Lowering and raising operators for some special\\ orthogonal polynomials
Tom H. Koornwinder
This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system $BC_2$ a new pair of lowering and raising operators is obtained.
math0505378.tex
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\usepackage{amssymb}
\newcommand\CC{\mathbb{C}}
\newcommand\RR{\mathbb{R}}
\newcommand\ZZ{\mathbb{Z}}
\newcommand\FSD{\mathcal{D}}
\newcommand\al\alpha
\newcommand\be\beta
\newcommand\ga\gamma
\newcommand\tha\theta
\newcommand\la\lambda
\newcommand\Om\Omega
\newcommand\half{\frac12}
\newcommand\iy\infty
\newcommand\pa\partial
\newcommand\thalf{\textstyle\half}
\newcommand\LHS{left-hand side}
\newcommand\RHS{right-hand side}
\newcommand\const{\mbox{const.}\,}
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  \genfrac{}{}{0pt}{}{#3}{#4};#5\right)}
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\begin{document}
%
\title[Lowering and raising operators]
{Lowering and raising operators for some special\\
orthogonal polynomials}
%
%    Information for first author
\author{Tom H. Koornwinder}
%    Address of record for the research reported here
\address{Korteweg-de Vries Institute, University of Amsterdam,
Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands}
%    Current address
%\curraddr{}
\email{thk@science.uva.nl}
%    \thanks will become a 1st page footnote.
%\thanks{}
%
%    General info
\subjclass{Primary 33C45, 33D45, 33C52}
\date{February 21, 2005.}
%
\dedicatory{This paper is dedicated to Ian Macdonald on the occasion of his
75th birthday.}
%
\keywords{lowering and raising operators, continuous q-ultraspherical
polynomials, ultraspherical polynomials, Jacobi polynomials,
Jacobi polynomials for root system $BC_2$}
%
\begin{abstract}
This paper discusses operators lowering or raising the degree
but preserving the parameters of special orthogonal
polynomials. Results for one-variable classical (q-)orthogonal
polynomials are surveyed. For Jacobi polynomials associated with
root system $BC_2$ a new pair of lowering and raising operators
is obtained.
\end{abstract}
%
\maketitle
%
%until sec:06
\section{Introduction} 
\label{sec:05}
Kirillov and Noumi \cite{10} gave explicit $q$-difference lowering and raising
operators for $A_{n-1}$ type Macdonald polynomials
$J_\la(x;q,t)=c_\la(q,t) P_\la(x;q,t)$ (see \cite[(VI.8.3)]{11}).
These operators don't change the parameter $t$, they only lower or
raise $\la$. This is quite different from
Opdam's \cite{05} shift operators acting on Jacobi polynomials associated
with root systems, which do change parameters.
In \cite[Remark 5.3]{10} the interesting problem is
mentioned to find lowering and raising operators for
Macdonald-Koornwinder polynomials (see \cite{12}). As far as we know,
such operators have not yet been given in literature until now,
and neither in the corresponding $q=1$ case of $BC_n$ type
Jacobi polynomials (see \cite{03}, \cite{04}).

The present paper makes only a minor step in the direction of a general
answer to the problem raised in \cite[Remark 5.3]{10}. In the $BC_2$,
$q=1$ case, for parameters $(\al,\be,\ga)$ with $\al=\be$,
and for partition $\la=(n,0)$ of length 1, a raising and lowering operator
in explicit form are obtained (sections \ref{sec:03} and
\ref{sec:04}).
In the earlier sections \ref{sec:06}, \ref{sec:02}, \ref{sec:01}
lowering and raising
formulas in the rank 1 case
(continuous $q$-ultraspherical, ultraspherical and Jacobi polynomials,
but not yet Askey-Wilson polynomials) are discussed.
Some formulas scattered in the literature are brought here together,
and also the explicit specialization of the Kirillov-Noumi operators
to the $A_1$ case is given. Some of the formulas in
sections \ref{sec:06}--\ref{sec:01}
may be new in one-variable special function theory.
%
\subsubsection*{Acknowledgement}
I thank Hechun Zhang (Tsinghua University, Bejing) for helpful
discussions in an early stage of writing this manuscript.
%
\section{Lowering and raising continuous $q$-ultraspherical polynomials}
\label{sec:06}
%until 06.16
See \cite{14} or \cite[\S3.10.1]{13} for the definition of the
\textit{continuous $q$-ultraspherical polynomials} $C_n(x;t|q)$.
A pleasant explicit formula for them is as a finite Fourier series:
\begin{equation}
C_n(\cos\tha;t|q)=\sum_{k=0}^n \frac{(t;q)_k\,(t;q)_{n-k}}
{(q;q)_k\,(q;q)_{n-k}}\,e^{i(n-2k)\tha}.
\label{eq:06.15}
\end{equation}
(Throughout see \cite{15} for definition of ($q$-)hypergeometric series
and ($q$-)Pochhammer symbols.)
The $A_1$ type Macdonald polynomials can be expressed in terms
of continuous $q$-ultraspherical polynomials:
\begin{equation}
P_{m,n}(x,y;q,t)=
\frac{(q;q)_{m-n}}{(t;q)_{m-n}}\,(xy)^{\half(m+n)}\,
C_{m-n}\biggl(\frac{x+y}{2(xy)^\half};t|q\biggr)\quad
(m\ge n\ge0).
\end{equation}
For the renormalized polynomials we have
\begin{multline*}
J_{m,n}(x,y;q,t)=c_{m,n}(q,t) P_{m,n}(x,y;q,t),\\
\mbox{where}\quad c_{m,n}(q,t)=(t^2 q^{m-n};q)_n\,(t;q)_{m-n}\,(t;q)_n,
\quad\mbox{so:}
\end{multline*}
\begin{multline}
J_{m,n}(x,y;q,t)=(t^2 q^{m-n};q)_n\,(t;q)_n\,(q;q)_{m-n}\\
\times(xy)^{\half(m+n)}\,
C_{m-n}\biggl(\frac{x+y}{2(xy)^\half};t|q\biggr)
\quad(m\ge n\ge0).
\end{multline}
In particular, if $(m,n)$ is replaced by $(n,0)$, and
if we use \eqref{eq:06.15}:
\begin{align}
J_{n,0}(x,y;q,t)&=(q;q)_n\,(xy)^{\half n}\,
C_n\biggl(\frac{x+y}{2(xy)^\half};t|q\biggr)
\label{eq:06.03}\\
&=(q;q)_n\,\sum_{k=0}^n
\frac{(t;q)_k\,(t;q)_{n-k}}{(q;q)_k\,(q;q)_{n-k}}\,x^{n-k}y^k.
\label{eq:06.14}
\end{align}

