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\begin{document}
\begin{center}
{\Large \bf 
Orthogonal polynomial identities from\\[3mm]
tensor product decompositions
}\\[15mm]
{\bf J.\ Van der Jeugt\footnote{Research
Associate of the Fund for Scientific Research -- Flanders (Belgium).}
}\\[1cm] 
Department of Applied Mathematics and Computer Science,\\
University of Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium.\\[2mm]
E-mail : Joris.VanderJeugt@rug.ac.be
\end{center}

\vskip 15mm

\noindent
We consider the Lie algebra $su(1,1)$ and its positive discrete series
representations. A general self-adjoint element $X$ of $su(1,1)$ then acts
as a second order difference operator in a representation. For such a
second order difference operator, there exist the associated orthogonal
polynomials of the first kind~\cite{Berez}. In the present case, these
are Laguerre or Meixner-Pollaczek polynomials. When studying the tensor
product of positive discrete series representations, overlap
coefficients of generalised eigenvectors of $X$ can again be expressed
in terms of orthogonal polynomials; in our case Jacobi or continuous
Hahn polynomials. New identities for these orthogonal polynomials,
involving the Clebsch-Gordan or Racah coefficients of $su(1,1)$, can
then be obtained.

The idea originates in work of Granovskii and Zhedanov~\cite{GZ}. The
first application to Laguerre polynomials associated with $su(1,1)$,
and to Hermite polynomials associated with the oscillator algebra, was
given in~\cite{V}. The general case of $su(1,1)$, and of polynomials
associated with the quantum algebra $U_q(su(1,1))$, was developed
in~\cite{KV1}. Further applications are related to generating functions
for orthogonal polynomials~\cite{VJ}, or to bilinear generating
functions~\cite{KV2}, see also Section~4.

\section{The Lie algebra $su(1,1)$}
\def\Zp{\Zah_+}
\def\Hi{\ell^2(\Zah_+)}

The Lie algebra $su(1,1)$ is generated by $H,B,C$ subject to
$$
[H,B] = 2B, \qquad [H,C] = -2C, \qquad [B,C]=H.
$$
There is a $\ast$-structure by $H^\ast=H$ and $B^\ast=-C$.
The positive discrete series representations $\pi_k$ of
$su(1,1)$ are unitary representations labelled by $k>0$.
The representation space is $\Hi$ equipped
with an orthonormal basis $\{ e^k_n\}_{\{ n\in\Zp\} }$, and the action
is given by
\beq
\begin{array}{ll}
\pi_k(H)\, e^k_n &= 2(k+n) \, e^k_n, \\[1mm]
\pi_k(B)\, e^k_n &= \sqrt{(n+1)(2k+n)}\, e^k_{n+1},\\[1mm]
\pi_k(C)\, e^k_n &= -\sqrt{n(2k+n-1)}\, e^k_{n-1}.
\end{array}
\label{actionsu11}
\eeq
The tensor product of two positive discrete series
representations decomposes as
\beq
\pi_{k_1}\otimes \pi_{k_2} = \bigoplus_{j=0}^\infty
\pi_{k_1+k_2+j},
\label{tensorproddec}
\eeq
and the corresponding intertwining operator can be expressed
by means of the Clebsch-Gordan coefficients
\beq
e^{(k_1k_2)k}_n = \sum_{n_1,n_2} C^{k_1,k_2,k}_{n_1,n_2,n}\,
e^{k_1}_{n_1}\otimes e^{k_2}_{n_2}.
\label{defCGC}
\eeq
The Clebsch-Gordan coefficients are non-zero only if $n_1+n_2=n+j$,
$k=k_1+k_2+j$ for $j,n_1,n_2,n\in\Zp$, and normalised
by $\langle e^k_0, e^{k_1}_0\otimes e^{k_2}_j\rangle >0$.
For the above results see Vilenkin and Klimyk~\cite[\S 8.7]{VileK},
or~\cite{KV1}. The Clebsch-Gordan coefficients can be expressed in terms
of a terminating hypergeometric series ${}_3F_2(1)$ of unit argument,
or, equivalently, in terms of a Hahn polynomial.

