% Latex file
% Representation theory and addition/convolution formulas for
% special functions
% J. Van der Jeugt
% in Proceedings of the Quantum Group Symposium at the XXI International
% Colloquium on Group Theoretical Methods in Physics, Goslar, 1996
% Editors. H.-D. Doebner and V.K. Dobrev
% Heron Press, Sofia, 1997; pp. 363-369.
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\documentclass{article}
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%\usepackage{times} % if available; if not - comment out,
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\title{Representation theory and addition/convolution formulas for
special functions}
\author{J. Van der Jeugt\footnote{Senior Research Associate of the
N.F.W.O. (National Fund for Scientific Research of Belgium)}\\
Department of Applied Mathematics,\\
University of Ghent,\\
Krijgslaan 281-S9, B9000 Gent, Belgium.}
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\def\bea{\begin{eqnarray}}
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\def\nn{\nonumber}
\def\ba{\begin{array}}
\def\ea{\end{array}}
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\la{\lambda}
\def\th{\theta}
\def\vp{\varphi}
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\def\Nat{\mathbb{N}}
\def\Zah{\mathbb{Z}}
\def\Real{\mathbb{R}}
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\def\C{\mathbb{C}}
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\begin{document}
\maketitle

\section{Introduction}

The representation theory of Lie algebras and quantum algebras, or
quantum enveloping algebras, is intimately linked to special functions
of (basic) hypergeometric type~\cite{VK}. 
In our approach, following an idea of
Granovski\u\i\ and Zhedanov~\cite{GZ}, orthogonal polynomials occur as
overlap coefficients of formal vectors in the positive discrete series
representations of $su(1,1)$ and $U_q[su(1,1)]$. These formal vectors
are generalized eigenvectors of some self-adjoint element of the
algebra. By considering the tensor product decomposition of two (or
three) representations, an identity involving these polynomials and the
Clebsch-Gordan coefficients of the algebra is obtained. This leads to
new summation/convolution identities for Laguerre, Meixner-Pollaczek,
Meixner, Jacobi, Hahn and continuous Hahn polynomials in the case of
$su(1,1)$, and for Al-Salam--Chihara and Askey-Wilson polynomials in
the case of $U_q[su(1,1)]$. Many of the results presented here have
been obtained in collaboration with H.T.\ Koelink~\cite{KV}.

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

The Lie algebra $su(1,1)$ is generated by $J_0, J_\pm$ subject to
the relations
\beq
[J_0, J_\pm]=\pm J_\pm,\qquad [J_+,J_-]=-2 J_0,
\label{defsu11}
\eeq
with the conditions $J_0^\dagger=J_0$ and $J_\pm^\dagger=J_\mp$. The
positive discrete representations are labeled by a positive real number
$k$. The representation space is $\ell^2(\Zah_+)$, with orthonormal
basis vectors denoted by $e^{(k)}_n$, with $n=0,1,2,\cdots$.
The explicit action of the generators in this representation $(k)$ is
given by~:
\beq
 \begin{array}{l}
 J_0 e^{(k)}_n = (n+k) e^{(k)}_n ,\\
 J_+ e^{(k)}_n = \sqrt{(n+1)(2k+n)} e^{(k)}_{n+1}\\
 J_- e^{(k)}_n = \sqrt{n(2k+n-1)} e^{(k)}_{n-1}.
 \end{array}
\label{su11action}
\eeq