Let us consider the Kirillov-Noumi operators \cite{10} in the two-variable
case while acting on the polynomials $J_{n,0}$, raising or lowering $n$.
There are two variants $K_1^+$ and $K_1^-$ of the raising operator,
and two variants $M_1^+$ and $M_1^-$ of the lowering operator.
The expressions for these operators will involve the operators $T_{q,x}$ and
$T_{q,y}$:
\begin{equation*}
(T_{q,x}f)(x,y):=f(qx,y),\quad
(T_{q,y}f)(x,y):=f(x,qy).
\end{equation*}
Then $K_1^\pm$ and $M_1^\pm$ and their actions are given by:
\begin{align*}
K_1^+&:=x\left(1-\frac{tx-y}{x-y}\,T_{q,x}\right)
+y\left(1-\frac{ty-x}{y-x}\,T_{q,y}\right),\\
K_1^-&:=x\left(-t T_{q,x} T_{q,y}+\frac{x-ty}{x-y}\,T_{q,y}\right)+
y\left(-t T_{q,x} T_{q,y}+\frac{y-tx}{y-x}\,T_{q,x}\right),
\end{align*}
\begin{equation}
K_1^\pm\,J_{n,0}(x,y;q,t)=J_{n+1,0}(x,y;q,t);
\label{eq:06.01}
\end{equation}
\begin{align*}
M_1^+&:=\frac1x\left(1-\frac{tx-y}{x-y}\,T_{q,x}\right)
+\frac1y\left(1-\frac{ty-x}{y-x}\,T_{q,y}\right),\\
M_1^-&:=\frac1x\left(-t T_{q,x} T_{q,y}+\frac{x-ty}{x-y}\,T_{q,y}\right)+
\frac1y\left(-t T_{q,x} T_{q,y}+\frac{y-tx}{y-x}\,T_{q,x}\right),
\end{align*}
\begin{equation}
M_1^\pm\,J_{n,0}(x,y;q,t)=(1-q^n)(1-t^2q^{n-1})\,J_{n-1,0}(x,y;q,t).
\label{eq:06.02}
\end{equation}
For the moment I will take \eqref{eq:06.01} and \eqref{eq:06.02} for
granted from \cite{10}. At the end of this section I will prove these
formulas independently.

We can write \eqref{eq:06.01} and \eqref{eq:06.02} more explicitly
by substitution of
\eqref{eq:06.03} and the explicit expressions for the operators
$K_1^\pm$ and $M_1^\pm$. Then put $x=z$, $y=z^{-1}$. In the resulting
formulas it is
convenient to assume $t,q$ fixed and to use the notation
\begin{equation*}
C_n[z]:=C_n(\thalf(z+z^{-1});t|q),\qquad
TC_n[z]:=C_n[q^\half z],\qquad
T^{-1}C_n[z]:=C_n[q^{-\half} z].
\end{equation*}
We obtain:
\begin{multline}
-\frac{tz^2-1}{z-z^{-1}}\,TC_n[z]+
\frac{tz^{-2}-1}{z-z^{-1}}\,T^{-1}C_n[z]+
q^{-\half n}(z+z^{-1})\,C_n[z]\\
=(q^{-\half n}-q^{\half n+1})\,C_{n+1}[z],
\label{eq:06.04}
\end{multline}
%
\begin{multline}
-\frac{z^{-2}-t}{z-z^{-1}}\,TC_n[z]+
\frac{z^2-t}{z-z^{-1}}\,T^{-1}C_n[z]-
q^{\half n}t(z+z^{-1})\,C_n[z]\\
=(q^{-\half n}-q^{\half n+1})\,C_{n+1}[z],
\label{eq:06.05}
\end{multline}
%
\begin{multline}
-\frac{t-z^{-2}}{z-z^{-1}}\,TC_n[z]+
\frac{t-z^2}{z-z^{-1}}\,T^{-1}C_n[z]+
q^{-\half n}(z+z^{-1})\,C_n[z]\\
=(q^{-\half n}-t^2q^{\half n-1})\,C_{n-1}[z],
\label{eq:06.06}
\end{multline}
%
\begin{multline}
-\frac{1-tz^2}{z-z^{-1}}\,TC_n[z]+
\frac{1-tz^{-2}}{z-z^{-1}}\,T^{-1}C_n[z]-
q^{\half n}t(z+z^{-1})\,C_n[z]\\
=(q^{-\half n}-t^2q^{\half n-1})\,C_{n-1}[z].
\label{eq:06.07}
\end{multline}

If we subtract \eqref{eq:06.05} from \eqref{eq:06.04} or
\eqref{eq:06.07} from \eqref{eq:06.06}, and if we divide the resulting
second order $q^\half$-difference formula by a suitable factor which
all terms have in common, then we obtain
\begin{equation}
\frac{1-tz^2}{1-z^2}\,TC_n[z]+
\frac{1-tz^{-2}}{1-z^{-2}}\,T^{-1}C_n[z]-
(q^{-\half n}+q^{\half n} t)\,C_n[z]=0,
\label{eq:06.08}
\end{equation}
which is a known formula for continuous
$q$-ultraspherical polynomials.
Indeed, rewrite continuous $q$-ultraspherical polynomials
as Askey-Wilson polynomials by
\begin{multline*}
C_n(x;q^{\al+\half}|q)=
\const P_n^{(\al,\al)}(x;q^\half)\\
=\const p_n(x;q^{\frac14},q^{\half\al+\frac14},-q^{\half\al+\frac14},
-q^{\frac14}|q^\half),
\end{multline*}
(see \cite[(7.5.34), (7.5.25), (7.5.1)]{15}), and then use the
second order $q$-difference formula
\cite[(3.1.7)]{13}.
Thus \eqref{eq:06.04} is equivalent with \eqref{eq:06.05} modulo
\eqref{eq:06.08}, and similarly for \eqref{eq:06.06} and \eqref{eq:06.07}.

