambda_{2}+p,\dots,\lambda_{N}+p)\in\mathbb{GT}^{+}_{N}$.
By homogeneity of Macdonald polynomials, we have

$\displaystyle
P_{\widetilde{\lambda}}\big{(}x_{1},\dots,x_{m};N,q,q^{\theta}\big{)}=P_{\lambda}\big{(}x_{1},\dots,x_{m};N,q,q^{\theta}\big{)}(x_{1}\cdots
x_{m})^{p}q^{-\theta pNm}q^{\theta pm(m+1)/2}$

and

$\displaystyle
P_{\widetilde{\lambda}}\big{(}x_{i};N,q,q^{\theta}\big{)}=P_{\lambda}\big{(}x_{i};N,q,q^{\theta}\big{)}x_{i}^{p}q^{-\theta
pN}q^{\theta p}\qquad\forall\,1\leq i\leq m$ $\displaystyle\Rightarrow\
T_{q,x_{i}}^{(i-1)\theta+\tau_{i}^{+}-\tau_{i}^{-}}P_{\widetilde{\lambda}}\big{(}x_{i};N,q,q^{\theta}\big{)}=q^{-\theta
pN}q^{p(i\theta+\tau_{i}^{+}-\tau_{i}^{-})}x_{i}^{p}$
$\displaystyle\hphantom{\Rightarrow\
T_{q,x_{i}}^{(i-1)\theta+\tau_{i}^{+}-\tau_{i}^{-}}P_{\widetilde{\lambda}}\big{(}x_{i};N,q,q^{\theta}\big{)}=}{}\times
T_{q,x_{i}}^{(i-1)\theta+\tau_{i}^{+}-\tau_{i}^{-}}P_{\lambda}\big{(}x_{i};N,q,q^{\theta}\big{)}\qquad\forall\,1\leq
i\leq m.$

We know that (4.1) holds for $\widetilde{\lambda}$; from the expressions
above, it holds also for $\lambda$ provided that (the powers of $q$ match)

$\displaystyle-\theta pNm+\frac{\theta pm(m+1)}{2}=\sum_{i=1}^{m}{(-\theta
pN+p(i\theta+\tau_{i}^{+}-\tau_{i}^{-}))}.$

The latter equation is easy to check, and the proof of Theorem 4.1 is
therefore finished.

##  5 Asymptotics of Macdonald characters

In the remaining of the paper, starting here, we denote the Macdonald
polynomial $P_{\lambda}(x_{1},\dots,x_{N};$ $q,t)$ simply by
$P_{\lambda}(x_{1},\dots,x_{N})$.

In this section, assume that $\theta\in\mathbb{N}$ and let $t=q^{\theta}$. We
study the asymptotics of (certain normalization of) Macdonald characters of a
fixed number $m$ of variables, as the rank $N$ tends to infinity and the
signatures $\lambda(N)$ stabilize in certain way that we define next.

######  Definition 5.1.

Let

$\displaystyle\mathcal{N}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\\{\nu=(\nu_{1},\nu_{2},\dots)\in\mathbb{Z}^{\infty}\colon\nu_{1}\leq\nu_{2}\leq\dots\\}$

be the set of weakly increasing integer sequences. We say that the sequence
$\\{\lambda(N)\\}_{N\geq 1}$ of signatures, $\lambda(N)\in\mathbb{GT}_{N}$,
stabilizes to $\nu\in\mathcal{N}$ if we have the following limits

$\displaystyle\lim_{N\rightarrow\infty}{\lambda_{N-i+1}(N)}=\nu_{i},\qquad\forall\,i=1,2,\dots.$

In other words $\\{\lambda(N)\\}_{N\geq 1}$ stabilizes to $\nu$ if there
exists a sequence of integers $0<N_{1}<N_{2}<\cdots$ such that
$\lambda_{N-i+1}(N)=\nu_{i}$, $\forall\,N>N_{i}$, $i=1,2,\dots$.

###  5.1 Asymptotics of Macdonald characters of one variable

######  Theorem 5.2.

Let $t=q^{\theta}$, $\theta\in\mathbb{N}$, and $\\{\lambda(N)\\}_{N\geq 1}$,
$\lambda(N)\in\mathbb{GT}_{N}$, be a sequence of signatures that stabilizes to
$\nu\in\mathcal{N}$. Then

$\displaystyle\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}}=\Phi^{\nu}(x;q,t),$
(5.1)

where

$\displaystyle\Phi^{\nu}(x;q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{(q;q)_{\infty}}{(xq;q)_{\infty}}\frac{\ln{q}}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{\infty}{\frac{\big{(}q^{-z+\nu_{i}}t^{i};q\big{)}_{\infty}}{\big{(}q^{-z+\nu_{i}}t^{i-1};q\big{)}_{\infty}}}{\rm
d}z},$ (5.2)

and the contour $\mathcal{C}^{+}$ is the infinite positive contour consisting
of
$\big{[}{-}M+\frac{\pi\sqrt{-1}}{\ln{q}},-M-\frac{\pi\sqrt{-1}}{\ln{q}}\big{]}$
and the lines
$\big{[}{-}M+\frac{\pi\sqrt{-1}}{\ln{q}},+\infty+\frac{\pi\sqrt{-1}}{\ln{q}}\big{)}$,
$\big{[}{-}M-\frac{\pi\sqrt{-1}}{\ln{q}},+\infty-\frac{\pi\sqrt{-1}}{\ln{q}}\big{)}$
for an arbitrary $M>\max\\{0,-\nu_{1}\\}$, see Fig. 2. The function
$\Phi^{\nu}(x;q,t)$ is defined by the formula above in the domain

$\displaystyle\mathcal{U}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\bigcap_{k\geq 1}{\big{\\{}x\neq q^{-k}\big{\\}}}\cap{\\{x\neq
0\\}},$

and admits an analytic continuation to the domain $\\{x\neq
0\\}=\mathbb{C}\setminus\\{0\\}$. Moreover if $\nu_{1}\geq 0$, the function
$\Phi^{\nu}(x;q,t)$ can be analytically continued to $\mathbb{C}$.

The convergence (5.1) is uniform on compact subsets of