# Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal
Polynomials

Maria das Neves Rebocho1, Nicholas Witte2,3 1 Departamento de Matemática and
CMA-UBI, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama,
6201-001 Covilhã, Portugal  2 Department of Computer Science, Texas Tech
University, Lubbock TX 79409-3104, USA  3 School of Mathematics and
Statistics, Victoria University of Wellington,Wellington 6012, New Zealand
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Our work studies sequences of orthogonal polynomials
$\\{P_{n}(x)\\}_{n=0}^{\infty}$ of the Laguerre-Hahn class, whose Stieltjes
functions satisfy a Riccati type differential equation with polynomial
coefficients, are subject to a deformation parameter $t$. We derive systems of
differential equations and give Lax pairs, yielding non-linear differential
equations in $t$ for the recurrence relation coefficients and Lax matrices of
the orthogonal polynomials. A specialisation to a non semi-classical case
obtained via a Möbius transformation of a Stieltjes function related to a
modified Jacobi weight is studied in detail, showing this system is governed
by a differential equation of the Painlevé type P${}_{\textrm{VI}}$. The
particular case of P${}_{\textrm{VI}}$ arising here has the same four
parameters as the solution found by Magnus [31] but differs in the boundary
conditions.

###### Key words and phrases:

Orthogonal polynomials; matrix Sylvester equations; modified weight; semi-
classical weight; Painlevé equations

###### 2000 Mathematics Subject Classification:

33C47, 39A99

##  1\. Introduction

###  1.1. Integrable Probability and Mathematical Physics

Integrability, in one form or another, is central in soluble statistical
mechanics models [3],[54], the one-dimensional impenetrable Bose gas [25],
[15], the enumerative combinatorics of the longest increasing subsequences of
random permutations [2], the Gaussian, Laguerre and Jacobi unitary ensembles
of random matrix theory [47], [50], [49], non-equilibrium particle hopping
models such as the Asymmetric Simple Exclusion Process [48] and models in the
Kardar-Parisi-Zhang universality class [38], [45], and the modelling of
L-functions in analytic number theory [26], [10]. In particular, for the
ensembles of random matrices mentioned above, which have joint eigenvalue
probability distribution function given in terms of certain modifications of
the classical Gaussian, Laguerre or Jacobi weights, it is well-known that
certain statistical quantities related to the eigenspectrum can be expressed
in terms of semi-classical polynomials or of the corresponding Hankel
determinants, and are, in turn, characterised by certain special non-linear
differential equations (see [14, Chap. 8], [4, 16] and their references).
Additional connections with integrable Hamiltonians, Bäcklund transformations,
Lax pairs, Toda lattices and Painlevé equations are well-known in the
literature and we refer the interested reader to the reference list of [31,
Sec. 1].

Integrability arises in these models and applications either explicitly as a
semi-classical weight $w(x)$ (here a deformation of one of the classical
weights in the Askey Tableaux [27] according to a precise definition given
below), or equivalently by $w(x)$ being characterised as the solution of a
Pearson equation - the first order homogeneous differential equation

(1) $A(x)\partial_{x}w(x)=C(x)w(x)\,,$

with $A,C$ being co-prime polynomials. Alternatively, this characterisation
may be implied by the fact that the Stieltjes transform of the weight $w(x)$
with support $I$

(2) $f(z):=\int_{I}dx\;\frac{w(x)}{z-x},\qquad z\in\mathbb{C}\backslash I\,,$

satisfies the inhomogeneous form of the Pearson equation,

(3) $A(x)\partial_{x}f(x)=C(x)f(x)+D(x).$

This latter form can also be taken as the defining characteristic [43].

However it is known within approximation theory that the semi-classical class
is subsumed within a broader class - known as the Laguerre-Hahn class - where
the defining equation is the Ricatti generalisation of (3), namely [29]

(4) $A(x)\partial_{x}f(x)=B(x)f^{2}(x)+C(x)f(x)+D(x)\,.$

When the leading coefficient $B(x)$ vanishes we are back to the semi-classical
class. The class of Laguerre-Hahn orthogonal polynomials - the ones
corresponding to Stieltjes functions satisfying (4), contains not only the
semi-classical polynomials but also their standard modifications [1, 11, 21,
22, 23, 40, 41, 42, 44, 55]. In the viewpoint of orthogonal polynomial theory,
special emphasis has been given to the problem of classification, that is,
describing the recurrence relation coefficients of the orthogonal polynomial
system, given the polynomials $A,B,C,D$ in (4) (see, amongst many others, [12,
13, 30]).

In the present paper we take Stieltjes functions satisfying (4) subject to a
dependence on an additional variable, say, a deformation parameter $t$. In
general terms, the main goal is to analyse the recurrence relation
coefficients of the corresponding orthogonal polynomials under such a
dependence. These kind of problems have been analysed from different
perspectives and we refer the interested reader to [7] and its references; see
also [37], where the equivalence between the equations of the infinite Toda
(Volterra) lattice and the Riccati differential equation for the Stieltjes
function is established .

The framework of our study relies on the matrix differential systems satisfied
by the polynomials. Here we adopt, to the Laguerre-Hahn setting, a technique
that can be traced back to the work of E. Laguerre in [28] \- the so-called
Laguerre method. Essentially, this method gives a mechanical way to obtain
differential equations satisfied by orthogonal polynomials provided one knows
the linear differential equation for the Stieltjes function. It has been
revisited in recent literature on semi-classical polynomials, see, for
instance, [5] and [31, Sec. 3], amongst many others. In the present paper, by
using a similar algorithm to [28], we deduce that the systems of orthogonal
polynomials corresponding to (4) satisfy difference-differential equations
(see Theorem 2), which are then coupled into the form of matrix Sylvester
equations for $Y_{n}(x,t)$ defined in (10) below

$\partial_{x}Y_{n}=\mathcal{A}_{n}Y_{n}-Y_{n}\mathcal{C}\,,\quad n\geq 0\,,$

with $\mathcal{A}_{n}$ and $\mathcal{C}$ matrices with rational entries.
Similarly, one has a differential equation for $f(x,t)$ with respect to the
deformation variable $t$, given as

$\widehat{A}\partial_{t}f=\widehat{B}f^{2}+\widehat{C}f+\widehat{D}\,,$

and a system of Sylvester equations

$\partial_{t}Y_{n}=\mathcal{B}_{n}Y_{n}-Y_{n}\widehat{\mathcal{C}}\,,\quad
n\geq 0\,,$

where $\mathcal{B}_{n}$ and $\widehat{\mathcal{C}}$ are matrices with rational
entries in the spectral variable $x$. The consistency of the two differential
equations for $Y_{n}$ combined with the consistency of the two differential
equations for the Stieltjes function yields (under the assumption that $f$ is
not algebraic) the compatibility conditions deduced in Theorem 5, which are
comparable to the Schlesinger equations in the semi-classical setting [9].

In Section 3, we specialize the results in Section 2 to a sequence of
Laguerre-Hahn polynomials. It is our aim to show by way of that specific
example that a natural deformation of an appropriate Laguerre-Hahn class
system directly related to the Jacobi weight does lead to the sixth Painlevé
equation, in fact to a solution of PVI with an identical parameter set to that
of the deformed (semi-classical) Jacobi weight and only differing in the
boundary conditions. Concluding remarks are given in Section 4.

###  1.2. Main Results

To present the main results of the paper some facts about the sixth Painlevé
equation and its Hamiltonian formulation are required. Painlevé VI - usually
denoted by P${}_{\textrm{VI}}$ \- is commonly presented as the sixth equation
in the Painlevé list [52], consisting of the six non-linear differential
equations having the special property that all movable singularities are
poles.

P${}_{\textrm{VI}}$ conventionally refers to the four-parameter second-order
nonlinear differential equation

(5)
$y^{\prime\prime}=\frac{1}{2}\left(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-t}\right)(y^{\prime})^{2}-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{y-t}\right)y^{\prime}\\\
+\frac{y(y-1)(y-t)}{t^{2}(t-1)^{2}}\left(\delta_{1}+\frac{\delta_{2}t}{y^{2}}+\frac{\delta_{3}(t-1)}{(y-1)^{2}}+\frac{\delta_{4}t(t-1)}{(y-t)^{2}}\right)\,.$

Here, and in the sequel, the dash denotes derivative with respect to $t$. In
the Hamiltonian formulation of P${}_{\textrm{VI}}$ the Hamiltonian
$H(q,p,t;{v_{k}})$, where ${v_{k}}$ are parameters, is given by Okamoto [35]
as

(6)
$t(t-1)H=q(q-1)(q-t)p^{2}-\left[(v_{3}+v_{4})(q-1)(q-t)+(v_{3}-v_{4})q(q-t)\right.\\\
\left.-(v_{1}+v_{2})q(q-1)\right]p+(v_{3}-v_{1})(v_{3}-v_{2})(q-t)\,.$

The Okamoto theory in the Hamiltonian approach to P${}_{\textrm{VI}}$ gives
that, after eliminating $p$ in the Hamilton equations

(7) $q^{\prime}=\frac{\partial H}{\partial p}\,,\;\;p^{\prime}=-\frac{\partial
H}{\partial q}\,,$

then $q$ satisfies (5) with parameters

$\delta_{1}=\frac{1}{2}(v_{1}-v_{2})^{2}\,,\;\delta_{2}=-\frac{1}{2}(v_{3}+v_{4})^{2}\,,\;\delta_{3}=\frac{1}{2}(v_{3}-v_{4})^{2}\,,\;\delta_{4}=\frac{1}{2}(1-(1-v_{1}-v_{2})^{2})\,.$

Let us now introduce some notations (the complete data is given in Sections 2
and 3). We start by considering the transformation giving a new Stieltjes
function $\tilde{f}$ (see (60))

(8) $\tilde{f}(x)=x-\beta_{0}-\frac{w_{0}}{f(x)}\,,$

where $w_{0}$ is the zero moment of the modified Jacobi weight and

(9) $w(x,t)=x^{\alpha}(1-x)^{\beta}(x-t)^{\mu}\,,$

in terms of the function $f(x)$ is the Stieltjes transform of $w(x,t)$, and
$\beta_{0}$ is the recurrence coefficient of the sequence of monic orthogonal
polynomials corresponding to $w(x,t)$.

The sequences of orthogonal polynomials corresponding to the weight (9) were
the subject of analysis in several works, showing relations with Painlevé
equations in the context of random matrix theory and linear isomonodromy
problems. We refer the interested reader to [19, 31]. In the present paper we
are mainly concerned with the analysis of the orthogonal polynomials related
to the Stieljtes function $\tilde{f}(x)$ defined by (8). Essentially, we will
deduce differential systems for these polynomials, and analyse their
dependence on the deformation parameter $t$ from the weight (9). At this stage
it is important to emphasise that, to the best of our knowledge, a closed
formula for the weight function $\tilde{w}(x,t)$ corresponding to
$\tilde{f}(x)$ is not known.

We shall denote by $\\{\tilde{P}_{n}\\}_{n\geq 0}$ the sequence of monic
orthogonal polynomials corresponding to $\tilde{f}$, and
$\tilde{\beta}_{n},\tilde{\gamma}_{n}$ its recurrence relation coefficients.
The matrix $\tilde{Y}_{n}$ given by

(10) $\tilde{Y}_{n}=\left[\begin{matrix}\tilde{P}_{n+1}&\tilde{P}_{n}^{(1)}\\\
\tilde{P}_{n}&\tilde{P}_{n-1}^{(1)}\\\ \end{matrix}\right]\,,$

where $\\{\tilde{P}_{n}^{(1)}\\}$ is the sequence of monic associated
polynomials (see Section 2), satisfies the matrix Sylvester equation

$\partial_{x}{\tilde{Y}}_{n}={\tilde{\mathcal{A}}}_{n}\tilde{Y}_{n}-\tilde{Y}_{n}{\tilde{\mathcal{C}}}\,,$

with

${\tilde{\mathcal{A}}}_{n}=\frac{1}{\tilde{A}}\begin{bmatrix}\tilde{l}_{n}&\tilde{{\Theta}}_{n}\\\
-\tilde{\Theta}_{n-1}/{\tilde{\gamma}}_{n}&\;\;-\tilde{l}_{n}\end{bmatrix},\;\;{\tilde{\mathcal{C}}}=\frac{1}{\tilde{A}}\begin{bmatrix}\tilde{C}/2&-\tilde{D}/\tilde{w}_{0}\\\
\tilde{w}_{0}\tilde{B}&-\tilde{C}/2\end{bmatrix},\;\;$

with $\tilde{A}=x(x-1)(x-t)$ and $\tilde{B},\tilde{C},\tilde{D}$ the remaining
polynomial coefficients of the $x$-Riccati equation for $\tilde{f}$ (see
(65)-(67)), and where $\tilde{w}_{0}$ is the zero moment in the asymptotic
expansion around $\infty$ of $\tilde{f}$. The matrix $\tilde{\mathcal{A}}_{n}$
is parameteriesd in terms of the polynomials
$\tilde{l}_{n},\tilde{\Theta}_{n}$, which now have degrees two and one,
respectively, by virtue of the bounds
$\deg(\tilde{B})=1,\deg(\tilde{C})=2,\deg(\tilde{D})=1$ (see Lemma 2),

$\tilde{l}_{n}(x)=\frac{(\nu_{n}-1)}{2}x^{2}+\tilde{l}_{n,1}x+\tilde{l}_{n,0}\,,\;\;\frac{\tilde{\Theta}_{n}(x)}{\tilde{\gamma}_{n+1}}=-\nu_{n}x-\vartheta_{n}\,,$

with

$\nu_{n}=2n+5+\alpha+\beta+\mu\,,\;\;\;\vartheta_{n}=\nu_{n}(\tilde{\beta}_{n+1}-1-t)+2\sum_{j=0}^{n}\tilde{\beta}_{j}+\tilde{\beta}_{n+1}+\mu
t+\beta+2\beta_{0}\,.$

As we shall see below in Section 3, the zero of the function
$\tilde{\Theta}_{n}$, i.e, the unique zero of the $(1,2)$ component of the
matrix $\tilde{\mathcal{A}}_{n}$ from the corresponding Sylvester equations,
that is, $q:=-\vartheta_{n}/\nu_{n}$, will be the Painlevé transcendent.