In addition to the operators $K^\pm$ and $L^\pm$ we introduce the operators
$A$ and $\Om$ given by:
\begin{equation*}
A:=T_{q,x}\,T_{q,y},\qquad
\Om:=\frac1{xy}\,
\left(1-\frac{tx-y}{x-y}\,T_{q,x}
-\frac{x-ty}{x-y}T_{q,y}+t\,T_{q,x}\,T_{q,y}
\right).
\end{equation*}
Since
\begin{equation*}
K_1^+-K_1^-=xy(x+y)\,\Om,\qquad
M_1^+-M_1^-=(x+y)\,\Om,
\end{equation*}
we will no longer consider $K_1^-$ and $M_1^-$, but we will concentrate
on $K_1^+$, $M_1^+$, $\Om$ and $A$. We can derive relations
\begin{equation}
A\,\Om=q^{-2}\,\Om\,A,
\qquad
A\,K_1^+=q\,K_1^+\,A,
\qquad
A\,M_1^+=\,q^{-1}\,M_1^+\,A,
\label{eq:06:16}
\end{equation}
\begin{equation*}
\Om\,K_1^+=q^2\,K_1^+\,\Om+(1-q)^2(x+y)\,\Om,
\qquad
\Om\,M_1^+=M_1^+\,\Om,
\end{equation*}
\begin{equation*}
q^2\,K_1^+\,M_1^+-M_1^+\,K_1^+=(q^2-1)+(1-q)(q+t^2)\,A
+(q^2-1)(x^2+xy+y^2)\,\Om,
\end{equation*}
%

A straightforward computation shows that the operators
$(x-y)\,\Om$, $(x-y)\,K_1^+$ and $(x-y)\,M_1^+$ send $x^my^n+x^ny^m$
to an antisymmetric polynomial in $x$ and $y$.
Therefore $\Om$, $K_1^\pm$, $M_1^\pm$ and (clearly) $A$ act on the space
of symmetric polynomials in $x$ and $y$. By the above relations,
the operators $K_1^+$, $M_1^+$ and $A$ also act on the subspace
of symmetric polynomials annihilated by $\Om$. The operators $K_1^+$,
$M_1^+$ and $A$ restricted to this subspace satisfy the relations
\begin{equation}
q^2\,K_1^+\,M_1^+-M_1^+\,K_1^+=(q^2-1)+(1-q)(q+t^2)\,A,
\label{eq:06.09}
\end{equation}
\begin{equation}
A\,K_1^+=q\,K_1^+\,A,
\qquad
A\,M^+=\,q^{-1}\,M_1^+\,A.
\label{eq:06.10}
\end{equation}

It does not seem that the relations \eqref{eq:06.09} and
\eqref{eq:06.10} are equivalent to the familiar relations
\begin{equation}
EF-FE=\frac{K-K^{-1}}{q-q^{-1}}\,,\qquad
KE=q^2 EK,\qquad
KF=q^{-2}FK,
\label{eq:06.11}
\end{equation}
for the
generators of $U_q(sl(2))$ as given, for instance, in
\cite[Definition VI.1.1]{16}.
Indeed, relations \eqref{eq:06.09} and \eqref{eq:06.10}
take after rescaling of the generators the form
\begin{equation}
q^2\,K_1^+\,M_1^+-M_1^+\,K_1^+=A-1,
\qquad
A\,K_1^+=q\,K_1^+\,A,
\qquad
A\,M^+=\,q^{-1}\,M_1^+\,A.
\label{eq:06.12}
\end{equation}
while relations \eqref{eq:06.11}, after substitution of $\tilde E:=EK$
and after rescaling,
become
\begin{equation}
q^2\tilde EF-F\tilde E=K^2-1,\qquad
K\tilde E=q^2 \tilde EK,\qquad
KF=q^{-2}FK.
\label{eq:06.13}
\end{equation}
Relations \eqref{eq:06.12} would match with relations \eqref{eq:06.13}
if the first relation in \eqref{eq:06.12} would have been
$q^\half K_1^+\,M_1^+-M_1^+\,K_1^+=A-1$.

Now I will give the promised independent proof of
\eqref{eq:06.01} and \eqref{eq:06.02}.
Because of the first relation in \eqref{eq:06:16},
the symmetric polynomials annihilated by $\Om$ have a basis of
homogeneous polynomials.
From $\Om\left(\sum_{k=0}^n c_k(x^{n-k}y^k+x^ky^{n-k})\right)=0$ with
$c_k=c_{n-k}$
one derives a recurrence relation for the $c_k$ which, on comparison
with \eqref{eq:06.14}, shows that the polynomials
$J_{n,0}(x,y;q,t)$ ($n\in\Znonneg$) given by \eqref{eq:06.03} span the
space of symmetric polynomials annihilated by $\Om$.
Thus, because $K_1^+$ resp.\ $M_1^+$ raise resp.\ lower the degree of
a homogeneous symmetric polynomial by 1, we find
\eqref{eq:06.01} for $K_1^+$ and \eqref{eq:06.02} for $M_1^+$
up to a constant factor. These constant factors are then obtained
by comparing terms of highest degree on the left and on the right.