In two previous papers~\cite{V,KV}, a recurrence operator in the Lie
algebra $su(1,1)$ was related to a Jacobi matrix. 
Here it will be more appropriate 
to use the (equivalent) notion of formal or generalized
eigenvectors. The formal vector
\beq
v^{(k)}(x)=\sum_{n=0}^\infty l^{(k)}_n(x) e^{(k)}_n
\label{formalv}
\eeq
is a generalized eigenvector of the self-adjoint operator $X=\sigma
J_0-J_+-J_-$ ($\sigma\in\Real$) for the eigenvalue
$\la(x)$ provided $l^{(k)}_n(x)$
satisfies a three-term recurrence relation.
This leads to an
identification of $l^{(k)}_n(x)$ with orthogonal polynomials in $x$.
These polynomials associated with $X$ are of different type for
$|\sigma|=2$, $|\sigma|<2$ or $|\sigma|>2$. Labelling the operator in
the three distinct cases as follows~:
\bea
X_2&=& 2J_0-J_+-J_- ,\label{X2}\\
X_\phi&=&-2\cos\phi\;J_0+J_++J_-,\qquad 0<\phi<\pi,\label{Xphi}\\
X_c&=&-(c+1/c)J_0+J_++J_-,\qquad 0<c<1, \label{Xc}
\eea
we give a summary of the polynomials in Table~1.

\begin{table}[htb]
\caption{Orthogonal polynomials appearing in the formal 
eigenvector $v^{(k)}(x)$.} 
\label{tab1}
\[
\begin{tabular}{|c|c|c|c|}
\hline
$X$ & $l^{(k)}_n(x)$ & polynomial & eigenvalue $\la(x)$ \\ \hline
& & & \\[-2mm]
$X_2$ & $\sqrt{n!\over(2k)_n}L^{(2k-1)}_n(x)$ &
generalized Laguerre & $x$ \\[3mm]
$X_\phi$ &$\sqrt{n!\over\Gamma(2k+n)}P^{(k)}_n(x;\phi)$ &
Meixner-Pollaczek & 
$2x\sin\phi$ \\[3mm]
$X_c$ & $\sqrt{(2k)_n\over n!}c^n M_n(x;2k;c^2)$ &
Meixner & $(c-1/c)(k+x)$ \\[3mm] \hline
\end{tabular}
\]
\end{table}

In Table~1, $(\alpha)_n=\Gamma(\alpha+n)/\Gamma(\alpha)$ is the common
notation for the Pochhammer symbol in terms of the classical $\Gamma$
function. The definition and notation of the orthogonal polynomials is
standard [see, e.g.\ Ref.~\cite{KS}].

By considering the tensor product decomposition for $su(1,1)$
representations, new summation formulas or convolution theorems are
obtained for these polynomials. The tensor product decomposes as
follows~: 
\beq
(k_1)\otimes(k_2) = \bigoplus_{j=0}^\infty (k_1+k_2+j).
\label{tensdec}
\eeq
The ``coupled basis vectors'' are written in terms of the uncoupled
ones 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{CGC}
\eeq
Herein, $k=k_1+k_2+j$ for some integer $j\geq 0$, and the sum is such
that $n_1+n_2=j+n$. 
Explicit expressions for the Clebsch-Gordan coefficients are given,
e.g.\ in Ref.~\cite{V}. 

In the tensor product space, the generalized eigenvectors of
$\Delta(X)=X\otimes 1+1\otimes X$ can be considered. Again, there are
``uncoupled'' eigenvectors $v^{(k_1)}(x_1) v^{(k_2)}(x_2)$ with
eigenvalue $\la(x_1+x_2)$, but also ``coupled'' eigenvectors
$v^{(k_1k_2)k}(x_1+x_2)$ defined as follows~:
\beq
v^{(k_1k_2)k}(x_1+x_2)=\sum_{n=0}^\infty l^{(k)}_n(x_1+x_2)
e^{(k_1k_2)k}_n,
\label{vcoup}
\eeq
where $k=k_1+k_2+j$ for some integer $j\geq0$. It can be
shown~\cite{V,KV} 
that
\beq
v^{(k_1)}(x_1) v^{(k_2)}(x_2) = \sum_{j=0}^\infty S_j(x_1,x_2;k_1,k_2) 
v^{(k_1k_2)k_1+k_2+j}(x_1+x_2),
\label{vv}
\eeq
where $S$ is again an orthogonal (Jacobi, continuous Hahn, or Hahn)
polynomial given in Table~2, according 
to which of the three cases is considered. The constants in this table 
are determined by
\bea
C&=&\left(j!/((2k_1)_j(2k_2)_j(2k_1+2k_2+j-1)_j)\right)^{1/2},
\label{C1}\\ 
C'&=&\left({j!(2k_1+2k_2+2j-1)\Gamma(2k_1+2k_2+j-1) \over
 \Gamma(2k_1+j)\Gamma(2k_2+j) }\right)^{1/2},\label{C2}\\
C''&=&C' (2k_1)_j/j!.\label{C3}
\eea