Let us also define $\varphi$ \- a polynomial in the variable $x$ \- in terms
of the polynomials $\tilde{A},\tilde{B},\tilde{C},\tilde{D}$ and
$\tilde{w}_{0}$, as

$\varphi=\frac{\tilde{D}}{\tilde{w}_{0}}\left(\tilde{A}+\tilde{w}_{0}\tilde{B}\right)-\frac{\tilde{C}^{2}}{4}\,.$

The evaluation of $\varphi$ at the zeroes of $\tilde{A}$ will have a
fundamental role in the derivation of P${}_{\textrm{VI}}$. Our main result can
now be stated as follows.

######  Theorem 1.

Under the above notations, let $v_{1},v_{2},v_{3},v_{4}$ be given in terms of
the function $\varphi$ as

(11)
$(v_{1}-v_{2})^{2}=1\,,\;(1-v_{1}-v_{2})^{2}=-4\frac{\varphi(t)}{t^{2}(t-1)^{2}}\,,\;(v_{3}+v_{4})^{2}=-4\frac{\varphi(0)}{t^{2}}\,,\;(v_{3}-v_{4})^{2}=-4\frac{\varphi(1)}{(t-1)^{2}}\,.\;$

In terms of the coefficients ($\tilde{\beta}_{n},\tilde{\gamma}_{n}$) in the
three term recurrence relation satisfied by the sequence of monic orthogonal
polynomials $\\{\tilde{P}_{n}\\}_{n\geq 0}$, let $\nu_{n}$ and $\vartheta_{n}$
be defined by the above equations,

$\displaystyle\nu_{n}$ $\displaystyle=2n+5+\alpha+\beta+\mu\,,$
$\displaystyle\vartheta_{n}$
$\displaystyle=\nu_{n}(\tilde{\beta}_{n+1}-1-t)+2\sum_{j=0}^{n}\tilde{\beta}_{j}+\tilde{\beta}_{n+1}+\mu
t+\beta+2\beta_{0}\,.$

The Hamilton equations are satisfied by

(12) $q=-\vartheta_{n}/\nu_{n}\,,$

and $p$, which is given by

(13)
$p=\frac{\nu_{n}q^{2}+2q\tilde{l}_{n,1}-q+2\tilde{l}_{n,0}}{2q(q-1)(q-t)}+\frac{v_{3}+v_{4}}{2q}+\frac{v_{3}-v_{4}}{2(q-1)}-\frac{v_{1}+v_{2}}{2(q-t)}\,.$

######  Remark 1.

It turns out that the quantities $\varphi(0),\varphi(1),\varphi(t)$ are such
that

$\;\;\frac{\varphi(0)}{t^{2}}=-\tfrac{1}{4}\alpha^{2}\,,\;\;\frac{\varphi(1)}{(t-1)^{2}}=-\tfrac{1}{4}\beta^{2}\,,\;\;\frac{\varphi(t)}{t^{2}(t-1)^{2}}=-\tfrac{1}{4}\mu^{2}\,,$

and hence, $q$ satisfies (5) with parameters
$(\delta_{1},\delta_{2},\delta_{3},\delta_{4})=(\tfrac{1}{2},-\tfrac{1}{2}\alpha^{2},\tfrac{1}{2}\beta^{2},\tfrac{1}{2}(1-\mu^{2}))$.

##  2\. Lax Pairs for Laguerre-Hahn Orthogonal Polynomials

###  2.1. The Laguerre Method

In this subsection we shall consider general weight functions (possibly
deformed and complex - however this is not relevant for our current argument
and is notationally suppressed) supported on an interval $I\subset\mathbb{R}$,
for which all the moments

$w_{n}=\int_{I}dx\,x^{n}w(x)\,,\quad n\geq 0\,,$

exist and satisfy the regularity condition that the Hankel determinants

$\Delta_{n}:=\det[w_{j+k-2}]_{j,k=1,\dots,n}\,,\quad\Delta_{0}:=1\,,$

are such that

$\Delta_{n}\neq 0,\;\;\;n\geq 0\,.$

A central object is the Stieltjes transform of the weight, in the most general
situation taken as the formal series

(14) $f(x)=f(w,x):=\displaystyle\sum_{n=0}^{+\infty}\frac{w_{n}}{x^{n+1}}\,.$

The corresponding sequences of orthogonal polynomials, which we will take
monic, $\\{P_{n}(x)=x^{n}+\textrm{lower degree terms}\\}_{n\geq 0}$, defined
via the orthogonality relation

(15) $\int_{I}dx\,P_{n}(x)P_{m}(x)w(x)=h_{n}\delta_{m,n}\,,\quad n,m\geq
0\,,\;\;h_{n}\neq 0\,,\quad
h_{n}=\displaystyle\frac{\Delta_{n+1}}{\Delta_{n}}\,,\quad$

are fully characterised by the three-term recurrence relation [46]

(16) $xP_{n}(x)=P_{n+1}(x)+\beta_{n}P_{n}(x)+\gamma_{n}P_{n-1}(x)\,,\quad
n=1,2,\dots\,,$

with the initial conditions $P_{0}(x)=1,\;P_{1}(x)=x-\beta_{0}$. The numbers
$\beta_{n},\gamma_{n},$ commonly called the recurrence relation coefficients,
can be expressed as

(17) $\beta_{n}=\frac{1}{h_{n}}\int_{I}dx\,xP_{n}^{2}(x)w(x)\,,\;n\geq
0\,,\;\;\;\gamma_{n}=\frac{1}{h_{n-1}}\int_{I}dx\,P^{2}_{n}(x)w(x)\,,\;\;n\geq
1\,.$

Thus, the norms $h_{n}$ in (15) satisfy

(18) $h_{0}=w_{0}\,,\;\;\gamma_{n}=h_{n}/h_{n-1}\,,\;n\geq 1\,,$

hence, the recurrence coefficients $\gamma_{n}$ are given in terms of the
Hankel determinants as

$\gamma_{n}=\frac{\Delta_{n-1}\Delta_{n+1}}{\Delta^{2}_{n}}\,,\quad n\geq
1\,.$

Given the sequence of monic polynomial $\\{P_{n}\\}_{n\geq 0}$, corresponding
to a weight $w(x)$, one defines the sequence of monic associated polynomials,
denoted by $\\{P_{n}^{(1)}\\}_{n\geq 0}$, as [8, 46]

(19)
$P_{n}^{(1)}(x):=\frac{1}{w_{0}}\int_{0}^{1}dy\,\frac{P_{n+1}(x)-P_{n+1}(y)}{x-y}w(y)\,,\quad
n=0,1,\dots\,.$

As an immediate consequence of (16), it follows that $\\{P_{n}^{(1)}\\}_{n\geq
0}$ satisfies the three-term recurrence relation

(20)
$xP^{(1)}_{n-1}(x)=P^{(1)}_{n}(x)+\beta_{n}P^{(1)}_{n-1}(x)+\gamma_{n}P^{(1)}_{n-2}(x)\,,\quad
n=1,2,\dots\,,$

with initial conditions $P^{(1)}_{-1}(x)=0,\;P^{(1)}_{0}(x)=1$. The following
relation is well-known [46]:

(21) $P_{n}^{(1)}P_{n}-P_{n+1}P_{n-1}^{(1)}=\prod_{k=1}^{n}\gamma_{k}\,,\quad
n\geq 1\,.$

The functions $P_{n},P_{n}^{(1)}$ and the Stieltjes function $f$ are related
via [46]

$P_{n+1}(x)f(x)-w_{0}P_{n}^{(1)}(x)=\mathcal{O}(x^{-n-1})\,,\;\;\;n\geq 0\,.$

This formula is commonly referred to as the Hermite-Padé formula, and will be
written as

(22) $P_{n+1}(x)f(x)-w_{0}P_{n}^{(1)}(x)=:\varepsilon_{n+1}(x)\,,\;\;\;n\geq
0\,.$

Note that the functions $\varepsilon_{n}(x)$ satisfy the same recurrence
relation (20) with $\varepsilon_{0}(x)=f(x)$.

The first main result of this section is given as follows.

######  Theorem 2.

Let $f$ be a Stieltjes function given by (14), let $\\{P_{n}\\}_{n\geq 0}$ and
$\\{P_{n}^{(1)}\\}_{n\geq 0}$ be the corresponding sequences of monic
orthogonal polynomials and associated polynomials, respectively. If $f$
satisfies

(23) $A\partial_{x}f=Bf^{2}+Cf+D\,,$

with $A,B,C,D$ co-prime polynomials, then the difference-differential
equations hold, for all $n\geq 0$,

(24) $\displaystyle
A\partial_{x}P_{n+1}=(l_{n}-C/2)P_{n+1}-w_{0}BP_{n}^{(1)}+\Theta_{n}P_{n}\,,$
(25) $\displaystyle
A\partial_{x}P^{(1)}_{n}=\frac{D}{w_{0}}P_{n+1}+(l_{n}+C/2)P^{(1)}_{n}+\Theta_{n}P^{(1)}_{n-1}\,,$

with $l_{n},\Theta_{n}$ polynomials of $x$ with degrees given by

$\deg(l_{n})=\max\\{\max\\{\deg(A),\deg(B)\\}-1,\deg(C)\\}\,,\quad\deg(\Theta_{n})=\deg(l_{n})-1,$

and satisfying the initial conditions

(26)
$\frac{\Theta_{-1}}{w_{0}}=D\,,\;\Theta_{0}=A+(x-\beta_{0})(\frac{C}{2}-l_{0})+w_{0}B\,,\;l_{-1}=\frac{C}{2}\,,\;l_{0}=-\frac{C}{2}-(x-\beta_{0})\frac{D}{w_{0}}\,.$

Here, $\beta_{0}$ is the recurrence relation coefficient of
$\\{P_{n}\\}_{n\geq 0}.$

###### Proof.

Using (22) in (23) we obtain, after standard computations,

(27)
$A\left(\partial_{x}P_{n}^{(1)}P_{n+1}-\partial_{x}P_{n+1}P^{(1)}_{n}\right)-Bw_{0}(P^{(1)}_{n})^{2}-CP^{(1)}_{n}P_{n+1}-\frac{D}{w_{0}}P_{n+1}^{2}\\\
=-\frac{A}{w_{0}}\left(\partial_{x}\varepsilon_{n+1}P_{n+1}-\partial_{x}P_{n+1}\varepsilon_{n+1}\right)+2BP_{n}^{(1)}\varepsilon_{n+1}+\frac{C}{w_{0}}\varepsilon_{n+1}P_{n+1}\,.$

Thus, the left hand side of the previous equation is a polynomial, say
$\check{\Theta}_{n}$,

(28)
$A\left(\partial_{x}P_{n}^{(1)}P_{n+1}-\partial_{x}P_{n+1}P^{(1)}_{n}\right)-Bw_{0}(P^{(1)}_{n})^{2}-CP^{(1)}_{n}P_{n+1}-\frac{D}{w_{0}}P_{n+1}^{2}=\check{\Theta}_{n}\,,$

and its degree is bounded by a constant, as

$\check{\Theta}_{n}=-\frac{A}{w_{0}}\left(\partial_{x}\varepsilon_{n+1}P_{n+1}-\partial_{x}P_{n+1}\varepsilon_{n+1}\right)+2BP_{n}^{(1)}\varepsilon_{n+1}+\frac{C}{w_{0}}\varepsilon_{n+1}P_{n+1}$

is bounded by a power less or equal than
$\max\\{\deg(A)-2,\deg(B)-2,\deg(C)-1\\}$.

From (21), replace $\check{\Theta}_{n}$ in (28) by
$\displaystyle\frac{P_{n}^{(1)}P_{n}-P_{n+1}P_{n-1}^{(1)}}{\prod_{k=1}^{n}\gamma_{k}}\check{\Theta}_{n}$,
and after rearranging and using the notation
$\Theta_{n}=-\check{\Theta}_{n}/(\prod_{k=1}^{n}\gamma_{k})$, we get

(29)
$P_{n}^{(1)}\left[A\partial_{x}P_{n+1}+\frac{C}{2}P_{n+1}-\Theta_{n}P_{n}+w_{0}BP_{n}^{(1)}\right]\\\
=P_{n+1}\left[A\partial_{x}P_{n}^{(1)}-\frac{C}{2}P_{n}^{(1)}-\Theta_{n}P_{n-1}^{(1)}-\frac{D}{w_{0}}BP_{n+1}\right]\,.$

As $P_{n}^{(1)}$ and $P_{n+1}$ do not have common zeroes (this property
follows, for instance, from (21)), then both hand sides of (29) must therefore
have the form $l_{n}P_{n}^{(1)}P_{n+1}$, where $l_{n}$ is a new auxiliary
polynomial of bounded degree, thus we have

(30) $\displaystyle
A\partial_{x}P_{n+1}+\tfrac{1}{2}CP_{n+1}-\Theta_{n}P_{n}+w_{0}BP_{n}^{(1)}=l_{n}P_{n+1}\,,$
(31) $\displaystyle
A\partial_{x}P_{n}^{(1)}-\tfrac{1}{2}CP_{n}^{(1)}-\Theta_{n}P_{n-1}^{(1)}-\frac{D}{w_{0}}BP_{n+1}=l_{n}P_{n}^{(1)}\,,$

hence we get (24)-(25).