Up to a scalar multiple, the most general self-adjoint element in
$su(1,1)$ is of the form
\beq
X=B-C-\al H,\qquad \al\in\Real,
\label{X}
\eeq
thus $X^*=X$. The action of $X$ on a basis vector $e^k_n$ in the
representation $\pi_k$ is then given by
\bea
&&\pi_k(X) e^k_n = a_{n} e^k_{n+1} + b_n e^k_n + a_{n-1} e^k_{n-1}
\label{actionX} \\
&&\hbox{where } a_n=\sqrt{(n+1)(2k+n)} \hbox{ and } b_n=-2\al(k+n). \nn
\eea
Thus $\pi_k(X)$ act as a recurrence operator, or second order
difference operator. Since the coefficients $a_n$ and $b_n$ in
(\ref{actionX}) satisfy $a_n>0$ and $b_n\in\Real$, there exist
orthogonal polynomials associated with $\pi_k(X)$ defined
by~\cite{Berez} 
\bea
&&x p_n(x) = a_{n} p_{n+1}(x) + b_n p_n(x) + a_{n-1} p_{n-1}(x)
\label{recurrence} \\
&&\hbox{with } p_{-1}(x)=0, \quad p_0(x)=1. \nn
\eea
Instead of $xp_n(x)$ in the lhs of (\ref{recurrence}) it is sometimes
more convenient to write a linear combination $(ux+v)p_n(x)$; this is a
trivial extension.

\section{Laguerre and Jacobi polynomials}

In this section, we choose the parameter $\al$ in (\ref{X}) as $\al=1$.
Consider the classical Laguerre polynomials
\beq
L^{(\al)}_n(x)={(\al+1)_n \over n!}{\ }_1F_1\left({-n\atop \al+1};
x\right), 
\eeq
with the usual notation for Pochhammer symbols and hypergeometric
series, and let for $k>0$,
\beq
l^{(k)}_n(x) = \sqrt{ n!\over (2k)_n}\, L^{(2k-1)}_n(x).
\eeq
{}From the three-term recurrence relation for Laguerre polynomials, it
follows that the normalised Laguerre polynomials $l^{(k)}_n(x)$ satisfy
\beq
-x l^{(k)}_n(x) = a_{n} l^{(k)}_{n+1}(x) -2(k+n) l^{(k)}_n(x) + a_{n-1}
l^{(k)}_{n-1}(x).
\eeq
If we denote the orthogonality measure of the normalised Laguerre
polynomials $l^{(k)}_n(x)$ by $d\mu_k(x)$, comparing this with
(\ref{actionX}) implies~: 
\begin{prop}
$\La : \Hi \rightarrow L^2(\Real,d\mu_k(x))$, mapping $e_n^k$ onto
$l^{(k)}_n(x)$, is a unitary mapping intertwining $\pi_k(X)$ acting in
$\Hi$ with $M_{-x}$ on $L^2(\Real,d\mu_k(x))$.
\end{prop}
Herein, $M_g$ denotes multiplication by the function $g$, so $M_g f(x)
= g(x)f(x)$. The proposition states that $v^k(x)=\sum_{n=0}^\infty
l^{(k)}_n(x) e^k_n$ is a generalised eigenvector for $\pi_k(X)$ for the
eigenvalue $-x$.