\begin{table}[htb]
\caption{Orthogonal polynomial appearing in the expansion (10).}
\label{tab2}
\[
\begin{tabular}{|c|c|c|}
\hline
$X$ & $S_j(x_1,x_2;k_1,k_2)$ & polynomial \\ \hline
 & & \\[-2mm]
$X_2$ & $C (-1)^j (x_1+x_2)^j P_j^{(2k_1-1,2k_2-1)}\left({x_2-x_1\over 
x_2+x_1} \right)$ & Jacobi \\[3mm] 
$X_\phi$ & $C' (-2\sin\phi)^j
p_j(x_1;k_1,k_2-i(x_1+x_2),k_1,k_2+i(x_1+x_2))$ & 
contin.\ Hahn \\[3mm] 
$X_c$ & $C''(-c+1/c)^j (-x_1-x_2)_j \; Q_j(x_1;2k_1-1,2k_2-1,x_1+x_2)$
& Hahn\\[3mm] \hline
\end{tabular}
\]
\end{table}

A number of interesting new addition or convolution identities for
orthogonal 
polynomials can be constructed from the relation ($k=k_1+k_2+j$)
\beq
\sum_{n_1+n_2=n+j} 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_j(x_1,x_2;k_1,k_2),
\label{ident}
\eeq
together with the explicit form of the Clebsch-Gordan coefficient in
terms of a Hahn polynomial.
For example, for the case $X_2$ the relation (\ref{ident}) reduces to 
a new addition formula for Laguerre polynomials involving a Hahn and a
Jacobi polynomial~\cite{V}~:
\bea
&&\sum_{k=0}^{n+j} Q_j(k;a,b,j+n) L_k^{(a)}(x_1) L_{n+j-k}^{(b)}(x_2)
\nn\\ 
&&\qquad = {(-1)^j n!j!\over (a+1)_j(n+j)!} (x_1+x_2)^j
L_n^{(a+b+2j+1)}(x_1+x_2) P_j^{(a,b)}({x_2-x_1\over x_2+x_1}).
\label{intro1}
\eea
For $j=0$, this reduces to the known addition formula for Laguerre
polynomials. The relation (\ref{intro1}) can also be viewed as yielding
connection coefficients between two sets of orthogonal polynomials in
the variables $(x_1,x_2)$ with respect to the same orthogonality
measure. 

In a similar fashion, new convolution identities involving the Hahn
polynomial can be constructed for
Meixner-Pollaczek and Meixner polynomials~\cite{KV}. 
By considering the tensor
product of three representations and the explicit form of the Racah
coefficient, new convolution identities for the polynomials of Table~2 
involving a Racah polynomial~\cite{KV} can be obtained.