Note that, using again (21) in (30)-(31), we deduce the polynomial $l_{n}$ is
given by

(32)
$\left(\prod_{k=1}^{n}\gamma_{k}\right)l_{n}=A\left(\partial_{x}P_{n}^{(1)}P_{n}-\partial_{x}P_{n+1}P_{n-1}^{(1)}\right)-\tfrac{1}{2}C\left(P_{n}^{(1)}P_{n}+P_{n+1}P_{n-1}^{(1)}\right)\\\
-\frac{D}{w_{0}}P_{n+1}P_{n}-w_{0}BP_{n}^{(1)}P_{n-1}^{(1)}\,.$

This concludes the proof. ∎

Using the three-term recurrence relations of $\\{P_{n}\\}_{n\geq 0}$ and
$\\{P^{(1)}_{n}\\}_{n\geq 0}$, (16) and (20), in (24)-(25), two further
equations that give $A\partial_{x}P_{n+1}$ and $A\partial_{x}P^{(1)}_{n}$ in
terms of $P_{n},P_{n-1},P^{(1)}_{n-1}$ and $P^{(1)}_{n-2}$ are obtained, and
such four equations yield a system that can be coupled into the form of matrix
Sylvester equations, as given in the next theorem.

######  Theorem 3.

(see [6]) Let the notations of the previous theorem hold. Let the Stieltjes
function $f$ given by (14) satisfy the Riccati type differential equation
(23), $A\partial_{x}f=Bf^{2}+Cf+D$. Then, the following matrix Sylvester
equations hold:

(33) $\partial_{x}Y_{n}=\mathcal{A}_{n}Y_{n}-Y_{n}\mathcal{C}\,,\quad n\geq
0\,,\quad Y_{n}=\left[\begin{matrix}P_{n+1}&P_{n}^{(1)}\\\
P_{n}&P_{n-1}^{(1)}\\\ \end{matrix}\right]\ $

where

(34) $\mathcal{A}_{n}=\frac{1}{A}\begin{bmatrix}l_{n}&\Theta_{n}\\\
-\Theta_{n-1}/\gamma_{n}&\;\;l_{n-1}+(x-\beta_{n})\Theta_{n-1}/\gamma_{n}\end{bmatrix}\,,\;\;\mathcal{C}=\frac{1}{A}\begin{bmatrix}C/2&-D/w_{0}\\\
w_{0}B&-C/2\end{bmatrix}\,.$

The differential systems of type (33) will play a relevant role in the sequel.
Before continuing, let us emphasize the differences regarding the semi-
classical case. Let us use the notation

(35) $\breve{\mathcal{A}}_{n}=\begin{bmatrix}l_{n}&\Theta_{n}\\\
-\Theta_{n-1}/\gamma_{n}&\;\;l_{n-1}+(x-\beta_{n})\Theta_{n-1}/\gamma_{n}\end{bmatrix}\,.$

Firstly, we stress that the sequences
$\mathcal{Q}_{n}:=\left[\begin{array}[]{cc}\varepsilon_{n+1}\\\
\varepsilon_{n}\end{array}\right],\;n\geq 0$, with the $\varepsilon_{n}$
defined in (22), corresponding to Stieltjes functions such that (23) holds,
satisfy the differential system [6, Th. 1]

(36)
$A\partial_{x}\mathcal{Q}_{n}=\left(\breve{\mathcal{A}}_{n}+(Bf+C/2)I\right)\mathcal{Q}_{n}\,,\quad
n\geq 0\,,$

where $I$ denotes the $2\times 2$ identity matrix. In the semi-classical case,
taking $B\equiv 0$ in (36) and combining with the spectral differential
equation for the weight, $\partial_{x}\ln w=C/A,$ then a second solution is
obtained,

$A\partial_{x}\left[\begin{array}[]{c}\varepsilon_{n+1}/w\\\
\varepsilon_{n}/w\end{array}\right]=\left(\breve{\mathcal{A}}_{n}-\frac{C}{2}I\right)\left[\begin{array}[]{c}\varepsilon_{n+1}/w\\\
\varepsilon_{n}/w\end{array}\right]\,,$

which gives [31, Sec. 3]

(37) $\partial_{x}\left[\begin{array}[]{cc}P_{n+1}&\varepsilon_{n+1}/w\\\
P_{n}&\varepsilon_{n}/w\end{array}\right]=\frac{1}{A}\left(\breve{\mathcal{A}}_{n}-\frac{C}{2}I\right)\left[\begin{array}[]{cc}P_{n+1}&\varepsilon_{n+1}/w\\\
P_{n}&\varepsilon_{n}/w\end{array}\right]\,.$

Systems of the type (37) are a standard tool when studying semi-classical
orthogonal polynomials (see [4, 19] amongst many others). Let us remark that,
in the non semi-classical Laguerre-Hahn case, in general, explicit formulae
for the weights are not available. Indeed, it is an open problem to get the
general form of the weight given a Riccati differential equation for the
Stieltjes function (see [55]).

###  2.2. Bilinear Recurrence Relations

It is useful to write the recurrence relations (16) and (20) in the matrix
form

(38)
$Y_{n}=\mathcal{M}_{n}Y_{n-1}\,,\quad\mathcal{M}_{n}=\begin{bmatrix}x-\beta_{n}&-\gamma_{n}\\\
1&0\\\ \end{bmatrix}\,,\quad n\geq 1\,,$

with initial conditions $Y_{0}=\begin{bmatrix}x-\beta_{0}&-\gamma_{0}\\\
1&0\\\ \end{bmatrix}$. The matrix $\mathcal{M}_{n}$ is known as the transfer
matrix. The consistency of (33) and (38) yields the equation

(39)
$\partial_{x}\mathcal{M}_{n}={\mathcal{A}}_{n}\mathcal{M}_{n}-\mathcal{M}_{n}{\mathcal{A}}_{n-1}\,,\quad
n\geq 1\,.$

Equation (39) implies two non-trivial equations from entries $(1,1)$ and
$(1,2)$, respectively:

(40) $\displaystyle A+\gamma_{n}\frac{\Theta_{n-2}}{\gamma_{n-1}}-\Theta_{n}$
$\displaystyle=(x-\beta_{n})(l_{n}-l_{n-1})\,,$ (41) $\displaystyle
l_{n}+(x-\beta_{n})\frac{\Theta_{n-1}}{\gamma_{n}}$
$\displaystyle=l_{n-2}+(x-\beta_{n-1})\frac{\Theta_{n-2}}{\gamma_{n-1}}\,.$

After basic computations (41) implies

(42) $l_{n}+l_{n-1}=-(x-\beta_{n})\frac{\Theta_{n-1}}{\gamma_{n}}\,.$

The multiplication of (40) and (42) gives us

$-l_{n}^{2}+l_{n-1}^{2}=A\frac{\Theta_{n-1}}{\gamma_{n}}+\Theta_{n-1}\frac{\Theta_{n-2}}{\gamma_{n-1}}-\Theta_{n}\frac{\Theta_{n-1}}{\gamma_{n}}\,.$

Summing on $n$ and using the initial conditions (26), we get

(43)
$-l_{n}^{2}+\Theta_{n}\frac{\Theta_{n-1}}{\gamma_{n}}=\frac{D}{w_{0}}\left(A+w_{0}B\right)-\frac{C^{2}}{4}+A\sum_{k=1}^{n}\frac{\Theta_{k-1}}{\gamma_{k}}\,.$

Note that equation (42) reads as

$\operatorname{tr}{\mathcal{A}}_{n}=0\,,\quad n\geq 0\,,$

thus it gives us a parameterisation of the matrices ${\mathcal{A}}_{n}$, given
by

${\mathcal{A}}_{n}=\frac{1}{A}\begin{bmatrix}l_{n}&\Theta_{n}\\\
-\Theta_{n-1}/\gamma_{n}&\;\ -l_{n}\end{bmatrix}\,.$

Consequently, recalling the notation (35), equation (43) reads as

(44)
$\det\breve{\mathcal{A}}_{n}=\det\breve{\mathcal{A}}_{0}+A\displaystyle\sum_{k=1}^{n}\frac{\Theta_{k-1}}{\gamma_{k}}\,,\quad
n\geq 1\,,$

with
$\det\breve{\mathcal{A}}_{0}=\displaystyle\frac{D}{w_{0}}\left(A+w_{0}B\right)-C^{2}/4.$

###  2.3. Lax Pairs for Deformed Laguerre-Hahn Orthogonal Polynomials

Let us now assume that the Stieltjes function depends on a deformation
parameter $t$, say $f=f(x,t)$, and it satisfies a Riccati type differential
equation

(45) $\widehat{A}\partial_{t}f=\widehat{B}f^{2}+\widehat{C}f+\widehat{D}\,,$

where $\widehat{A},\widehat{B},\widehat{C},\widehat{D}$ are co-prime
polynomials in $x$. Proceeding analogously to Theorem 2 we get the results
that follows.

######  Theorem 4.

Let $f$ be a Stieltjes function given by (14), let $\\{P_{n}\\}_{n\geq 0}$ be
the corresponding sequence of monic orthogonal polynomials, and let
$\beta_{n},\gamma_{n}$ be its recurrence relation coefficients. Assuming that
$f=f(x,t)$ is such that it satisfies the Riccati-type differential equation
(45), then the matrices $Y_{n}$ given in (33) satisfy the Sylvester equation

(46)
$\partial_{t}Y_{n}=\mathcal{B}_{n}Y_{n}-Y_{n}\widehat{\mathcal{C}}\,,\quad
n\geq 0\,,$

where

(47)
$\mathcal{B}_{n}=\frac{1}{\widehat{A}}\begin{bmatrix}\widehat{l}_{n}&\widehat{\Theta}_{n}\\\
-\widehat{\Theta}_{n-1}/\gamma_{n}&\;\;\widehat{l}_{n-1}+(x-\beta_{n})\widehat{\Theta}_{n-1}/\gamma_{n}\end{bmatrix}\,,\;\;\widehat{\mathcal{C}}=\frac{1}{\hat{A}}\begin{bmatrix}\widehat{C}/2&-\widehat{D}/w_{0}\\\
w_{0}\widehat{B}&-\widehat{C}/2+\partial_{t}(\ln
w_{0})\widehat{A}\end{bmatrix}$

with $\widehat{l}_{n},\widehat{\Theta}_{n}$ polynomials of degrees given by

$\displaystyle\deg(\widehat{l}_{n})=\max\\{\max\\{\deg(\widehat{A})-1,\deg(\widehat{B})-1\\},\deg(\widehat{C})\\}\,,$
$\displaystyle\deg(\widehat{\Theta}_{n})=\max\\{\max\\{\deg(\widehat{A})-1,\deg(\widehat{B})-2\\},\deg(\widehat{C})-1\\}\,,$

satisfying the initial conditions

$\displaystyle\frac{\widehat{\Theta}_{-1}}{w_{0}}=\widehat{D}\,,\widehat{\Theta}_{0}=-\frac{d}{dt}(\beta_{0})\widehat{A}+(x-\beta_{0})(\widehat{C}/2-\widehat{l}_{0})+w_{0}\widehat{B}\,,\;$
$\displaystyle\widehat{l}_{-1}=\widehat{C}/2\,,\;\widehat{l}_{0}=-\widehat{C}/2-(x-\beta_{0})\widehat{D}/w_{0}+\partial_{t}(\ln
w_{0})\widehat{A}\,.$

The matrices $\mathcal{B}_{n}$ satisfy further identities, given in the next
corollary.

######  Corollary 4.1.

The following statements hold:
(a) the transfer matrix $\mathcal{M}_{n}$ from (38) satisfies the equation

(48)
$\partial_{t}\mathcal{M}_{n}=\mathcal{B}_{n}\mathcal{M}_{n}-\mathcal{M}_{n}{\mathcal{B}}_{n-1}\,,\quad
n\geq 1\,;$

(b) the matrix
$\breve{\mathcal{B}}_{n}:=\begin{bmatrix}\widehat{l}_{n}&\widehat{\Theta}_{n}\\\
-\widehat{\Theta}_{n-1}/\gamma_{n}&\;\;\widehat{l}_{n-1}+(x-\beta_{n})\widehat{\Theta}_{n-1}/\gamma_{n}\end{bmatrix}$
satisfies

$\displaystyle\operatorname{tr}\breve{\mathcal{B}}_{n}=\widehat{A}\,\sum_{k=0}^{n}\partial_{t}\ln(\gamma_{k})\,,\quad
n\geq 0\,,$
$\displaystyle\det\breve{\mathcal{B}}_{n}=\det\breve{\mathcal{B}}_{0}+\widehat{A}\,\displaystyle\sum_{k=1}^{n}\left(\partial_{t}\ln(\gamma_{k})\widehat{l}_{k-1}-\partial_{t}(\beta_{k})\frac{\widehat{\Theta}_{k-1}}{\gamma_{k}}\right)\,,\quad
n\geq 1\,,\;$

with $\det\breve{\mathcal{B}}_{0}=\left(-\frac{\displaystyle 1}{\displaystyle
w_{0}}\partial_{t}\beta_{0}\widehat{A}+\widehat{B}\right)\displaystyle\widehat{D}+\tfrac{1}{2}\partial_{t}(\ln
w_{0})\widehat{A}\widehat{C}-\tfrac{1}{4}\left(\widehat{C}\right)^{2}\,.$

###### Proof.

Equation (48) follows from the Lax pair

$\left\\{\begin{array}[]{l}{Y}_{n}=\mathcal{M}_{n}Y_{n-1}\\\
\partial_{t}{Y}_{n}=\mathcal{B}_{n}Y_{n}-Y_{n}\widehat{\mathcal{C}}\,,\quad
n\geq 1\,.\end{array}\right.$

The equations for the trace and determinant of $\breve{\mathcal{B}}_{n}$ are
deduced using similar techniques to [7, Cor. 2]. ∎

####  2.3.1. The Compatibility Conditions

Here we would like to stress that, in the Laguerre-Hahn setting, the
compatibility conditions (indeed, the Schlesinger equations in (49)) follow
from the consistency of the two linear differential equations for $Y_{n}$
combined with the consistency of the two differential equations for the
Stieltjes function. The complete result is given as follows.