Interesting applications follow from studying the action of $X$ in the
tensor product representation $\pi_{k_1}\otimes \pi_{k_2}$. In the
tensor product, the action of every element of the Lie algebra, and in
particular of $X$, is given by the trivial comultiplication
$\De(X)=1\otimes X + X\otimes 1$. Then, we have,
\begin{prop}
$\Up\colon \Hi\otimes\Hi \to
L^2(\Real^2,d\mu_{k_1}(x_1)d\mu_{k_2}(x_2))$,
defined by $e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \mapsto
l_{n_1}^{(k_1)}(x_1) l_{n_2}^{(k_2)}(x_2)$
is a unitary mapping intertwining
$\pi_{k_1}\otimes \pi_{k_2}(\De(X))$
with $M_{-(x_1+x_2)}$.
\end{prop}
The sum $x_1+x_2$ follows from the trivial comultiplication.

A crucial property is that under the mapping of Proposition~2 the
``coupled basis vectors'' $e^{(k_1 k_2)k}_n$ are mapped into a polynomial
that factorises. For a proof, see~\cite{KV1}.

\begin{prop} For $k=k_1+k_2+j$, $j\in\Zp$, we have
\beq
\Up e^{(k_1 k_2)k}_n = l^{(k)}_n(x_1+x_2) \, S_j^{(k_1,k_2)}(x_1,x_2),
\eeq
where 
\bea
S^{(k_1,k_2)}_j(x_1,x_2)&=&(-1)^j \sqrt{ {{ j!}\over{(2k_1)_j(2k_2)_j
(2k_1+2k_2+j-1)_j}} }\nn\\
&&\times (x_1+x_2)^j P^{(2k_1-1,2k_2-1)}_j\Bigl(
{{x_2-x_1}\over{x_1+x_2}}\bigr).
\label{SexplicitL}
\eea
\end{prop}
Herein, $P^{(a,b)}_j(z)=(a+1)_j/j!{\ }_2F_1(-j,j+a+b+1;a+1;(1-z)/2)$ is
a classical Jacobi polynomial. 

A consequence of the above formula is the identity
\beq
\sum_{n_1,n_2} C^{k_1,k_2,k}_{n_1,n_2,n}\
 l^{(k_1)}_{n_1}(x_1)  l^{(k_2)}_{n_2}(x_2)  
= l^{(k)}_n(x_1+x_2)\, S^{(k_1,k_2)}_j(x_1,x_2).
\label{cgcident}
\eeq
Using the explicit expression for $su(1,1)$ Clebsch-Gordan
coefficients, and relabelling the parameters, 
this gives rise to the following identity for
Laguerre polynomials~\cite{V}:

\begin{theo}
\bea
&&\sum_{l=0}^{n+j} Q_j(l;a,b,j+n) L_l^{(a)}(x) L_{n+j-l}^{(b)}(y) \nn\\
&&\qquad = {(-1)^j n!j!\over (a+1)_j(n+j)!} (x+y)^j
L_n^{(a+b+2j+1)}(x+y) P_j^{(a,b)}({y-x\over y+x});
\label{res1}
\eea
\end{theo}
Herein, we have used the common notation for the Hahn polynomials defined by
\beq
Q_n(x;a,b,N) = {}_3F_2\left(
{{-n,n+a+b+1,-x}\atop{a+1,\ -N}};1\right)
\eeq
for $N\in\Zp$, $0\leq n\leq N$. For $j=0$, the identity (\ref{res1})
reduces to the classical addition formula for Laguerre polynomials. 