\section{Lowering and raising ultraspherical polynomials}
\label{sec:02}
%until 02.16
%
The \emph{ultraspherical polynomials} $C_n^{(\la)}(x)$
(see for instance \cite{01} or \cite{13})
can be obtained from
\eqref{eq:06.15} by putting $t=q^\la$ and letting $q\uparrow1$:
\begin{equation*}
C_n^{(\la)}(\cos\tha)=\sum_{k=0}^n \frac{(\la)_k\,(\la)_{n-k}}
{k!\,(n-k)!}\,e^{i(n-2k)\tha}.
\end{equation*}
They are special cases of Jacobi polynomials (see \eqref{eq:01.13}):
\begin{equation*}
C_n^{(\la)}(x)=\frac{(2\la)_n}{(\la+\thalf)_n}\,P_n^{(\la-\half,\la-\half)}(x).
\end{equation*}

In principle, one could take limits for $q\uparrow1$
of all formulas in \S\ref{sec:06}, but I prefer to present the
results on lowering and raising operators for ultraspherical polynomials,
more classical than the results in the $q$-case, here independently from
\S\ref{sec:06}. Lowering and raising formulas can be given in three
different forms.

The first form (see \cite[10.9(15)]{01}) is:
%
\begin{align}
\left((1-x^2)\frac d{dx}+nx\right) C_n^{(\la)}(x)=&
(n+2\la-1)\,C_{n-1}^{(\la)}(x),
\label{eq:02.06}\\
\left((1-x^2)\frac d{dx}-(n-1+2\la)x\right) C_{n-1}^{(\la)}(x)=&
-n\,C_n^{(\la)}(x).
\label{eq:02.07}
\end{align}
%
Note that substitution of \eqref{eq:02.06} into \eqref{eq:02.07}
causes $n$ to drop out from the terms with derivatives. There results
$(1-x^2)$ times the second order differential equation
\cite[10.9(14)]{01} for $C_n^{(\la)}(x)$.
Also, if $n$ is replaced by $n+1$ in \eqref{eq:02.07} and if
the term with first derivative is eliminated from the
resulting equation together with \eqref{eq:02.06}, then we obtain
the three-term recurrence relation \cite[10.9(13)]{01} for
$C_n^{(\la)}(x)$.

We get a second form of lowering and raising formulas by rewriting
\eqref{eq:02.06} and \eqref{eq:02.07} into an equivalent form:
\begin{equation}
\frac d{dx}\left((1+x^2)^{\half n}\,
C_n^{(\la)}\left(\frac x{\sqrt{1+x^2}}\right) \right)
=(n+2\la-1)\,(1+x^2)^{\half(n-1)}\,
C_{n-1}^{(\la)}\left(\frac x{\sqrt{1+x^2}}\right),
\label{eq:02.01}
\end{equation}
%
\begin{equation}
\frac d{dx}\left((1+x^2)^{-\half(n-1)-\la}\,
C_{n-1}^{(\la)}\left(\frac x{\sqrt{1+x^2}}\right) \right)
=-n\,(1+x^2)^{-\half n-\la}\,
C_n^{(\la)}\left(\frac x{\sqrt{1+x^2}}\right).
\label{eq:02.02}
\end{equation}
%
Iteration of \eqref{eq:02.02} yields the Rodrigues type formula
%
\begin{equation}
C_n^{(\la)}\left(\frac x{\sqrt{1+x^2}}\right)=
\frac{(-1)^n}{n!}\,(1+x^2)^{\half n+\la}\,
\frac{d^n}{dx^n}\left((1+x^2)^{-\la}\right).
\label{eq:02.03}
\end{equation}
%
A formula equivalent to \eqref{eq:02.03} (by analytic continuation)
is given in \cite[10.9(37)]{01}, where the formula is ascribed
to F. Tricomi, Ann.\ Mat.\ Pura Appl.\ (4) \textbf{28} (1949), 283--300
(but I could not find the formula there).

Tranformation of the generating function \cite[10.9(29)]{01} for
$C_n^{(\la)}(x)$ yields
%
\begin{multline*}
\frac 1{(1+(x-z)^2)^\la}=
\sum_{n=0}^\iy (1+x^2)^{-\half n-\la}\,
C_n^{(\la)}\left(\frac x{\sqrt{1+x^2}}\right)z^n\\
(z\in\CC,\;x\in\RR,\;|z|<\sqrt{1+x^2}).
\end{multline*}
%
Then \eqref{eq:02.03} follows by considering Taylor coefficients
in the above formula.

We obtain a third form of lowering and raising operators by
rewriting \eqref{eq:02.06} and \eqref{eq:02.07}
in an equivalent form as follows:
%
\begin{multline}
\left(\frac\pa{\pa x}+\frac\pa{\pa y}\right)
\left((xy)^{\half n}\,
C_n^{(\la)}\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right)\right)\\
=(n+2\la-1)\,(xy)^{(\half(n-1)}\,
C_{n-1}^{(\la)}\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right),
\label{eq:02.10}
\end{multline}
\begin{multline}
\left(x^2\frac\pa{\pa x}+y^2\frac\pa{\pa y}+\la(x+y)\right)
\left((xy)^{\half(n-1)}\,
C_{n-1}^{(\la)}\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right)\right)\\
=n\,(xy)^{\half n}\,
C_n^{(\la)}\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right).
\label{eq:02.11}
\end{multline}
%
Iteration of \eqref{eq:02.11} yields the Rodrigues type formula
%
\begin{equation}
(xy)^{\half n}\,
C_n^{(\la)}\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right)
=\frac 1{n!}\left(x^2\frac\pa{\pa x}+y^2\frac\pa{\pa y}+\la(x+y)\right)^n(1).
\label{eq:02.16}
\end{equation}

Formulas \eqref{eq:02.10} and \eqref{eq:02.11}
may be rewritten in terms of the following
special \emph{Jack polynomials} in
two variables:
\begin{equation*}
J_{n,0}^{1/\la}(x,y)=\frac{(\la)_n}{\la^n}P_{n,0}^{1/\la}(x,y)=
\frac{n!}{\la^n}\,(xy)^{\half n}\,
C_n^\la\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right)
\end{equation*}
(see also \eqref{eq:03.06}, \eqref{eq:03.07}).
Thus \eqref{eq:02.10} and \eqref{eq:02.11} can be seen to be special cases
of formualas (5.14) resp.\ (2.16) in \cite{10} (there put
$n=2$, $m=1$).