######  Theorem 5.

Let the systems (33) and (46) hold, with the corresponding Stieltjes function
not algebraic. Then the following compatibility conditions hold, for all
$n\geq 1$:

(49)
$\partial_{t}\mathcal{A}_{n}=\partial_{x}{\mathcal{B}}_{n}+{\mathcal{B}}_{n}{\mathcal{A}_{n}}-{\mathcal{A}_{n}}{\mathcal{B}}_{n}\,.$

###### Proof.

Taking derivatives in (46) with respect to $x$ and derivatives in (33) with
respect to $t$ we get, on account of the consistency
$\partial_{t}\partial_{x}Y_{n}=\partial_{x}\partial_{t}Y_{n}\,,$

(50)
$\left[\partial_{t}\mathcal{A}_{n}-\partial_{x}{{\mathcal{B}}_{n}}+{\mathcal{A}_{n}}{\mathcal{B}}_{n}-{\mathcal{B}}_{n}{\mathcal{A}_{n}}\right]Y_{n}=Y_{n}\left[\partial_{t}{\mathcal{C}}-\partial_{x}\widehat{\mathcal{C}}+\mathcal{C}\widehat{\mathcal{C}}-\widehat{\mathcal{C}}\mathcal{C}\right]\,.$

Let us now deduce that the matrix

(51)
$\partial_{t}{\mathcal{C}}-\partial_{x}\widehat{\mathcal{C}}+\mathcal{C}\widehat{\mathcal{C}}-\widehat{\mathcal{C}}\mathcal{C},$

is null. This follows from the compatibility of the differential equations
(45) and (23). Indeed, by taking derivatives in (45) with respect to $x$ and
derivatives in (23) with respect to $t$ we get, on account of the consistency
$\partial_{t}\partial_{x}f=\partial_{x}\partial_{t}f\,,$

$\partial_{t}\left(\frac{B}{A}\right)f^{2}+2\frac{B}{A}f\partial_{t}f+\partial_{t}\left(\frac{C}{A}\right)f+\left(\frac{C}{A}\right)\partial_{t}f+\partial_{t}\left(\frac{D}{A}\right)\\\
=\partial_{x}\left(\frac{\widehat{B}}{\widehat{A}}\right)f^{2}+2\frac{\widehat{B}}{\widehat{A}}f\partial_{x}f+\partial_{x}\left(\frac{\widehat{C}}{\widehat{A}}\right)f+\left(\frac{\widehat{C}}{\widehat{A}}\right)\partial_{x}f+\partial_{x}\left(\frac{\widehat{D}}{\widehat{A}}\right).$

The use of (45) and (23) in the previous equation yields, after
simplifications,

$\left[\partial_{t}\left(\frac{B}{A}\right)-\partial_{x}\left(\frac{\widehat{B}}{\widehat{A}}\right)+\frac{B\widehat{C}-\widehat{B}C}{A\widehat{A}}\right]f^{2}+\left[\partial_{t}\left(\frac{C}{A}\right)-\partial_{x}\left(\frac{\widehat{C}}{\widehat{A}}\right)+\frac{2(B\widehat{D}-\widehat{B}D)}{A\widehat{A}}\right]f\\\
+\partial_{t}\left(\frac{D}{A}\right)-\partial_{x}\left(\frac{\widehat{D}}{\widehat{A}}\right)+\frac{C\widehat{D}-\widehat{C}D}{A\widehat{A}}=0\,.$

As $f$ is not algebraic, then it must hold

(52)
$\displaystyle\partial_{t}\left(\frac{B}{A}\right)-\partial_{x}\left(\frac{\widehat{B}}{\widehat{A}}\right)+\frac{B\widehat{C}-\widehat{B}C}{A\widehat{A}}$
$\displaystyle=0\,,$ (53)
$\displaystyle\partial_{t}\left(\frac{C}{A}\right)-\partial_{x}\left(\frac{\widehat{C}}{\widehat{A}}\right)+\frac{2(B\widehat{D}-\widehat{B}D)}{A\widehat{A}}$
$\displaystyle=0\,,$ (54)
$\displaystyle\partial_{t}\left(\frac{D}{A}\right)-\partial_{x}\left(\frac{\widehat{D}}{\widehat{A}}\right)+\frac{C\widehat{D}-\widehat{C}D}{A\widehat{A}}$
$\displaystyle=0\,.$

Now, the $(2,1)$ component of the matrix (51) is zero by virtue of (52); the
$(1,1)$ and $(2,2)$ components of (51) are zero by virtue of (53); the $(1,2)$
component of (51) is zero by virtue of (54). Hence, from (50), we are left
with

$\left[\partial_{t}\mathcal{A}_{n}-\partial_{x}{{\mathcal{B}}_{n}}+{\mathcal{A}_{n}}{\mathcal{B}}_{n}-{\mathcal{B}}_{n}{\mathcal{A}_{n}}\right]Y_{n}=0_{2\times
2}\,.$

The term $Y_{n}$ cancels out, as $\det(Y_{n})\neq 0$ by virtue of (21). Thus,
we obtain (49). ∎

##  3\. Möbius Transformation of the Stieltjes Function $f$

###  3.1. The System of Matrix Sylvester Equations

We begin by specialising our weight to the modified Jacobi weight defined in
terms of parameters $\alpha,\beta,\mu$ by (9),

$w(x,t)=x^{\alpha}(1-x)^{\beta}(x-t)^{\mu}\,,$

defined on a support that joins $0,1,t$ (see [31, Sec. 5]). It satisfies the
differential equation

$A\partial_{x}w=Cw\,,$

with the polynomials $A,C$ given by

(55) $A(x)=x(x-1)(x-t)\,,\;\;C(x)=\alpha(x-1)(x-t)+\beta x(x-t)+\mu x(x-1)\,.$

Let us consider the sequence of monic orthogonal polynomials,
$\\{P_{n}\\}_{n\geq 0}$ defined by $w$, as well as their associated
polynomials, $\\{P_{n}^{(1)}\\}_{n\geq 0}$. According to the classification in
[32], the weight $w$ and $\\{P_{n}\\}_{n\geq 0}$ are semi-classical of class
one as $\deg(A)=3,\,\deg(C)=2$.

Note that $w$ also satisfies the differential equation

$(x-t)\partial_{t}w=-\mu w\,.$

Therefore, by integrating, we have the following relation for the moments,

$\partial_{t}w_{1}=-\mu w_{0}+t\partial_{t}w_{0}\,.$

Using the general relation $w_{1}=\beta_{0}w_{0}$ in the equation above, where
$\beta_{0}$ is the recurrence relation coefficient of $\\{P_{n}\\}_{n\geq 0}$,
we obtain

(56) $(t-\beta_{0})\partial_{t}\ln(w_{0})=\partial_{t}\beta_{0}+\mu\,.$

The equivalence between an homogeneous differential equation for the weight
and the non-homogeneous linear first order differential equation for the
corresponding Stieltjes function is well-known [32, 43]. The precise
differential equations for the Stieltjes function of the weight (9) are given
in the following lemma.

######  Lemma 1.

The Stieltjes function of the modified Jacobi weight (9), henceforth denoted
by $f$, satisfies the inhomogeneous differential equations

(57) $\displaystyle x(x-1)(x-t)\partial_{x}f$ $\displaystyle=Cf+D\,,$ (58)
$\displaystyle(x-t)\partial_{t}f$ $\displaystyle=-\mu f+\hat{D}_{0}\,,$

with $C$ the polynomial given in (55) and $D,\hat{D}_{0}$ the polynomials
given as follows: $D(x)=d_{1}x+d_{0}$, with

$d_{1}=-(1+\alpha+\beta+\mu)w_{0}\,,d_{0}=\left[-(2+\alpha+\beta+\mu)\beta_{0}+1+\alpha+\beta+\mu+(1+\alpha+\beta)t\right]w_{0}\,,$

and

$\hat{D}_{0}=\partial_{t}w_{0}\,.$

###### Proof.

The polynomials $D,\hat{D}_{0}$ are obtained by substituting the asymptotic
expansion (14) of $f$ into the differential equations (57) and (58),
respectively . ∎

######  Remark 2.

For further purposes, we note the following relation,

(59) $D(t)+t(t-1)\hat{D}_{0}=0\,,$

which follows from the compatibility of equations (57)-(58): taking
derivatives of (57) with respect to $t$ and derivatives of (58) with respect
to $x$, and using $\partial_{t}\partial_{x}f=\partial_{x}\partial_{t}f$.

The function $f(x)$ has a Jacobi continued fraction expansion given by (see
[8, pp. 85, eq. (4.1)] combined with the notation $w_{0}=\gamma_{0}$ from [8,
Th. 4.4., Ch. I])

$f(x)=\frac{w_{0}}{x-\beta_{0}-\displaystyle\frac{\gamma_{1}}{x-\beta_{1}-\frac{\gamma_{2}}{\ddots}}}\,,$

where the $\gamma_{n}$’s and the $\beta_{n}$’s are the coefficients in the
three-term recurrence relation for the corresponding sequences of monic
orthogonal polynomials. Recalling that the continued fraction expansion for
the Stieltjes function corresponding to the sequence $\\{P_{n}^{(1)}\\}_{n\geq
0}$ is

$\tilde{f}(x)=\frac{\gamma_{1}}{x-\beta_{1}-\displaystyle\frac{\gamma_{2}}{x-\beta_{2}-\frac{\gamma_{3}}{\ddots}}}\,,$

an immediate relation between $f$ and $\tilde{f}$ is

(60) $f(x)=\frac{w_{0}}{x-\beta_{0}-\tilde{f}(x)}\,.$

Also we note the asymptotic expansion of $\tilde{f}$ in terms of the
corresponding moments,

(61)
$\tilde{f}(x)=\displaystyle\sum_{n=0}^{+\infty}\frac{\tilde{w}_{n}}{x^{n+1}}\,.$

A consequence of (60), taking into account the asymptotic expansions of $f$
and $\tilde{f}$, (14) and (61), is the relation for the moments,

(62)
$\tilde{w}_{n}=\frac{w_{n+2}-\beta_{0}w_{n+1}-\sum_{k=1}^{n}w_{k}\tilde{w}_{n-k}}{w_{0}}\,,\quad
n\geq 0\,,$

with the convention $\sum_{i}^{j}\cdot=0$ whenever $i>j$.

Following from (60) we shall now take the function $\tilde{f}$ to be given by
(8)

$\tilde{f}(x)=x-\beta_{0}-\frac{w_{0}}{f(x)}\,.$

The quantities $w_{0}$ and $\beta_{0}$, acting as parameters defining the new
function $\tilde{f}$ via (8),will appear in the fundamental formulae to be
subsequently deduced.

The function $\tilde{f}$ will play a central role in the sequel, as the
corresponding sequence of orthogonal polynomials is a member of the non semi-
classical, Laguerre-Hahn class. Indeed, using standard computations combining
(8) with (57), we deduce the Riccati differential equation for $\tilde{f}$,

(63)
$\tilde{A}\partial_{x}\tilde{f}=\tilde{B}\tilde{f}^{2}+\tilde{C}\tilde{f}+\tilde{D}\,,$

with coefficients

(64) $\displaystyle\tilde{A}(x)$ $\displaystyle=x(x-1)(x-t)$ (65)
$\displaystyle\tilde{B}(x)$
$\displaystyle=-(1+\alpha+\beta+\mu)x-(2+\alpha+\beta+\mu)\beta_{0}+1+\alpha+\mu+(1+\alpha+\beta)t\,,$
$\displaystyle\tilde{C}(x)$
$\displaystyle=(2+\alpha+\beta+\mu)x^{2}+[-(2+\alpha+\mu)-(2+\alpha+\beta)t+2\beta_{0}]x\,,$
(66) $\displaystyle\hskip 261.76526pt-\alpha t+2B_{1}(0)\beta_{0}\,,$
$\displaystyle\tilde{D}(x)$
$\displaystyle=[(3+\alpha+\beta+\mu)\beta_{0}^{2}-(2+\alpha+\mu+(2+\alpha+\beta)t)\beta_{0}+(1+\alpha)t]x$
(67) $\displaystyle\hskip
39.83368pt-(2+\alpha+\beta+\mu)\beta_{0}^{3}+(1+\alpha+\mu+(1+\alpha+\beta)t)\beta_{0}^{2}-\alpha
t\beta_{0}\,.$

In order to proceed with our theory of the deformed Laguerre-Hahn polynomials,
we need to identify the role of the deformation parameter within the spectral
differential system, i.e. as the zero $x=t$ of the polynomial $\tilde{A}$
given above.