In a similar way, one can consider the tensor product of three
representations $\pi_{k_1}\otimes \pi_{k_2} \otimes \pi_{k_3}$. The
Racah coefficients of $su(1,1)$ are transformation coefficients
between two orthonormal bases in the tensor product space~:
\beq
e^{((k_1k_2)k_{12}k_3)k}_n = \sum_{k_{23}}
U^{k_1,k_2,k_{12}}_{k_3,k,k_{23}} \ e^{(k_1(k_2k_3)k_{23})k}_n .
\eeq
Herein,
\beas
&&k_{12}=k_1+k_2+j_{12}, \qquad k_{23}=k_2+k_3+j_{23}, \\
&&k=k_{12}+k_3+j =k_1+k_{23}+j',\qquad
j_{12},j,j_{23},j' \in \Zp,\ \hbox{and}\ j_{12}+j=j_{23}+j'.
\eeas
Thus the above sum is finite. The identity that can now be obtained is
the following~:
\beq
S_{j_{12}}^{(k_1,k_2)}(x_1,x_2)  S_{j}^{(k_{12},k_3)}(x_1+x_2,x_3) =
\sum_{j_{23}=0}^{j_{12}+j} U^{k_1,k_2,k_{12}}_{k_3,k,k_{23}}\ 
S_{j_{23}}^{(k_2,k_3)}(x_2,x_3)
S_{j_{12}+j-j_{23}}^{(k_1,k_{23})}(x_1,x_2+x_3) .
\eeq
Using the explicit expression~(\ref{SexplicitL} for the $S$-function, a
convolution identity for Jacobi polynomials involving Racah polynomials
is obtained~\cite{V,KV1}. 

So far, we have considered only the case $\al=1$ in (\ref{X}). In the
following section the case $|\al|<1$ is discussed; the case $|\al|>1$
is similar~\cite{KV1}.

\section{Meixner-Pollaczek and continuous Hahn polynomials}

The Meixner-Polla\-czek polynomials are defined by
\beq
P_n^{(\la)}(x;\phi) = {{(2\la)_n}\over{n!}} e^{in\phi}
\, {}_2F_1\left( {{-n,\la+ix}\atop{2\la}};1-e^{-2i\phi}\right).
\eeq
For $\la>0$ and $0<\phi<\pi$ these are orthogonal polynomials
with respect to a positive measure on $\Real$, see \cite{KoekS}.
Let $k>0$; then the orthonormal Meixner-Pollaczek polynomials
\beq
p_n(x) = p_n^{(k)}(x;\phi) = \sqrt{ {{n!}\over{\Gamma(n+2k)}}}
P_n^{(k)}(x;\phi)
\eeq
satisfy the three-term recurrence relation
\beq
2x\, \sin\phi\, p^{(k)}_n(x) = a_n\, p^{(k)}_{n+1}(x) - 2(n+\la)\cos\phi\,
p^{(k)}_n(x) + a_{n-1}\, p^{(k)}_{n-1}(x).
\eeq
Comparing with (\ref{actionX}) leads to identifying $\al$ with
$\cos\phi$, and with
\beq
X_\phi = B-C-\cos\ph H,
\eeq
we have the following (cfr.\ Proposition~1)~:
\begin{prop}
Let $d\mu_{k,\phi}(x)$ denote the orthogonality measure of the
orthonormal Meixner-Pollaczek polynomials $p_n^{(k)}(x;\phi)$. Then
$\Lambda\colon \Hi \to L^2(\Real,d\mu_{k,\phi}(x))$,
$e^k_n\mapsto p_n^{(k)}(x;\phi)$,
is a unitary mapping
intertwining $\pi_k(X_\phi)$ acting in $\Hi$ with $M_{2x\sin\phi}$
on $L^2(\Real,d\mu_{k,\phi}(x))$.
\end{prop}
Similarly, in the tensor product $\pi_{k_1}\otimes\pi_{k_2}$,
\begin{prop}
$\Up\colon \Hi\otimes\Hi \to
L^2(\Real^2,d\mu_{k_1,\phi}(x_1)d\mu_{k_2,\phi}(x_2))$,
defined by $e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \mapsto
p_{n_1}^{(k_1)}(x_1;\phi)p_{n_2}^{(k_2)}(x_2;\phi)$
is a unitary mapping intertwining
$\pi_{k_1}\otimes \pi_{k_2}(\De(X_\phi))$
with $M_{2(x_1+x_2)\sin\phi}$.
\end{prop}