Formulas \eqref{eq:02.10} and \eqref{eq:02.11} are realizations of a
representation of the Lie algebra $sl(2)$. Indeed, put
%
\begin{equation*}
H:=2\left( x\frac\pa{\pa x}+y\frac\pa{\pa y}+\la\right),\;\;
E:=x^2\frac\pa{\pa x}+y^2\frac\pa{\pa y}+\la(x+y),\;\;
F:=-\left(\frac\pa{\pa x}+\frac\pa{\pa y}\right),
\end{equation*}
\begin{equation*}
f_n(x,y):=(xy)^{\half n}\,
C_n^{(\la)}\left(\thalf\bigl((x/y)^\half+(y/x)^\half\bigr)\right).
\end{equation*}
%
Then
\begin{align*}
&[H,E]=2E,\quad
[H,F]=-2F,\quad
[E,F]=H,\\
&Ff_n=-(n+2\la-1) f_{n-1},\quad
Ef_{n-1}=n f_n,\quad
Hf_n=2(n+\la) f_n.
\end{align*}
%
\section{Lowering and raising Jacobi polynomials}
\label{sec:01}
%
% until 01.13
%
\emph{Jacobi polynomials} $P_n^{(\al,\be)}(x)$ and their normalized version
$R_n^{(\al,\be)}(x)$ (see for instance \cite{01} or \cite{13})
can be defined in terms
of hypergeometric functions by
\begin{equation}
\begin{split}
P_n^{(\al,\be)}(x)&=
\frac{(\al+1)_n}{n!}\,R_n^{(\al,\be)}(x)\\
&=\frac{(\al+1)_n}{n!}\hyp21{-n,n+\al+\be+1}{\al+1}{\thalf(1-x)}.
\end{split}
\label{eq:01.13}
\end{equation}
They satisfy the symmetry
%
\begin{equation}
P_n^{(\al,\be)}(-x)=(-1)^n P_n^{(\be,\al)}(x),
\label{eq:01.11}
\end{equation}
%
and the second order differential equation
(see \cite[10.8(14)]{01})
\begin{equation}
\left(
(1-x^2)\frac{d^2}{dx^2}+\bigl(\be-\al-(\al+\be+2)x\bigr)\frac d{dx}+
n(n+\al+\be+1)
\right)
P_n^{(\al,\be)}(x)=0.
\label{eq:01.04}
\end{equation}

As in \S\ref{sec:02}, lowering and raising formulas can be given in
three different forms. I~start with analogues of
\eqref{eq:02.01} and \eqref{eq:02.02} (the second form in \S\ref{sec:02}):
%
\begin{multline}
\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)
\left((1+x^2)^n\,P_n^{(\al,\be)}\left(\frac{1-x^2}{1+x^2}\right)\right)\\
=-4(n+\al)(n+\be)\,(1+x^2)^{n-1}
P_{n-1}^{(\al,\be)}\left(\frac{1-x^2}{1+x^2}\right),
\label{eq:01.01}
\end{multline}
%
\begin{multline}
\left(\frac{d^2}{dx^2}+\frac{2\al+1}x\,\frac d{dx}\right)
\left((1+x^2)^{-n-\al-\be}\,P_{n-1}^{(\al,\be)}\left(\frac{1-x^2}{1+x^2}
\right)\right)\\
=-4n(n+\al+\be)\,(1+x^2)^{-n-\al-\be-1}
P_n^{(\al,\be)}\left(\frac{1-x^2}{1+x^2}\right).
\label{eq:01.06}
\end{multline}
%
These formulas were first given in \cite[(2.10), (2.11)]{02}.

From \eqref{eq:01.01} and \eqref{eq:01.06} one can derive analogues
of \eqref{eq:02.06} and \eqref{eq:02.07} (the first form in \S\ref{sec:02}):
%
\begin{multline}
\left(
(2n+\al+\be)(1-x^2)\frac d{dx}+n\bigl((2n+\al+\be)x+\be-\al\bigr)
\right)
P_n^{(\al,\be)}(x)\\
=2(n+\al)(n+\be)\,P_{n-1}^{(\al,\be)}(x).
\label{eq:01.05}
\end{multline}
%
\begin{multline}
\left(
(2n+\al+\be)(1-x^2)\frac d{dx}
-(n+\al+\be)\bigl((2n+\al+\be)x+\alpha-\beta\bigr)
\right)
P_{n-1}^{(\al,\be)}(x)\\
=-2n(n+\al+\be)\,P_n^{(\al,\be)}(x).
\label{eq:01.09}
\end{multline}
%
The lowering formula \eqref{eq:01.05} was earlier given in
\cite[10.8(15)]{01}.

In order to obtain \eqref{eq:01.05} from \eqref{eq:01.01},
first rewrite \eqref{eq:01.01} as
%
\begin{multline*}
\left(
(1-x^2)\frac{d^2}{dx^2}-2(x+\al)\frac d{dx}
\right)
\left(
(1+x)^{-n} P_n^{(\al,\be)}(x)
\right)\\
=-2(n+\al)(n+\be)\,(1+x)^{-n-1} P_{n-1}^{(\al,\be)}(x),
\end{multline*}
and next as
%
\begin{multline}
\Bigl(
(1+x)(1-x^2)\frac{d^2}{dx^2}+2(1+x)\bigl((n-1)x-n-\al\bigr)\frac d{dx}\\
+n\bigl(-(n-1)x+n+2\al+1\bigr)
\Bigr)P_n^{(\al,\be)}(x)
=-2(n+\al)(n+\be)\,P_{n-1}^{(\al,\be)}(x).
\label{eq:01.03}
\end{multline}
%
Subtract $(1+x)$ times
the second order differential equation \eqref{eq:01.04}
for Jacobi polynomials from identity \eqref{eq:01.03}
in order to remove its term with a
second order derivative. Then we obtain \eqref{eq:01.05}.