The Riccati equation in the variable $t$ exhibits a dependence on the
parameters $\beta_{0}$ and $w_{0}$ from (8). Indeed, we have

(68)
$\hat{A}\partial_{t}\tilde{f}=\hat{B}\tilde{f}^{2}+\hat{C}\tilde{f}+\hat{D}\,,$

with coefficients

(69) $\displaystyle\hat{A}(x)$ $\displaystyle=x-t\,,$ (70)
$\displaystyle\hat{B}$ $\displaystyle=\partial_{t}\ln w_{0}\,,$ (71)
$\displaystyle\hat{C}(x)$ $\displaystyle=-\partial_{t}(\ln
w_{0})x+(2\beta_{0}-t)\partial_{t}\ln(w_{0})+\mu\,,$ (72)
$\displaystyle\hat{D}$ $\displaystyle=-(\beta_{0}-t)\partial_{t}\beta_{0}\,.$

The next step is to deduce the differential systems satisfied by the
polynomials corresponding to $\tilde{f}$. In order to simplify the text, we
will use the following notations: we denote by $\\{\tilde{P}_{n}\\}_{n\geq 0}$
the sequence of monic orthogonal polynomials corresponding to $\tilde{f}$, and
we denote by $(\tilde{\beta}_{n})_{n\geq 0},(\tilde{\gamma}_{n})_{n\geq 0}$
its sequences of recurrence relation coefficients. Let $\tilde{Y}_{n}$ be
defined by (10),

$\tilde{Y}_{n}=\left[\begin{matrix}\tilde{P}_{n+1}&\tilde{P}_{n}^{(1)}\\\
\tilde{P}_{n}&\tilde{P}_{n-1}^{(1)}\\\ \end{matrix}\right]\,.$

Upon utilising equations (63) and (68) we develop the results of Theorems 3
and 4, now applied to the Stieltjes function $\tilde{f}$ and the corresponding
sequence $\\{\tilde{Y}_{n}\\}_{n\geq 0}$. We have the matrix Sylvester
equations given in the following lemma.

######  Lemma 2.

Let $\tilde{f}$ be the Stieltjes function defined by (8), and let
$\\{\tilde{P}_{n}\\}_{n\geq 0}$ be the corresponding sequence of monic
orthogonal polynomials. Under the previous notations, the following
differential systems hold, for all $n\geq 0$:

(73) $\displaystyle\partial_{x}{\tilde{Y}}_{n}$
$\displaystyle={\tilde{\mathcal{A}}}_{n}\tilde{Y}_{n}-\tilde{Y}_{n}{\tilde{\mathcal{C}}}\,,$
(74) $\displaystyle\partial_{t}\tilde{Y}_{n}$
$\displaystyle=\tilde{\mathcal{B}}_{n}\tilde{Y}_{n}-\tilde{Y}_{n}\hat{\mathcal{C}}\,,$

where

(75)
${\tilde{\mathcal{A}}}_{n}=\frac{1}{\tilde{A}}\begin{bmatrix}\tilde{l}_{n}&\tilde{{\Theta}}_{n}\\\
-\tilde{\Theta}_{n-1}/{\tilde{\gamma}}_{n}&\;\;-\tilde{l}_{n}\end{bmatrix},\;\;{\tilde{\mathcal{C}}}=\frac{1}{\tilde{A}}\begin{bmatrix}\tilde{C}/2&-\tilde{D}/\tilde{w}_{0}\\\
\tilde{w}_{0}\tilde{B}&-\tilde{C}/2\end{bmatrix},$

and

(76)
$\tilde{\mathcal{B}}_{n}=\frac{1}{\hat{A}}\begin{bmatrix}\hat{l}_{n}&\hat{\Theta}_{n}\\\
-\hat{\Theta}_{n-1}/{\tilde{\gamma}}_{n}&\;\;\hat{l}_{n-1}+(x-{\tilde{\beta}}_{n})\hat{\Theta}_{n-1}/{\tilde{\gamma}}_{n}\end{bmatrix},$
(77)
$\hat{\mathcal{C}}=\frac{1}{\hat{A}}\begin{bmatrix}\hat{C}/2&-\hat{D}/\tilde{w}_{0}\\\
\tilde{w}_{0}\hat{B}&-\hat{C}/2+\partial_{t}(\ln\tilde{w}_{0})\hat{A}\end{bmatrix},$

with $\tilde{A},\tilde{B},\tilde{C},\tilde{D}$ the polynomials in (63),
$\hat{A},\hat{B},\hat{C},\hat{D}$ the polynomials in (68), and $\tilde{w}_{0}$
denoting the zero moment in the asymptotic expansion (61) of $\tilde{f}$.

The entries of $\tilde{\mathcal{A}}_{n}$ and $\tilde{\mathcal{B}}_{n}$ are
polynomials in the variable $x$, and given by

$\tilde{l}_{n}(x)=\tilde{l}_{n,2}x^{2}+\tilde{l}_{n,1}x+\tilde{l}_{n,0}\,,\;\;\tilde{\Theta}_{n}(x)=\tilde{\Theta}_{n,1}x+\tilde{\Theta}_{n,0}\,,\;\;\hat{l}_{n}(x)=\hat{l}_{n,1}x+\hat{l}_{n,0}\,,\;\;\hat{\Theta}_{n}(x)=\hat{\Theta}_{n}\,,$

where, for all $n\geq 1,$

(78) $\displaystyle\tilde{l}_{n,2}=$ $\displaystyle(\nu_{n}-1)/2\,,$ (79)
$\displaystyle\tilde{l}_{n,1}=$
$\displaystyle\sum_{j=0}^{n}\tilde{\beta}_{j}-\tfrac{1}{2}(\nu_{n}-1)(1+t)+\tfrac{1}{2}(\beta+\mu
t)+\beta_{0}\,,$ $\displaystyle\tilde{l}_{n,0}=$
$\displaystyle\sum_{j=0}^{n}(\tilde{\beta}^{2}_{j}-(1+t)\tilde{\beta}_{j}+2\tilde{\gamma}_{j})-\tfrac{1}{2}(t+1)(\alpha+\mu
t)+\tfrac{1}{2}(\nu_{n}-1)t+\tfrac{1}{2}(\alpha+\mu t^{2})$ (80)
$\displaystyle\hskip
76.82234pt+\nu_{n}\tilde{\gamma}_{n+1}+\beta_{0}(\beta_{0}-1-\beta-t)-\tfrac{1}{2}(\beta+\mu)t-\tfrac{1}{2}(\alpha+\mu
t^{2})\,,$ (81) $\displaystyle\tilde{\Theta}_{n,1}=$
$\displaystyle-\nu_{n}\tilde{\gamma}_{n+1}\,,$ (82)
$\displaystyle\tilde{\Theta}_{n,0}=$
$\displaystyle-\vartheta_{n}(t)\tilde{\gamma}_{n+1}\,,$ (83)
$\displaystyle\hat{l}_{n,1}=$ $\displaystyle-\tfrac{1}{2}\partial_{t}\ln
w_{0}\,,$ (84) $\displaystyle\hat{l}_{n,0}=$
$\displaystyle\tfrac{1}{2}\left((2\beta_{0}-t)\partial_{t}\ln
w_{0}+\mu)\right)-\sum_{j=0}^{n}\partial_{t}\tilde{\beta}_{j}$ (85)
$\displaystyle{\hat{\Theta}_{n}}=$ $\displaystyle\left(\partial_{t}\ln
w_{0}+\sum_{j=0}^{n+1}\partial_{t}\ln\tilde{\gamma}_{j}\right)\tilde{\gamma}_{n+1}\,,$

with

(86) $\displaystyle\nu_{n}$ $\displaystyle=2n+5+\alpha+\beta+\mu\,,$ (87)
$\displaystyle\vartheta_{n}(t)$
$\displaystyle=\nu_{n}(\tilde{\beta}_{n+1}-1-t)+2\sum_{j=0}^{n}\tilde{\beta}_{j}+\tilde{\beta}_{n+1}+\mu
t+\beta+2\beta_{0}\,.$

The following initial conditions hold:

$\displaystyle\frac{\tilde{\Theta}_{-1}}{\tilde{w}_{0}}=\tilde{D}\,,\tilde{\Theta}_{0}=\tilde{A}+(x-\tilde{\beta}_{0})(\tilde{C}/2-\tilde{l}_{0})+\tilde{w}_{0}\tilde{B}\,,$
$\displaystyle\tilde{l}_{-1}=\tilde{C}/2\,,\;\tilde{l}_{0}=-\tilde{C}/2-(x-\tilde{\beta}_{0})\tilde{D}/\tilde{w}_{0}\,,$
$\displaystyle\frac{\hat{\Theta}_{-1}}{\tilde{w}_{0}}=\hat{D}\,,\hat{\Theta}_{0}=-\partial_{t}(\tilde{\beta}_{0})\hat{A}+(x-\tilde{\beta}_{0})(\hat{C}/2-\hat{l}_{0})+\tilde{w}_{0}\hat{B}\,,\;$
$\displaystyle\hat{l}_{-1}=\hat{C}/2\,,\;\hat{l}_{0}=-\hat{C}/2-(x-\tilde{\beta}_{0})\hat{D}/\tilde{w}_{0}+\partial_{t}(\ln\tilde{w}_{0})\hat{A}\,.$

###### Proof.

The coefficients (78)-(82) follow from the differential equations in (73)
using the expansions

(88) $\displaystyle\tilde{P}_{n+1}(x)$
$\displaystyle=x^{n+1}-(\tilde{\eta}_{n}+\tilde{\beta}_{0})x^{n}+(\tilde{\kappa}_{n}+\tilde{\beta}_{0}\tilde{\eta}_{n}-\tilde{\gamma}_{1})x^{n-1}+\dots+P_{n+1}(0)\,,\phantom{00}$
(89) $\displaystyle\tilde{P}_{n}^{(1)}(x)$
$\displaystyle=x^{n}-\tilde{\eta}_{n}x^{n-1}+\tilde{\kappa}_{n}x^{n-2}+\dots+P_{n}^{(1)}(0)\,,$

where

$\tilde{\eta}_{n}=\sum_{k=1}^{n}\tilde{\beta}_{k}\,,\;\;\tilde{\kappa}_{n}=\sum_{1\leq
i<j\leq
n}^{n}\tilde{\beta}_{i}\tilde{\beta}_{j}-\sum_{k=2}^{n}\tilde{\gamma}_{k}\,.$

The coefficients (83)-(85) follow from the equations in (74) this time using
the expansions (88)-(89). ∎

The data from $\hat{l}_{n},\hat{\Theta}_{n}$ will be a fundamental tool to
deduce the Toda equations in Theorem 6 and to deduce the derivatives in Lemma
4. We now give further relations to be used later on.

######  Corollary 5.1.

Under the previous notations, we have the following equation for
$\hat{\Theta}_{n}$:

(90)
$\hat{\Theta}_{n}=-\sum_{j=1}^{n}\partial_{t}\tilde{\gamma}_{j}+t\sum_{j=1}^{n}\partial_{t}\tilde{\beta}_{j}-\tfrac{1}{2}\sum_{j=1}^{n}\partial_{t}\tilde{\beta}^{2}_{j}+(t-\tilde{\beta}_{0})\partial_{t}\tilde{\beta}_{0}+\tilde{w}_{0}\partial_{t}\ln
w_{0}\,.$

###### Proof.

The identity for $\hat{\Theta}_{n}$ follows from the coefficient of $x^{n}$ in
the equation from the $(1,1)$ component in (74), simplified using standard
computations. ∎

###  3.2. Fundamental Formulae and Derivatives of Related Quantities

In this section we give relations satisfied by the quantities
$\tilde{l}_{n,1},\tilde{l}_{n,0}$ and $\tilde{\Theta}_{n,0}$ that define the
polynomial entries of the matrix $\tilde{\mathcal{A}}_{n}$ for the system
(73). Such relations will be a fundamental tool in the derivation of the sixth
Painlevé equation in Section 3.3. Note that the poles of the matrix
$\tilde{\mathcal{A}}_{n}$ are the zeroes of the polynomial $\tilde{A}$ given
in (64).

####  3.2.1. Toda-type Equations for the Recurrence Coefficients

We start by obtaining relations between the polynomial entries of the matrix
of the systems (73) and (74). The first result shows how
$\hat{l}_{n},\hat{\Theta}_{n}$ and $\tilde{l}_{n},\tilde{\Theta}_{n}$ are
related when evaluated at the pole $x=t$.