The $S$-function of Proposition~3 now becomes a continuous Hahn
polynomial, see~\cite{KV1}~:
\begin{prop} For $k=k_1+k_2+j$, $j\in\Zp$, we have
\beq
\Up e^{(k_1 k_2)k}_n = p^{(k)}_n(x_1+x_2;\phi) \,
S_j^{(k_1,k_2)}(x_1,x_2;\phi), 
\eeq
where 
\bea
S^{(k_1,k_2)}_j(x_1,x_2;\phi)&=& (-2\sin\phi)^j
\sqrt{ {{j!\, (2j+2k_1+2k_2-1)\Gamma(j+2k_1+2k_2-1)}\over
{\Gamma(2k_1+j) \Gamma(2k_2+j)}} } \nn\\
&&\times\  p_j(x_1;k_1, k_2-i(x_1+x_2),k_1,k_2+i(x_1+x_2)).
\label{SexplicitMP}
\eea
\end{prop}
Herein, $p_j$ is a continuous Hahn polynomial~\cite{KoekS}
\beq
p_n(x;a,b,c,d)= i^n {{(a+c)_n(a+d)_n}\over{n!}}\,
{}_3F_2\left( {{-n,n+a+b+c+d-1,a+ix}\atop{a+c,\ a+d}};1\right).
\eeq
The identity (\ref{cgcident}) now gives rise to the following~\cite{KV1}
\begin{theo}
With the notation for continuous Hahn, Meixner-Pollaczek and Hahn
polynomials as above,
the following convolution formula holds:
\bea
&&{{n+j}\choose n}
\sum_{l=0}^{n+j} Q_j(l;2k_1-1,2k_2-1,n+j)\, P_l^{(k_1)}(x_1;\phi)
\,  P_{n+j-l}^{(k_2)}(x_2;\phi) = \label{convol}\\
&&\qquad {{(-2\sin\phi)^j}\over{(2k_1)_j}}
\, P_n^{(k_1+k_2+j)}(x_1+x_2;\phi)\,
 p_j(x_1;k_1,
k_2-i(x_1+x_2),k_1,k_2+i(x_1+x_2)). \nn
\eea
\label{res2}
\end{theo}
The case $j=0$ gives back the classical convolution identity for the
Meixner-Pollaczek polynomials. Formula~(\ref{convol}) also
has an interpretation as a connection coefficient formula between
orthogonal polynomials in two variables with the same orthogonality
measure. 

\section{Bilinear generating functions}

Consider again the polynomials discussed in Section~2. From
Proposition~1 it follows that the elements $H$, $B$ and $C$ of
$su(1,1)$ have a realisation in $L^2(\Real,d\mu_k(x))$ with standard
action~(\ref{actionsu11}) on the basis $l^{(k)}_n(x)$. For fixed
$y\in\Real$ and $-1<t<1$, the functions
\beq
v^k_t(x;y) = \sum_{n=0}^\infty l^{(k)}_n(x) l^{(k)}_n(y) t^n.
\label{vL}
\eeq
are also in $L^2(\Real,d\mu_k(x))$, and the action of the $su(1,1)$
basis elements on such functions is easy to write down. An explicit
expression for $v^k_t(x;y)$ follows from the Poisson kernel for
Laguerre polynomials, also known as the Hille-Hardy formula
\cite[10.12 (20)]{HTFtwee}~:
\beq
v^k_t(x;y) = e^{-t(x+y)/(1-t)}(1-t)^{-2k} 
{\ }_0F_1\left( {-\atop{2k}}; {{xyt}\over{(1-t)^2}} \right).
\label{vexplicitL}
\eeq
Studying the tensor product $\pi_{k_1}\otimes \pi_{k_2}$ in this
context gives rise~\cite{KV2} to the following identity, involving the
Poisson kernel functions $v^k_t(x;y)$ and the
$S$-function~(\ref{SexplicitL})~: 
\beq
v^{k_1}_t(x_1;y_1)v^{k_2}_t(x_2;y_2) = \sum_{j=0}^\infty t^j
v^{k_1+k_2+j}_t(x_1+x_2;y_1+y_2)\, S_j^{(k_1,k_2)}(x_1,x_2)\,
S_j^{(k_1,k_2)}(y_1,y_2).
\label{SpoissonL}
\eeq
This can be interpreted as a bilinear generating function for the
orthogonal polynomials associated with $S_j$. In the present case,
after relabelling
$(2k_1-1, 2k_2-1)$ by $(a,b)$, $((x_2-x_1)/(x_2+x_1),
(y_2-y_1)/(y_2+y_1) )$ by $(x,y)$, and $t(x_1+x_2)(y_1+y_2)/(1-t)^2$ by
$r$, we obtain~\cite{KV2}
\bea
&&\sum_{j=0}^\infty {{ r^j\, j!}\over{ (a+1)_j (b+1)_j
(a+b+j+1)_j}}
P_j^{(a,b)}(x) P_j^{(a,b)}(y)\, {}_0F_1(-;
a+b+2j+2;r) \nn\\ 
&&\qquad = {}_0F_1(-;a+1; {r\over 4}(1-x)(1-y))\, 
{}_0F_1(-;b+1; {r\over 4}(1+x)(1+y)).
\eea