The derivation of \eqref{eq:01.09} from \eqref{eq:01.06} is similar,
with the two intermediate formulas
%
\begin{multline*}
\left(
(1-x^2)\frac{d^2}{dx^2}-2(x+\al)\frac d{dx}
\right)
\left(
(1+x)^{n+\al+\be} P_{n-1}^{(\al,\be)}(x)
\right)\\
=2n(n+\al+\be)\,(1+x)^{n+\al+\be-1} P_n^{(\al,\be)}(x),
\end{multline*}
%
\begin{multline}
\Bigl(
(1+x)(1-x^2)\frac{d^2}{dx^2}+2(1+x)\bigl(-(n+\al+\be+1)x+n+\be\bigr)
\frac d{dx}\\
+(n+\al+\be)\bigl(-(n+\al+\be+1)x+n-\al+\be-1\bigr)
\Bigr)
 P_{n-1}^{(\al,\be)}(x)\\
=-2n(n+\al+\be)\,P_n^{(\al,\be)}(x).
\label{eq:01.08}
\end{multline}

The third form of the lowering and raising formulas is:
%
\begin{multline}
\left(\left(\frac\pa{\pa z}+\frac\pa{\pa w}\right)^2+
\frac{4\be+2}{z+w}\left(\frac\pa{\pa z}+\frac\pa{\pa w}\right)\right)
\left((zw)^n P_n^{(\al,\be)}\left(\thalf(z/w+w/z)\right)\right)\\
=4(n+\al)(n+\be) (zw)^{n-1} P_{n-1}^{(\al,\be)}\left(\thalf(z/w+w/z)\right),
\label{eq:01.12}
\end{multline}
%
\begin{multline}
\Biggl(\left(z^2\frac\pa{\pa z}+w^2\frac\pa{\pa w}\right)^2+
\left((\al+\be+1)(z+w)-\frac{(2\be+1)zw}{z+w}\right)
\left(z^2\frac\pa{\pa z}+w^2\frac\pa{\pa w}\right)
\\
+(\al+\be+1)\bigl((\al+\be+2)(z^2+w^2)+2(\al-\be)zw\bigr)\Biggr)
\left((zw)^{n-1} P_{n-1}^{(\al,\be)}\left(\thalf(z/w+w/z)\right)\right)
\\
=4n(n+\al+\be)\,(zw)^n P_n^{(\al,\be)}\left(\thalf(z/w+w/z)\right).
\label{eq:01.16}
\end{multline}
The lowering formula \eqref{eq:01.12} can be obtained by rewriting
\eqref{eq:01.03} (use the symmetry \eqref{eq:01.11}).
Similarly, the raising formula \eqref{eq:01.16} is obtained from
\eqref{eq:01.08}.

The cases $\be=\pm\half$ of \eqref{eq:01.12} correspond to iterated cases of
\eqref{eq:02.10} in view of the quadratic transformations
%
\begin{align}
C_{2n}^{(\la)}(x)=&\frac{(\la)_n}{(\thalf)_n}\,
P_n^{(\la-\half,-\half)}(2x^2-1),
\label{eq.01.13}\\
C_{2n+1}^{(\la)}(x)=&\frac{(\la)_{n+1}}{(\thalf)_{n+1}}\,
x\, P_n^{(\la-\half,\half)}(2x^2-1).
\label{eq.01.14}
\end{align}
%
\section{Jacobi polynomials for root system $BC_2$}
\label{sec:03}
%until 03.08
Jacobi polynomials for root system $BC_2$ are a very special case
(in fact one of the motivating examples) of the Jacobi polynomials
associated with root systems of Heckman and Opdam \cite[Theorem 8.3]{03},
\cite{05}, \cite{04}. (Of course, the $BC_1$ case is given by
the classical Jacobi polynomials of \S\ref{sec:01}.)
They were introduced by the author in
\cite{06}, and further elaborated in
\cite{07} and in (my main reference) \cite{08}.

The \emph{$BC_2$ Jacobi polynomials} $R_{n,k}^{\al,\be,\ga}(\xi,\eta)$
($n\ge k\ge 0$)
are obtained by orthogonalizing the sequence
$1,\xi,\eta,\xi^2,\xi\eta,\eta^2,\xi^3,\ldots,\xi^n,\xi^{n-1}\eta,\ldots,
\xi^{n-k}\eta^k,\ldots$ with respect to the weight function
$\eta^\al(1-\xi+\eta)^\be(\xi^2-4\eta)^\ga\,d\xi\,d\eta$ on the region
in the $(\xi,\eta)$ plane
bounded by the straight lines $\eta=0$ and $1-\xi+\eta=0$ and by
the parabola $\xi^2-4\eta=0$ (so the region has vertices $(0,0)$, $(1.0)$ and
$(2,1)$). Furthermore, the polynomials are normalized such that
$R_{n,k}^{(\al,\be,\ga)}(0,0)=1$.
In fact, it can be shown that $R_{n,k}^{\al,\be,\ga}(\xi,\eta)$ is
only a linear combination of the monomials $\xi^{m-l}\eta^l$
for $m\le n$ and $m+l\le n+k$.