######  Lemma 3.

Under the previous notations, the polynomial entries of the matrices (75) and
(76) satisfy

(91)
$\hat{l}_{n}(t)=-\frac{\tilde{l}_{n}(t)}{t(t-1)}\,,\;\;\;\hat{\Theta}_{n}=-\frac{\tilde{\Theta}_{n}(t)}{t(t-1)}\,.$

###### Proof.

The relation (91) is a direct consequence of the following equality for the
residue matrices at $x=t$:

(92)
$\left.\mathcal{R}es\;\tilde{\mathcal{B}}_{n}\right|_{x=t}=-\left.\mathcal{R}es\;\tilde{\mathcal{A}}_{n}\right|_{x=t}.$

Equation (92) is deduced as follows. Firstly, recall that
$\tilde{\mathcal{B}}_{n}$ has one single pole at $x=t$. To compute the residue
of $\tilde{\mathcal{B}}_{n}$ at $x=t,$ note that from (74) we have

$\tilde{\mathcal{B}}_{n}=\partial_{t}(\tilde{Y}_{n})\tilde{Y}_{n}^{-1}+(\tilde{Y}_{n})\hat{\mathcal{C}}\tilde{Y}_{n}^{-1}\,.$

Noting that $\partial_{t}(\tilde{Y}_{n})\tilde{Y}_{n}^{-1}$ is a matrix of
polynomial entries, thus entire, then the residue of $\tilde{\mathcal{B}}_{n}$
is given by

$\left.\mathcal{R}es\;\tilde{\mathcal{B}}_{n}\right|_{x=t}=\left.\mathcal{R}es\;\tilde{Y}_{n}\hat{\mathcal{C}}\tilde{Y}_{n}^{-1}\right|_{x=t}\,.$

As $\tilde{Y}_{n}\hat{\mathcal{C}}\tilde{Y}_{n}^{-1}$ is a matrix of rational
entries with a single pole at $x=t$, then

$\left.\mathcal{R}es\;\tilde{Y}_{n}\hat{\mathcal{C}}\tilde{Y}_{n}^{-1}\right|_{x=t}=\tilde{Y}_{n}(t)\hat{\hat{\mathcal{C}}}(t)\tilde{Y}_{n}^{-1}(t)\,,\quad\hat{\hat{\mathcal{C}}}=\begin{bmatrix}\hat{C}/2&-\hat{D}/\tilde{w}_{0}\\\
\tilde{w}_{0}\hat{B}&-\hat{C}/2+\partial_{t}(\ln\tilde{w}_{0})\hat{A}\end{bmatrix}.$

For matters of simplicity, we only give details for the computations
concerning the $(1,1)$ component. We have, in the account of
$\det\tilde{Y}_{n}=-\prod_{k=}^{n}\tilde{\gamma}_{k}$,

(93)
$\left[\tilde{Y}_{n}(t)\hat{\hat{\mathcal{C}}}(t)\tilde{Y}_{n}^{-1}(t)\right]_{(1,1)}=-\frac{1}{\prod_{k=}^{n}\tilde{\gamma}_{k}}\left\\{\left(\tilde{P}_{n+1}(t)\tilde{P}^{(1)}_{n-1}(t)+\tilde{P}_{n}(t)\tilde{P}^{(1)}_{n}(t)\right)\tfrac{1}{2}\hat{C}(t)\right.\\\
\left.+\tilde{P}_{n+1}(t)\tilde{P}_{n}(t)\frac{\hat{D}}{\tilde{w}_{0}}+\tilde{w}_{0}\tilde{P}_{n}(t)\tilde{P}^{(1)}_{n-1}(t)\hat{B}\right\\}\,.$

We now compare (93) with the residue at $x=t$ of the $(1,1)$ component in
$\tilde{\mathcal{A}}_{n}$. Note that

$\left[\left.\mathcal{R}es\;\tilde{\mathcal{A}}_{n}\right|_{x=t}\right]_{(1,1)}=\frac{\tilde{l}_{n}(t)}{t(t-1)}\,.$

The computation of $\tilde{l}_{n}$, in accordance with the definition in (32),
yields

(94)
$\tilde{l}_{n}(t)=-\frac{1}{\prod_{k=}^{n}\tilde{\gamma}_{k}}\left\\{\left(\tilde{P}_{n+1}(t)\tilde{P}^{(1)}_{n-1}(t)+\tilde{P}_{n}(t)\tilde{P}^{(1)}_{n}(t)\right)\tfrac{1}{2}\tilde{C}(t)\right.\\\
\left.+\tilde{P}_{n+1}(t)\tilde{P}_{n}(t)\frac{\tilde{D}(t)}{w_{0}^{(1)}}+w_{0}^{(1)}\tilde{P}_{n}(t)\tilde{P}^{(1)}_{n-1}(t)\tilde{B}(t)\right\\}\,.$

Taking into account the data contained in
$\tilde{B},\tilde{C},\tilde{D},\hat{B},\hat{C}$ and $\hat{D},$ from equations
(93) and (94), we get

$\left[\left.\mathcal{R}es\;\tilde{\mathcal{A}}_{n}\right|_{x=t}\right]_{(1,1)}+\left[\left.\mathcal{R}es\;\tilde{\mathcal{B}}_{n}\right|_{x=t}\right]_{(1,1)}\\\
=-\frac{1}{w_{0}\prod_{k=}^{n}\tilde{\gamma}_{k}}\left\\{-(t-\beta_{0})\left(\tilde{P}_{n+1}(t)\tilde{P}^{(1)}_{n-1}(t)+\tilde{P}_{n}(t)\tilde{P}^{(1)}_{n}(t)\right)\right.\\\
\left.+\frac{(t-\beta_{0})^{2}}{\tilde{w}_{0}}\tilde{P}_{n+1}(t)\tilde{P}_{n}(t)+\tilde{w}_{0}\tilde{P}_{n}(t)\tilde{P}^{(1)}_{n-1}(t)\right\\}\left(\hat{D}_{0}+\frac{D(t)}{t(t-1)}\right)\,,$

with $D,\hat{D}_{0}$ the polynomials given in (57) and (58), respectively. As
$\hat{D}_{0}+\frac{D(t)}{t(t-1)}=0$ (recalling (59)), then we conclude that

$\left[\left.\mathcal{R}es\;\tilde{\mathcal{A}}_{n}\right|_{x=t}\right]_{(1,1)}+\left[\left.\mathcal{R}es\;\tilde{\mathcal{B}}_{n}\right|_{x=t}\right]_{(1,1)}=0.$

Similar computations lead to

$\left[\left.\mathcal{R}es\;\tilde{\mathcal{A}}_{n}\right|_{x=t}\right]_{(i,j)}+\left[\left.\mathcal{R}es\;\tilde{\mathcal{B}}_{n}\right|_{x=t}\right]_{(i,j)}=0,\quad
i,j=1,2\,,$

hence (92) follows. ∎

As a consequence, we can show that the derivatives of the recurrence
coefficients are expressible in terms of the functions
$\tilde{\Theta}_{n},\tilde{l}_{n}$ via the following Toda-type equations.

######  Theorem 6.

Under the conditions and notations of Lemma 2, the recurrence relation
coefficients $(\tilde{\beta}_{n})_{n\geq 0},(\tilde{\gamma}_{n})_{n\geq 0}$
satisfy the following Toda-type equations, for all $n\geq 1$ :

(95) $\displaystyle t(t-1)\frac{d}{dt}\tilde{\beta}_{n}$
$\displaystyle=\tilde{\beta_{n}}(\tilde{\beta}_{n}-1)+(\nu_{n-1}+2)\tilde{\gamma}_{n+1}-(\nu_{n-1}-2)\tilde{\gamma}_{n}\,,$
(96) $\displaystyle t(t-1)\frac{d}{dt}\ln\tilde{\gamma}_{n}$
$\displaystyle=-2+(\nu_{n-1}+1)\tilde{\beta}_{n}-(\nu_{n-1}-3)\tilde{\beta}_{n-1}\,.$

###### Proof.

Because of (91), equations (95)-(96) are obtained, respectively, from

$\displaystyle t(t-1)\left(\hat{l}_{n}(t)-\hat{l}_{n-1}(t)\right)$
$\displaystyle=\tilde{l}_{n-1}(t)-\tilde{l}_{n}(t)\,,$ $\displaystyle
t(t-1)\left(\frac{\hat{\Theta}_{n}}{\tilde{\gamma_{n+1}}}-\frac{\hat{\Theta}_{n-1}}{\tilde{\gamma_{n}}}\right)$
$\displaystyle=\frac{\tilde{\Theta}_{n-1}}{\tilde{\gamma}_{n}}(t)-\frac{\tilde{\Theta}_{n}}{\tilde{\gamma}_{n+1}}(t)\,,$

by using the data given in Lemma 2. ∎

####  3.2.2. The Schlesinger-type Equations

We shall now analyse the compatibility conditions that result from the
consistency of the two systems for $\tilde{Y}_{n}$, (73) and (74). The
compatibility conditions stated in Theorem 5 are now given by

(97)
$\partial_{t}\tilde{\mathcal{A}}_{n}=\partial_{x}\tilde{\mathcal{B}}_{n}+\tilde{\mathcal{B}}_{n}\tilde{\mathcal{A}}_{n}-\tilde{\mathcal{A}}_{n}\tilde{\mathcal{B}}_{n}\,.$

Denoting the poles of $\tilde{\mathcal{A}}_{n}$ in (75) by
$x_{1}=0,x_{2}=1,x_{3}=t,$ then we may write

(98)
$\tilde{\mathcal{A}}_{n}=\sum_{j=1}^{3}\frac{\tilde{\mathcal{A}}_{n,j}}{x-x_{j}}\,,$

where the residue matrices are

${\tilde{\mathcal{A}}}_{n,j}=\frac{1}{\partial_{x}\tilde{A}(x_{j})}\begin{bmatrix}\tilde{l}_{n}(x_{j})&\tilde{{\Theta}}_{n}(x_{j})\\\
-\frac{\tilde{\Theta}_{n-1}}{{\tilde{\gamma}}_{n}}(x_{j})&\;\;-\tilde{l}_{n}(x_{j})\end{bmatrix}\,,\quad
j=1,2,3\,.$

The matrix $\tilde{\mathcal{B}}_{n}$, as implied by (76) and (92) has the form

(99)
$\tilde{\mathcal{B}}_{n}=\tilde{\mathcal{B}}_{n,\infty}-\frac{\tilde{\mathcal{A}}_{n,3}}{x-t}\,,$

with residue matrix

$\tilde{\mathcal{B}}_{n,\infty}=\begin{bmatrix}\hat{l}_{n,1}&0\\\
0&\;\;\hat{l}_{n-1,1}+\frac{\hat{\Theta}_{n-1}}{\tilde{\gamma}_{n}}\end{bmatrix}.$

Due to the forms of (98) and (99), (97) becomes a Schlesinger-type equation

(100)
$\sum_{j=1}^{3}\frac{\partial_{t}\tilde{\mathcal{A}}_{n,j}}{x-x_{j}}=\sum_{j=1}^{3}\frac{[\hat{\mathcal{\mathcal{B}}}_{n\infty}\,,\,\tilde{\mathcal{A}}_{n,j}]}{x-x_{j}}+\frac{[{\mathcal{\mathcal{A}}}_{n,1}\,,\,\tilde{\mathcal{A}}_{n,3}]}{(x-x_{1})(x-x_{3})}+\frac{[{\mathcal{\mathcal{A}}}_{n,2}\,,\,\tilde{\mathcal{A}}_{n,3}]}{(x-x_{2})(x-x_{3})}\,.$

The above equation (100) will now be used to obtain the derivatives of the
quantities $\tilde{l}_{n,1},\tilde{l}_{n,0}$ and
$\Theta_{n,0}/\tilde{\gamma}_{n+1}$ that parameterise the matrix
$\tilde{\mathcal{A}}_{n}$.

######  Lemma 4.

Under the previous notations, the following equations hold, for all $n\geq 1$:

(101) $\displaystyle\partial_{t}\tilde{l}_{n,0}$
$\displaystyle=\frac{\tilde{l}_{n,0}}{t}-\frac{\tilde{\gamma}_{n+1}(\nu_{n}\vartheta_{n-1}-\vartheta_{n}\nu_{n-1})}{t(t-1)}\,,$
(102) $\displaystyle\partial_{t}\tilde{l}_{n,1}$
$\displaystyle=\frac{\nu_{n}-1}{2(t-1)}+\frac{\tilde{l}_{n,1}}{t-1}+\frac{\tilde{l}_{n,0}}{t(t-1)}\,,$
(103) $\displaystyle\partial_{t}\vartheta_{n}$
$\displaystyle=\frac{-\vartheta_{n}-\vartheta_{n}^{2}+2\xi_{n}}{t(t-1)}\,,$

with

(104) $\xi_{n}=\vartheta_{n}\tilde{l}_{n,1}-\nu_{n}\tilde{l}_{n,0}\,.$

###### Proof.

Equations (101)-(103) are obtained as follows: equation (101) follows from the
equation in the $(1,1)$ component of (100) at the residue $x=0$; equation
(102) follows from the equation in the $(1,1)$ component of (100) at the
residue $x=1$; equation (103) follows from the equation in the $(1,2)$
component of (100) at the residue $x=0$ combined with the equation in the
$(1,2)$ component at the residue $x=1$ and where the data (85) was used. ∎

####  3.2.3. Bilinear Relations: the Determinant of the Matrix
$\tilde{\mathcal{A}}_{n}$

Let us analyse the equations arising from the determinant of the matrix
$\tilde{\mathcal{A}}_{n}$ (see (43) and (44)), which in the present situation
are written as

(105)
$-\tilde{l}_{n}^{2}+\tilde{\Theta}_{n}\frac{\tilde{\Theta}_{n-1}}{\tilde{\gamma}_{n}}=\varphi+\tilde{A}\sum_{k=1}^{n}\frac{\tilde{\Theta}_{k-1}}{\tilde{\gamma}_{k}}\,,$

with

(106)
$\varphi=\frac{\tilde{D}}{\tilde{w}_{0}}\left(\tilde{A}+\tilde{w}_{0}\tilde{B}\right)-\frac{\tilde{C}^{2}}{4}\,.$