The realisation in terms of Meixner-Pollaczek polynomials instead of
Laguerre polynomials is even more interesting. In that case, the
Poisson kernel is given by~\cite{KV2}~:
\bea
v^k_t(x,\phi;y,\psi) &=& \sum_{n=0}^\infty p^{(k)}_n(x;\phi)
p^{(k)}_n(y;\psi) t^n \nn\\
&=&{1\over\Gamma(2k)}
(1-te^{i(\phi+\psi)})^{i(x+y)} 
(1-te^{i(\phi-\psi)})^{-k-iy}  
(1-te^{i(\psi-\phi)})^{-k-ix} \nn\\ 
&&\qquad\times 
\, {}_2F_1\left( {{k+ix, k+iy}\atop{2k}} ; r \right).
\eea
Since the $S$-function is in this case a continuous Hahn polynomial,
the identity (\ref{SpoissonL}) becomes now, after appropriate
relabellings~\cite{KV2},
\bea
&&\sum_{j=0}^\infty {(-1)^j j! \over (2a,b+d,2a+b+d+j-1)_j} 
{\ }_2F_1\left( {{a+d+j, a+d'+j}\atop{2a+b+d+2j}};r \right)\nn\\
&&\qquad\qquad\qquad\times\ p_j(x;a,b,a,d)\ p_j(y;a,b',a,d') \ r^j
\nn\\[1mm] 
&&= {}_2F_1\left( {{a+ix, a+iy}\atop{2a}};r \right)
{\ }_2F_1\left( {{d-ix, d'-iy}\atop{b+d}}; r\right),
\label{hahn}
\eea
where $b+d=b'+d'$.

\section{The quantum algebra $U_q(su(1,1))$}

A similar theory can be developed for the quantum algebra
$U_q(su(1,1))$ \cite{KV1}. The role of Meixner-Pollaczek polynomials is
here taken by Al-Salam--Chihara polynomials, and the $S$-functions
become Askey-Wilson polynomials. Central results are new convolution
identities for the Al-Salam--Chihara and Askey-Wilson polynomials
involving $q$-Hahn and $q$-Racah polynomials respectively~\cite{KV1}.
The technique of Section~4 can also be applied, leading to a bilinear
generating function for Askey-Wilson polynomials~\cite{KV2}.

%\newpage

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\bibitem{HTFtwee}
A.~Erd\'elyi, W.~Magnus, F.~Oberhettinger, F.G.~Tricomi,
{\em Higher Transcendental Functions,}
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\bibitem{GZ}
Y.I.\ Granovskii and A.S.\ Zhedanov,
{\em J.\ Phys.\ A} {\bf 26} (1993) 4339--4344.

\bibitem{KoekS}
R.~Koekoek and R.F.~Swarttouw,
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\end{thebibliography}
\end{document}