Important special cases of the $BC_2$ Jacobi polynomials occur
for $\ga=\pm\thalf$, where they can be expressed in terms of classical
Jacobi polynomials $R_n^{(\al,\be)}(x)$ (the normalized form, see
\eqref{eq:01.13}):
\begin{multline}
R_{n,k}^{\al,\be,-\half}(x+y,xy)\\
=
\thalf\left(R_n^{(\al,\be)}(1-2x)\,R_k^{(\al,\be)}(1-2y)
+R_k^{(\al,\be)}(1-2x)\,R_n^{(\al,\be)}(1-2y)
\right),
\label{eq:03.03}
\end{multline}
\begin{multline}
R_{n,k}^{\al,\be,\half}(x+y,xy)=
\frac{-(\al+1)}{(n-k+1)(n+k+\al+\be+2)\,(x-y)}\\
\times\left(R_{n+1}^{(\al,\be)}(1-2x)\,R_k^{(\al,\be)}(1-2y)-
R_k^{(\al,\be)}(1-2x)\,R_{n+1}^{(\al,\be)}(1-2y)\right).
\label{eq:03.04}
\end{multline}

Many explicit formulas for $BC_2$ Jacobi polynomials with general values
for the parameters $\al,\be,\ga$ were found in \cite{06}, \cite{07},
\cite{08} by first deriving the desired formula for $\ga=\pm\thalf$,
next guessing the formula for general $\ga$ by interpolation between the
two known cases ($\ga=\pm\thalf$), and finally proving the conjectured
formula in some way. This method was for instance successful in the derivation
of explicit second order differential operators raising or lowering
some parameters
(so-called \emph{shift operators}, which were important
motivating examples for Opdam \cite{05}):
\begin{align*}
D_-^\ga&\colon R_{n,k}^{\al,\be,\ga}\to R_{n-1,k-1}^{\al+1,\be+1,\ga},&
D_+^{\al,\be,\ga}&\colon
R_{n-1,k-1}^{\al+1,\be+1,\ga}\to R_{n,k}^{\al,\be,\ga},\\
E_-^{\al,\be}&\colon R_{n,k}^{\al,\be,\ga}\to R_{n-1,k}^{\al,\be,\ga+1},&
E_+^{\al,\be,\ga}&\colon R_{n-1,k}^{\al,\be,\ga+1}\to R_{n,k}^{\al,\be,\ga}.
\end{align*}
For instance,
\begin{equation*}
D_-^\ga=\tfrac14\left(\pa_{\xi\xi}+\xi\,\pa_{\xi\eta}+\eta\,\pa_{\eta\eta}+
(\ga+\tfrac32)\,\pa_\eta\right).
\end{equation*}
(Here and in the following I use notation $\pa_x$ for the partial derivative
with respect to $x$, and similarly for other variables.)

For \emph{Jack polynomials in two variables} I will use the notation
\begin{equation}
Z_{m,n}^{\ga-\half}(x+y,xy):=P_{m,n}^{1/\ga}(x,y),
\label{eq:03.06}
\end{equation}
where the standard notation for Jack polynomials is used on the
\RHS. These polynomials can be expressed in terms of ultraspherical or
Jacobi polynomials by
\begin{equation}
\begin{split}
Z_{m,n}^{\ga-\half}(x+y,xy)&=
\frac{(m-n)!}{(\ga)_{m-n}}\,(xy)^{\half(m+n)}\,
C_{m-n}^\ga\left(\frac{x+y}{2(xy)^\half}\right)\\
&=\frac{(2\ga)_{m-n}}{(\ga)_{m-n}}\,(xy)^{\half(m+n)}\,
R_{m-n}^{(\ga-\half,\ga-\half)}\left(\frac{x+y}{2(xy)^\half}\right).
\end{split}
\label{eq:03.07}
\end{equation}

The $BC_2$ Jacobi polynomials can be explicitly expanded in terms of
Jack polynomials in two variables (see \cite[Corollary 6.6]{08}):
\begin{equation}
R_{n,k}^{\al,\be,\ga}=
\sum_{l=0}^k \sum_{m=l}^n c_{n,k;m.l}^{\al,\be,\ga}\,Z_{m,l}^\ga,
\label{eq:03.08}
\end{equation}
where
\begin{multline}
c_{n,k;m.l}^{\al,\be,\ga}=
\frac{(-k)_l(-n-\ga-\thalf)_l}{(-n)_l(\al+1)_l}\\
\times\frac{(-n)_m(n+\al+\be+\ga+\tfrac32)_m}
{(\al+\ga+\tfrac32)_m(\ga+\tfrac32)_m}\,
\frac{(k+\al+\be+1)_l(\ga+\tfrac32)_{m-l}}{l!\,(m-l)!}\\
\times\hyp 43{-m+l,-n+k,-n-k-\al-\be-1,\ga+\thalf}
{-n+l,-n-m-\al-\be-\ga-\thalf,2\ga+1}1.
\label{eq:03.02}
\end{multline}
%
More generally, $BC_n$ Jacobi polynomials can be expanded in terms of
Jack polynomials in $n$ variables with the expansion coefficients given
combinatorially. For this formula, due to Macdonald, see
\cite[(5.12), (5.13)]{09}.
For $k=0$ formulas \eqref{eq:03.02}, \eqref{eq:03.08} simplify to:
\begin{equation}
R_{n,0}^{\al,\be,\ga}=
\sum_{m=0}^n\frac{(-n)_m (n+\al+\be+2\ga+2)_m(\ga+\thalf)_m}
{(\al+\ga+\tfrac32)_m (2\ga+1)_m m!}\,Z_{m,0}^\ga.
\label{eq:03.05}
\end{equation}
%
\section{Lowering and raising $BC_2$ Jacobi polynomials in a special case}
\label{sec:04}
%until 04.12
Let us now look for lowering and raising operators
\[
M_n^{\al,\be,\ga}\colon R_{n,0}^{\al,\be,\ga}\to R_{n-1,0}^{\al,\be,\ga},
\qquad
K_n^{\al,\be,\ga}\colon R_{n,0}^{\al,\be,\ga}\to R_{n+1,0}^{\al,\be,\ga}.
\]
In this paper we will restrict to the case that
$\al=\be$. Let us first try to find such operators acting on
$R_n^{(\al,\al)}(x)\pm R_n^{\al,\al}(y)$
(slight variants of
the case $\al=\be$, $k=0$ of \eqref{eq:03.03} and \eqref{eq:03.04}).
From \eqref{eq:02.06} we obtain
\begin{equation}
\left((1-x^2)\pa_x+nx\right) R_n^{(\al,\al)}(x)=n R_{n-1}^{(\al,\al)}(x).
\label{e	q:04.01}
\end{equation}
Then
\begin{multline*}
\left((1-x^2)\pa_x+nx+(1-y^2)\pa_y+ny\right)
\left(R_n^{(\al,\al)}(x)+R_n^{(\al,\al)}(y)\right)\\
=
n\left(R_{n-1}^{(\al,\al)}(x)+R_{n-1}^{(\al,\al)}(y)\right)+
n\left(x R_n^{(\al,\al)}(y)+y R_n^{(\al,\al)}(x)\right).
\end{multline*}
Here we cannot express all occurrences of $R_m^{(\al,\be)}(x)$ and
$R_m^{(\al,\be)}(x)$ in terms of $R_m^{(\al,\be)}(x)+R_m^{(\al,\be)}(x)$.
The following trick will help us.