Evaluating (105) at the zeroes of $\tilde{A}$ we have, respectively,

(107) $\displaystyle\frac{\tilde{\Theta}_{n-1}(0)}{\tilde{\gamma}_{n}}$
$\displaystyle=\frac{\tilde{l}^{2}_{n}(0)+\varphi(0)}{\tilde{\Theta}_{n}(0)}\,,$
(108) $\displaystyle\frac{\tilde{\Theta}_{n-1}(1)}{\tilde{\gamma}_{n}}$
$\displaystyle=\frac{\tilde{l}^{2}_{n}(1)+\varphi(1)}{\tilde{\Theta}_{n}(1)}\,,$
(109) $\displaystyle\frac{\tilde{\Theta}_{n-1}(t)}{\tilde{\gamma}_{n}}$
$\displaystyle=\frac{\tilde{l}^{2}_{n}(t)+\varphi(t)}{\tilde{\Theta}_{n}(t)}\,.$

Also, for subsequent purposes, by utilising (81) and (82), we shall write
(107) and (108) respectively, as

(110) $\displaystyle\tilde{\gamma}_{n+1}\vartheta_{n-1}$
$\displaystyle=\frac{\tilde{l}^{2}_{n,0}+\varphi(0)}{\vartheta_{n}}\,,$ (111)
$\displaystyle\tilde{\gamma}_{n+1}\nu_{n-1}$
$\displaystyle=\frac{\tilde{l}^{2}_{n}(1)+\varphi(1)}{\nu_{n}+\vartheta_{n}}-\tilde{\gamma}_{n+1}\vartheta_{n-1}\,.$

######  Lemma 5.

Under the previous notations, the following equation holds, for all $n\geq 1$:

(112)
$\tilde{l}_{n,0}=-\frac{\varphi(0)}{t(\nu_{n}-1)}+\frac{\vartheta_{n}\varphi(1)}{(t-1)(\nu_{n}-1)(\nu_{n}+\vartheta_{n})}-\frac{\vartheta_{n}\varphi(t)}{t(t-1)(\nu_{n}-1)(\nu_{n}t+\vartheta_{n})}\\\
-\frac{\left[(\nu_{n}-1)\vartheta_{n}[t(t+1)\nu_{n}+(t^{2}+t+1)\vartheta_{n}]/4+[t\nu_{n}+(t+1)\vartheta_{n}]\xi_{n}+\xi_{n}^{2}/(\nu_{n}-1)\right]}{(\nu_{n}+\vartheta_{n})(\nu_{n}t+\vartheta_{n})}\,,$

with $\xi_{n}$ given by (104).

###### Proof.

The polynomial $\tilde{\Theta}_{n}$, being of degree one, satisfies the
following identity, for all $n\geq 1$:

$\frac{\tilde{\Theta}_{n-1}(0)}{t}+\frac{\tilde{\Theta}_{n-1}(1)}{1-t}+\frac{\tilde{\Theta}_{n-1}(t)}{t(t-1)}=0\,.$

Thus, due to (107)-(109) we have

$\frac{\tilde{l}^{2}_{n}(0)+\varphi(0)}{t\tilde{\Theta}_{n}(0)}-\frac{\tilde{l}^{2}_{n}(1)+\varphi(0)}{(t-1)\tilde{\Theta}_{n}(1)}+\frac{\tilde{l}^{2}_{n}(t)+\varphi(t)}{t(t-1)\tilde{\Theta}_{n}(t)}=0\,.$

The previous equation can be written as follows:

(113)
$\tilde{l}_{n,2}^{2}\left[\frac{t^{3}\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\right]+\tilde{l}_{n,1}^{2}\left[\frac{t\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\right]\\\
+\tilde{l}_{n,0}^{2}\left[\frac{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)-t\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(t)+\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(1)}{t(t-1)\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\right]\\\
+2\tilde{l}_{n,2}\tilde{l}_{n,1}\left[\frac{t^{2}\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\right]+2\tilde{l}_{n,2}\tilde{l}_{n,0}\left[\frac{t\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\right]+2\tilde{l}_{n,1}\tilde{l}_{n,0}\left[\frac{\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\right]\\\
+\frac{\varphi(0)}{t\tilde{\Theta}_{n}(0)}-\frac{\varphi(1)}{(t-1)\tilde{\Theta}_{n}(1)}+\frac{\varphi(t)}{t(t-1)\tilde{\Theta}_{n}(t)}=0\,.$

Using the identities

$\displaystyle\frac{\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{t-1}$
$\displaystyle=-\tilde{\Theta}_{n,1}\,,$
$\displaystyle\frac{t\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{t-1}$
$\displaystyle=\tilde{\Theta}_{n}(0)\,,$
$\displaystyle\frac{t^{2}\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{t-1}$
$\displaystyle=t\tilde{\Theta}_{n}(0)+\tilde{\Theta}_{n}(t)\,,$
$\displaystyle\frac{t^{3}\tilde{\Theta}_{n}(1)-\tilde{\Theta}_{n}(t)}{t-1}$
$\displaystyle=t^{2}\tilde{\Theta}_{n}(0)+(t+1)\tilde{\Theta}_{n}(t)\,,$
$\displaystyle\frac{(t-1)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)-t\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(t)+\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(1)}{t(t-1)}$
$\displaystyle=\tilde{\Theta}_{n,1}^{2}\,,$

then (113) can be simplified as

(114)
$\frac{2\tilde{l}_{n,2}\tilde{l}_{n,0}}{\tilde{\Theta}_{n}(0)}+\tilde{l}_{n,2}^{2}\tilde{\Theta}_{n}(0)\frac{[t^{2}\tilde{\Theta}_{n}(0)+(t+1)\tilde{\Theta}_{n}(t)]}{\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}+2\tilde{l}_{n,2}\xi_{n}\frac{[t\tilde{\Theta}_{n}(0)+\tilde{\Theta}_{n}(t)]}{\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}\\\
+\frac{\xi_{n}^{2}}{\tilde{\Theta}_{n}(0)\tilde{\Theta}_{n}(1)\tilde{\Theta}_{n}(t)}+\frac{\varphi(0)}{t\tilde{\Theta}_{n}(0)}-\frac{\varphi(1)}{(t-1)\tilde{\Theta}_{n}(1)}+\frac{\varphi(t)}{t(t-1)\tilde{\Theta}_{n}(t)}=0\,,$

with $\xi_{n}$ defined in (104). Solving (114) for $\tilde{l}_{n,0}$ yields
(112). ∎

###  3.3. Derivation of P${}_{\textrm{VI}}$

Now that the derivatives of $\vartheta_{n}$ and $\xi_{n}$ have been computed
our remaining task is to deduce the Painlevé equation and to obtain the
transformation that identifies the Hamiltonian equations for the
P${}_{\textrm{VI}}$ system. We now give the proof of Theorem 1. Recall that
dash means derivative with respect to $t$.

###### Proof.

Firstly, let us note that, in terms of the quantity $q$ defined in (12)

$q=-\vartheta_{n}/\nu_{n}\,,$

we have the following identities: equation (101) is given, taking into account
(110)-(111), by

(115)
$\tilde{l}^{\prime}_{n,0}=\frac{1}{t}\tilde{l}_{n,0}-\frac{1}{t(t-1)}\left[(\tilde{l}^{2}_{n,0}+\varphi(0))\displaystyle\left(\frac{q-1}{q}\right)+(\tilde{l}^{2}_{n}(1)+\varphi(1))\displaystyle\left(\frac{q}{1-q}\right)\right]\,;$

equation (103) by

(116) $t(t-1)q^{\prime}=-q+\nu_{n}q^{2}-2\xi_{n}/\nu_{n}\,;$

and equation (112) by

(117)
$\tilde{l}_{n,0}=-\frac{\varphi(0)}{(\nu_{n}-1)t}-\frac{\varphi(1)}{(\nu_{n}-1)(t-1)}\left(\frac{q}{1-q}\right)+\frac{\varphi(t)}{(\nu_{n}-1)t(t-1)}\left(\frac{q}{t-q}\right)\\\
-\frac{\left[-(\nu_{n}-1)q+[t(t+1)-(t^{2}+t+1)q]/4+[t-(t+1)q]\left(\xi_{n}/\nu_{n}\right)+\left(\xi_{n}/\nu_{n}\right)^{2}/(\nu_{n}-1)\right]}{(1-q)(t-q)}\,.$

Taking derivatives of $\xi_{n}/\nu_{n}=-q\tilde{l}_{n,1}-\tilde{l}_{n,0}$ (cf.
(104)) we have

(118)
$\left(\frac{\xi_{n}}{\nu_{n}}\right)^{\prime}=-q^{\prime}\tilde{l}_{n,1}-q\tilde{l}^{\prime}_{n,1}-\tilde{l}^{\prime}_{n,0}\,.$

Use of (102) and (115) in (118) gives us

$\left(\frac{\xi_{n}}{\nu_{n}}\right)^{\prime}=-q^{\prime}\tilde{l}_{n,1}-q\frac{(\nu_{n}-1)}{2(t-1)}-q\frac{\tilde{l}_{n,1}}{t}-q\frac{\tilde{l}_{n,0}}{t(t-1)}-\frac{\tilde{l}_{n,0}}{t}\\\
+\frac{\tilde{l}^{2}_{n,0}+\varphi(0)}{t(t-1)}\displaystyle\left(\frac{q-1}{q}\right)+\frac{\tilde{l}^{2}_{n}(1)+\varphi(1)}{t(t-1)}\left(\frac{q}{1-q}\right)\,.$

Using
$\tilde{l}_{n,1}=-\displaystyle\frac{\xi_{n}/\nu_{n}+\tilde{l}_{n,0}}{q}$ as
well as (116) and (117) in the previous equation, we get

(119)
$\left(\frac{\xi_{n}}{\nu_{n}}\right)^{\prime}=\left(\frac{1}{1-q}+\frac{1}{-q}+\frac{1}{t-q}\right)t(t-1)\left(\frac{q^{\prime}}{2}\right)^{2}+\left(2\nu_{n}q-1-\frac{t(t-1)}{t-q}\right)\left(\frac{q^{\prime}}{2}\right)\\\
+\frac{\varphi(1)}{t(t-1)^{2}}\frac{q(t-q)}{1-q}-\frac{\varphi(0)}{t^{2}(t-1)}\frac{(1-q)(t-q)}{q}-\left(\frac{1}{4}+\frac{\varphi(t)}{t^{2}(t-1)^{2}}\right)\frac{q(1-q)}{t-q}\\\
-\frac{q(1-q)(t-q)}{4t(t-1)}\,.$

Let us now compute the second derivative of $q$. By differentiating (116) we
get

(120)
$t(t-1)q^{\prime\prime}=-2tq^{\prime}+2\nu_{n}qq^{\prime}-2\left(\frac{\xi_{n}}{\nu_{n}}\right)^{\prime}\,.$

Employing (119) in (120) and taking into account that $\varphi$ as defined by
(106) has the evaluations

(121)
$\;\;\frac{\varphi(0)}{t^{2}}=-\frac{\alpha^{2}}{4}\,,\;\;\frac{\varphi(1)}{(t-1)^{2}}=-\frac{\beta^{2}}{4}\,,\;\;\frac{\varphi(t)}{t^{2}(t-1)^{2}}=-\frac{\mu^{2}}{4}\,,\,$

we finally obtain

(122)
$q^{\prime\prime}=\frac{1}{2}\left(\frac{1}{q}+\frac{1}{q-1}+\frac{1}{q-t}\right)\left({q^{\prime}}\right)^{2}-\left(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{q-t}\right)q^{\prime}\\\
+\frac{q(q-1)(q-t)}{t^{2}(t-1)^{2}}\left[\frac{1}{2}-\frac{\alpha^{2}t}{2q^{2}}+\frac{\beta^{2}(t-1)}{2(q-1)^{2}}+\frac{(1-\mu^{2})t(t-1)}{2(q-t)^{2}}\right]\,.$

Thus, the quantity $q(t)$ satisfies (5) with parameters
$(\delta_{1},\delta_{2},\delta_{3},\delta_{4})=(\frac{1}{2},-\frac{1}{2}\alpha^{2},\frac{1}{2}\beta^{2},\frac{1}{2}(1-\mu^{2}))$.
Correspondingly, the parameters $v_{k}$ in the Hamiltonian (6) are such that

$(v_{1}-v_{2})^{2}=1\,,\;(v_{3}+v_{4})^{2}=\alpha^{2}\,,\;(v_{3}-v_{4})^{2}=\beta^{2}\,,\;(1-v_{1}-v_{2})^{2}=\mu^{2}\,,$

which reads as (11). From the first of the Hamiltonian equations (7) combined
with (116) and taking into account that
$\xi_{n}/\nu_{n}=-q\tilde{l}_{n,1}-\tilde{l}_{n,0}$, we deduce the equation
for $p(t)$ as given by (13). ∎

It remains to specify the initial conditions for the differential equation to
uniquely characterize the quantity $q(t)$. This will be done in the next
subsection.

####  3.3.1. Boundary Values for the Laguerre-Hahn P${}_{\textrm{VI}}$
Solution

We first begin by establishing the initial conditions of the differential
equation satisfied by $q$.

######  Lemma 6.

The function $q(t)$ defined by (12), solution of the differential equation (5)
with parameters
$(\delta_{1},\delta_{2},\delta_{3},\delta_{4})=(\frac{1}{2},-\frac{1}{2}\alpha^{2},\frac{1}{2}\beta^{2},\frac{1}{2}(1-\mu^{2}))$,
is made unique via the specification of $\beta_{0}$ and $\tilde{\beta}_{0}$ in
the equations

(123) $\displaystyle q(0)$
$\displaystyle=-\left((2+\alpha+\beta+\mu)\tilde{\beta}_{0}(0)+2\beta_{0}(0)-3-\alpha-\mu\right)/\nu_{n}\,,$
(124) $\displaystyle q(1)$
$\displaystyle=-\left((2+\alpha+\beta+\mu)\tilde{\beta}_{0}(1)+2\beta_{0}(1)-\nu_{n}-3-\alpha\right)/\nu_{n}\,,$

as well as in

(125) $\displaystyle q^{\prime}(0)$
$\displaystyle=-\left(-g_{n}(0)+(2+\alpha+\beta+\mu)\tilde{\beta}^{\prime}_{0}(0)+2{\beta}^{\prime}_{0}(0)-\nu_{n}+\mu\right)/\nu_{n}\,,$
(126) $\displaystyle q^{\prime}(1)$
$\displaystyle=-\left(g_{n}(1)+(2+\alpha+\beta+\mu)\tilde{\beta}^{\prime}_{0}(1)+2{\beta}^{\prime}_{0}(1)-\nu_{n}+\mu\right)/\nu_{n}\,,$

where $g_{n}$ is defined in terms of the moments in the asymptotic expansion
(61) by

(127)
$g_{n}(t)=\left(\ln\left(\tilde{\Delta}_{n+2}/\tilde{\Delta}_{n+1}\right)-\ln\tilde{w}_{0}\right)^{\prime}\,,$

where $\tilde{\Delta}_{n}:=\det[\tilde{w}_{j+k-2}]_{j,k=1,\dots,n}\,,$ with
$\tilde{w}_{n}$ defined in terms of the moments of the weight (9) via relation
(62).

###### Proof.

Taking $t=t_{0}$ where $t_{0}=0$ or $t_{0}=1$ in (96) we get, respectively,

$-2+(\nu_{n-1}+1)\tilde{\beta}_{n}(t_{0})-(\nu_{n-1}-3)\tilde{\beta}_{n-1}(t_{0})=0\,.$

Using using $\nu_{n-1}-3=\nu_{n-2}-1,\;n\geq 1,$ and summing from one to $n+1$
we get

$\nu_{n}\tilde{\beta}_{n+1}(t_{0})-\nu_{-1}\tilde{\beta}_{0}(t_{0})+2\sum_{k=1}^{n}\tilde{\beta}_{k}(t_{0})+\tilde{\beta}_{n+1}(t_{0})+\tilde{\beta}_{0}(t_{0})=2n+2\,.$

Therefore, the quantity $\vartheta_{n}$ given by (87) satisfies (123)-(124).

Taking derivatives in (96) and using $\nu_{n-1}-3=\nu_{n-2}-1,\;n\geq 1,$ we
get

$(2t-1)\partial_{t}\ln\tilde{\gamma}_{n}+t(t-1)\partial_{t}^{2}\ln\tilde{\gamma}_{n}=\nu_{n-1}\tilde{\beta}^{\prime}_{n}-\nu_{n-2}\tilde{\beta}^{\prime}_{n-1}+\tilde{\beta}^{\prime}_{n}+\tilde{\beta}^{\prime}_{n-1}\,.$

Summing from one to $n+1$ and using the telescopic sum we get, respectively,

$\displaystyle-\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}(0)$
$\displaystyle=\nu_{n}\tilde{\beta}^{\prime}_{n+1}(0)-\nu_{-1}\tilde{\beta}^{\prime}_{0}(0)+2\sum_{k=1}^{n}\tilde{\beta}^{\prime}_{k}(0)+\tilde{\beta}^{\prime}_{0}(0)+\tilde{\beta}^{\prime}_{n+1}(0)\,,$
$\displaystyle\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}(1)$
$\displaystyle=\nu_{n}\tilde{\beta}^{\prime}_{n+1}(1)-\nu_{-1}\tilde{\beta}^{\prime}_{0}(1)+2\sum_{k=1}^{n}\tilde{\beta}^{\prime}_{k}(1)+\tilde{\beta}^{\prime}_{0}(1)+\tilde{\beta}^{\prime}_{n+1}(1)\,.$

Therefore, the quantity $\vartheta_{n}$ satisfies

(128) $\displaystyle\vartheta^{\prime}_{n}(0)$
$\displaystyle=-\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}(0)+(2+\alpha+\beta+\mu)\tilde{\beta}^{\prime}_{0}(0)+2{\beta}^{\prime}_{0}(0)-\nu_{n}+\mu\,,$
(129) $\displaystyle\vartheta^{\prime}_{n}(1)$
$\displaystyle=\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}(1)+(2+\alpha+\beta+\mu)\tilde{\beta}^{\prime}_{0}(1)+2{\beta}^{\prime}_{0}(1)-\nu_{n}+\mu\,.$

It remains to compute $\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}$.
Recalling (18), we now have

$\ln\tilde{\gamma}_{k}=\ln\tilde{h}_{k}-\ln\tilde{h}_{k-1}\,,$

thus, using the telescopic sum, we get

$\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}=\partial_{t}\left(\ln\tilde{h}_{n+1}-\ln\tilde{h}_{0}\right)\,,$

thus, in terms of the moments in the asymptotic expansion (61) of $\tilde{f}$,
we have

(130)
$\sum_{k=1}^{n+1}\partial_{t}\ln\tilde{\gamma}_{k}=\partial_{t}\left(\ln\left(\tilde{\Delta}_{n+2}/\tilde{\Delta}_{n+1}\right)-\ln\tilde{w}_{0}\right)\,,$

where $\tilde{\Delta}_{n}$ is the corresponding Hankel determinant, given by
$\tilde{\Delta}_{n}=\det[\tilde{w}_{j+k-2}]_{j,k=1,\dots,n}\,$. Therefore, in
the account of (128)-(130) we obtain (125)-(126). ∎

Furthermore, $\tilde{w}_{0}$ and $\tilde{\beta}_{0}$ may be expressed in terms
of the parameters $\beta_{0},w_{0}$, as follows.

######  Lemma 7.

The following relations hold:

(131)
$\displaystyle\tilde{w}_{0}=-\beta_{0}^{2}+\frac{(2+\alpha+\beta+\mu+(2+\alpha+\beta)t)\beta_{0}-(1+\alpha)t}{3+\alpha+\beta+\mu}\,,$
(132)
$\displaystyle\frac{(\tilde{\beta}_{0}-t)(\beta_{0}-t)\beta^{\prime}_{0}}{(\beta_{0}-t)(\ln
w_{0})^{\prime}-\beta^{\prime}_{0}-\tilde{\beta}^{\prime}_{0}}=\beta_{0}^{2}-\frac{(2+\alpha+\beta+\mu+(2+\alpha+\beta)t)\beta_{0}-(1+\alpha)t}{3+\alpha+\beta+\mu}\,.$

###### Proof.

Equation (131) follows from the two expressions for the linear term of the
polynomial $\tilde{D}$ in (67), say $d_{1,1}$: on the one hand, the expression
that is obtained by using the asymptotic expansion (61) in (63) gives

$d_{1,1}=-(3+\alpha+\beta+\mu)\tilde{w}_{0}\,;$

whilst on the other hand, from (67) we have

$d_{1,1}=(3+\alpha+\beta+\mu)\beta_{0}^{2}-(2+\alpha+\mu+(2+\alpha+\beta)t)\beta_{0}+(1+\alpha)t\,.$

The coefficient of $x^{n}$ in the equations in the $(1,2)$ component of (74)
gives us, by virtue of (90),

$\tilde{w}_{0}=\frac{[({\beta}_{0}-t)(\ln
w_{0})^{\prime}+\mu]({\beta}_{0}-t)(\tilde{\beta}_{0}-t)}{2(\beta_{0}-t)(\ln
w_{0})^{\prime}+\mu-\tilde{\beta}^{\prime}_{0}}\,.$

Using (56) in the previous equation we get

(133)
$\tilde{w}_{0}=-\frac{(\tilde{\beta}_{0}-t)(\beta_{0}-t)\beta^{\prime}_{0}}{(\beta_{0}-t)(\ln
w_{0})^{\prime}-\beta^{\prime}_{0}-\tilde{\beta}^{\prime}_{0}}\,.$

The equation (132) is then a consequence of (133) and (131). ∎

####  3.3.2. Relation with the Semi-classical Case

Now that the Painlevé equation P${}_{\textrm{VI}}$ for the sequences of
orthogonal polynomials corresponding to $\tilde{f}$ was derived, and given
that $\tilde{f}$ is a Möbius transformation of $f$ (see (8)),

$\tilde{f}(x)=x-\beta_{0}-\frac{w_{0}}{f(x)}\,,$

a natural question that can be posed is to relate the P${}_{\textrm{VI}}$
solution just derived with the Painlevé solution that governs the sequences of
polynomials corresponding to $f$. Recall that $f$ is the Stieltjes transform
of the modified Jacobi weight $w$ defined by (9). The sequences of orthogonal
polynomials corresponding to $w$, indeed taken orthonormal,
$\\{p_{n}\\}_{n\geq 0}$, were subject of analysis, e.g., in [31]. In [31, pp.
227-228] it is deduced that the quantity $q$ defined by the only zero of the
$(1,2)$ component of the matrix of the differential system

$\frac{d}{dx}\left[\begin{array}[]{cc}p_{n}&\varepsilon_{n+1}/w\\\
p_{n-1}&\varepsilon_{n}/w\end{array}\right]=\frac{1}{A}\begin{bmatrix}\Omega_{n}-C/2&-a_{n}\Theta_{n}\\\
a_{n}\Theta_{n-1}&\;-\Omega_{n}-C/2\end{bmatrix}\left[\begin{array}[]{cc}P_{n+1}&\varepsilon_{n+1}/w\\\
P_{n}&\varepsilon_{n}/w\end{array}\right]\,$

(here, we are using the notation $A,C$ for the polynomials $W,V$ of [31],
respectively, and $a_{n}$ are the recurrence relation coefficients defined by
$a_{n}^{2}=\gamma_{n}$), satisfies a P${}_{\textrm{VI}}$ equation (5) with
parameters
$(\delta_{1},\delta_{2},\delta_{3},\delta_{4})=(\frac{1}{2},-\frac{1}{2}\alpha^{2},\frac{1}{2}\beta^{2},\frac{1}{2}(1-\mu^{2})).$
Hence, the Laguerre-Hahn situation versus semi-classical differs in the
boundary conditions. On this topic we refer the interested reader to [19, Sec.
2], and the techniques used in the proof of Proposition 2.1, pages
12219-12220, relying in certain identities resulting from expansions of
Toeplitz determinants.

##  4\. Conclusions

We have demonstrated with an important example that the natural deformation of
a Laguerre-Hahn class system, which is directly related to the deformed Jacobi
weight, is governed by solutions to the sixth Painlevé equation with a
parameter (or monodromy exponents) set identical to the solution for this
semi-classical weight. The two solutions only differ in their boundary data,
or equivalently the off-diagonal elements of their connection matrices.

However there is an interesting question posed by our result. We have only an
implicit characterisation of this system by the fact that its Stieltjes
transform satisfies a certain Ricatti differential equation - we have not
given an explicit solution for this transform nor for the weight itself. The
construction of the transform or better still that of the weight, even in
special cases, would be a major achievement. And we would speculate that this
problem bears some relation to the problem of constructing the weight for the
associated Jacobi polynomials, a task performed by Wimp [53], via a
deformation of this situation.

Interestingly there is an apparent connection of this theme with the diagonal
spin-spin correlations of the Ising model on the anisotropic square lattice.
These are known to be characterised by a particular solution of PVI, through
the work of Jimbo and Miwa [24], and are also related to the Picard solution.
The correlations can be written as Toeplitz determinants of the first, second
and third kinds of Elliptic integrals with the deformation variable $t$
corresponding to the temperature, and these determinants possess a simple
algebraic symbol and are therefore semi-classical - see [34]. An observation
made by Mangazeev and Guttmann [33] was that these solutions have a one
parameter generalisation (denoted by $\lambda$) which enters only through the
boundary conditions. Using the Toda lattice equations appropriate to PVI they
observed that for explicit examples of small spin separations they could
evaluate these as Jacobi Elliptic functions for general $t$ and $\lambda$,
with the value $\lambda=1$ recovering the standard model. Furthermore, it has
been shown that the deformation differential equation for the Stieltjes
function has the Riccati form with the quadratic term only if $\lambda\neq 1$.
One of the authors (NSW) has also confirmed that these explicit examples
satisfy the other standard differential-difference relations of PVI. It is
also known that these solutions have a perturbation expansion about
$\lambda=0$ which coincides with the form-factor expansion of the
correlations. However it is not known what symbol or weight yields these
$\lambda$-correlations, nor the general formula for these at arbitrary
separation.

## Acknowledgments

The work of MNR is partially supported by CMA-UBI (grants UID-B-MAT/00212/2020
and UID-P-MAT/00212/2020), funded by FCT. The work of MNR is also partially
supported by Dirección General de Investigación e Innovación, Consejería de
Educación e Investigación of the Comunidad de Madrid (Spain), and Universidad
de Alcalá under grant CM/JIN/2021-014, Proyectos de I+D para Jóvenes
Investigadores de la Universidad de Alcalá 2021. NSW would like to acknowledge
the support of grant UID/MAT/00324/2019 provided by FCT/Portugal during the
visit to CMUC, Universidade de Coimbra, in October-November 2019.

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