Rewrite \eqref{e	q:04.01} as
\begin{equation}
\left((n+2\al+1)(1-x^2)\pa_x+n(n+2\al+1)x\right) R_n^{(\al,\al)}(x)=
n(n+2\al+1) R_{n-1}^{(\al,\al)}(x)
\label{eq:04.02}
\end{equation}
Then recognize $n(n+2\al+1)$ as the eigenvalue in the second order
differential equation for $R_n^{(\al,\al)}(x)$ (see \eqref{eq:01.04}):
\begin{equation}
\left((1-x^2)\pa_{xx}-2(\al+1)x\pa_x\right) R_n^{(\al,\al)}(x)=
-n(n+2\al+1) R_n^{(\al,\al)}(x).
\label{eq:04.03}
\end{equation}
From \eqref{eq:04.02} and \eqref{eq:04.03} we obtain
\begin{multline*}
\left((1-x^2)x\pa_{xx}-
((2\al+2)x^2+(n+2\al+1)(1-x^2))\pa_x\right)R_n^{(\al,\al)}(x)\\
=
-n(n+2\al+1)R_{n-1}^{(\al,\al)}(x).
\end{multline*}
This can be rewritten as
\begin{multline}
\left(x(1-x)(1-2x)\pa_{xx}+(\al+1+2(n-1)x-2(n-1)x^2)\pa_x\right)
R_n^{(\al,\be)}(1-2x)\\
=-n(n+2\al+1) R_{n-1}^{(\al,\al)}(1-2x).
\label{eq:04.05}
\end{multline}
%
If we add or subtract \eqref{eq:04.05}
and the same identity with $x$ replaced by $y$
then we obtain a lowering operator
acting on $R_n^{(\al,\al)}(1-2x)\pm R_n^{(\al,\al)}(1-2y)$.
There is still a lot of freedom here, since we can add
terms which end on $\pa_{xy}$. Thus, there are many ways to write down
lowering operators acting on $R_{n,0}^{\al,\be,\pm\half}(x+y,xy)$ and it will
be hard to decide in this way on a possible interpolation with respect to
the parameter $\ga$ of the lowering operators for $\ga=\pm\half$.

We can do better by the following approach. Put
\begin{equation}
\FSD_-:=\pa_x+\pa_y,\quad
\FSD_+^\ga:=x^2\pa_x+y^2\pa_y+(\ga+\thalf)(x+y),\quad
\FSD_0:=x\pa_x+y\pa_y.
\label{eq:04.12}
\end{equation}
From \eqref{eq:02.10}, \eqref{eq:02.11} and the homogeneity
of $Z_{m,0}^\ga(x+y,xy)$ in $x,y$ we obtain:
\begin{align}
\FSD_- Z_{m,0}^\ga(x+y,xy)&=
\frac{m(2\ga+m)}{\ga+m-\thalf}\,Z_{m-1,0}^\ga(x+y,xy),
\label{eq:04.06}\\[\smallskipamount]
\FSD_+^\ga Z_{m,0}^\ga(x+y,xy)&=(\ga+m+\thalf) Z_{m+1,0}^\ga(x+y,xy),
\label{eq:04.07}\\[\smallskipamount]
\FSD_0 Z_{m,0}^\ga(x+y,xy)&=
m Z_{m,0}^\ga(x+y,xy).
\label{eq:04.08}
\end{align}
Let us try to use the operators \eqref{eq:04.12} as building blocks
for a lowering operator acting on $R_{n,0}^{\al,\be,\ga}(x+y,xy)$
such that it reduces for $\ga=\pm\thalf$
to an operator we already know. I will work this out here only for the case
$\al=\be$. The following conjectured
lowering formula is obtained:
\begin{multline}
\Bigl(\FSD_-\FSD_0-3(\FSD_0)^2+2\FSD_+^\ga\FSD_0+(\al+\ga+\thalf)\FSD_-
+2(n-2\ga-\thalf)\FSD_0\\
-2n(\FSD_+^\ga-\ga-\thalf)\Bigr)
R_{n,0}^{\al,\al,\ga}(x+y,xy)=-n(n+2\al+1) R_{n-1,0}^{\al,\al,\ga}(x+y,xy).
\label{eq:04.10}
\end{multline}
Formula \eqref{eq:04.10} can indeed be verified
by using \eqref{eq:03.05},
\eqref{eq:04.06}, \eqref{eq:04.07}, \eqref{eq:04.08}.

Similarly as for \eqref{eq:04.10}, one can conjecture
and next prove the following raising formula:
\begin{multline}
\Bigl(\FSD_-\FSD_0-3(\FSD_0)^2+2\FSD_+^\ga\FSD_0+(\al+\ga+\thalf)\FSD_-
-2(n+2\al+4\ga+\tfrac52)\FSD_0\\
+2(n+2\al+2\ga+2)(\FSD_+^\ga-\ga-\thalf)\Bigr)
R_{n,0}^{\al,\al,\ga}(x+y,xy)\\
=-(n+2\ga+1)(n+2\al+2\ga+2) R_{n+1,0}^{\al,\al,\ga}(x+y,xy).
\label{eq:04.11}
\end{multline}
The computations to check \eqref{eq:04.10} and \eqref{eq:04.11}
are feasible on paper, but I have also checked the results
in {\em Mathematica}.
%
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\end{document}

math0505378.extracted_bib
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