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

Let $U_{q}(sl(2,\C))$ ($0<q<1$) be the complex unital
associative algebra generated by $A$, $B$, $C$, $D$ subject to the
relations
\beq
AD=1=DA, \quad AB=q^{1/2}BA,\quad AC=q^{-1/2}CA,\quad
BC-CB = {{A^2-D^2}\over{q^{1/2}-q^{-1/2}}}.
\label{defrel}
\eeq
This algebra can be equiped with a comultiplication, counit, and
antipode, turning it into a Hopf algebra. The Hopf $*$-algebra 
$U_q(su(1,1))$ has the following $*$-structure~:
\beq
A^*=A,\quad B^*=-C,\quad C^*=-B, \quad D^*=D.
\label{star}
\eeq
The positive discrete representations are labeled by a positive real
number $k$. The representation space is again $\ell^2(\Zah_+)$, with
orthonormal 
basis vectors denoted by $e^{(k)}_n$, with $n=0,1,2,\cdots$.
The explicit action of the generators in this representation $(k)$ is
given by~\cite{KV}
\bea
&&A\, e^{(k)}_n =q^{(k+n)/2}\, e^{(k)}_n, \label{A}\\
&&C\, e^{(k)}_n = q^{(1-2k-2n)/4}
{{\sqrt{(1-q^{n})(1-q^{2k+n-1})}}\over{q^{1/2}-q^{-1/2}}}\,
e^{(k)}_{n-1}, \label{C}\\
&&B \, e^{(k)}_n = q^{-(1+2k+2n)/4}
{{\sqrt{(1-q^{n+1})(1-q^{2k+n})}}\over{q^{-1/2}-q{1/2}}}\,
e^{(k)}_{n+1}. \label{B} 
\eea
Let $s\in\Real\backslash\{0\}$, and define
\beq
Y_s = q^{1/4}B-q^{-1/4}C + {{s^{-1}+s}\over{q^{-1/2}-q^{1/2}}}(A-D).
\label{Ys}
\eeq
Under the comultiplication
\def\De{\Delta}
\bea
\De(A)=A\otimes A,\quad \De(B)=A\otimes B+B\otimes D,\nn \\
\De(C) = A\otimes C+C\otimes D, \quad \De(D)=D\otimes D,
\label{comult}
\eea
$Y_s$ is twisted primitive, i.e.
\beq
\Delta(Y_s)=A\otimes Y_s + Y_s \otimes D,
\eeq
and $Y_sA$ is a self-adjoint element. 
We shall examine when the formal vectors 
\beq
v^{(k)}(x)=\sum_{n=0}^\infty l^{(k)}_n(x) e^{(k)}_n
\label{q-fv}
\eeq
are generalized eigenvectors of $Y_sA$. The results are in terms of 
Askey-Wilson polynomials ($x=\cos\th$)~:
\beq
p_m(x;a,b,c,d|q) = a^{-m} (ab,ac,ad;q)_m\, {}_4\vp_3
\left( {{q^{-m},abcdq^{m-1},ae^{i\th},ae^{-i\th}}\atop
{ab,\ ac,\ ad}}; q,q\right),
\label{a-w}
\eeq
and Al-Salam--Chihara polynomials~\cite{KS,GR} 
$$s_m(x;a,b|q) = p_m(x;a,b,0,0|q).$$ 
We follow~\cite{GR} for all notation related to $q$-shifted factorials
and basic hypergeometric series. 
Using the abbreviation $\mu(x)=(x+x^{-1})/2$, one can show~\cite{KV}
that for 
\beq
l^{(k)}_n(x) \equiv S_n(\mu(x);q^ks,q^k/s|q) = {1\over
\sqrt{(q,q^{2k};q)_n}} s_n(\mu(x);q^ks,q^k/s|q), 
\label{l-a-c}
\eeq
the vectors (\ref{q-fv}) are formal eigenvectors of $Y_sA$ for the
eigenvalue $$\la(x)=2(\mu(s)-\mu(x))/(q^{1/2}-q^{-1/2}).$$

The tensor product of two $U_q(su(1,1))$ representations
$(k_1)\otimes(k_2)$ is the same as in (\ref{tensdec}), and the
Clebsch-Gordan coefficients in 
\beq
e^{(k_1k_2)k}_n = \sum_{n_1,n_2} C^{k_1,k_2,k}_{n_1,n_2,n}(q)\ 
e^{(k_1)}_{n_1} \otimes e^{(k_2)}_{n_2},
\label{q-CGC}
\eeq
are known to be proportional to a $q$-Hahn polynomial. Explicitly,
\beq
C^{k_1,k_2,k_1+k_2+j}_{n_1,n_2,n}(q) =
C\ Q_j(q^{-n_1};q^{2k_1-1},q^{2k_2-1},n+j;q),
\eeq
with the constant $C$ given by
$$
{{ q^{k_1(n-n_1)}
(q;q)_{n+j}
\sqrt{(q^{2k_1};q)_{n_1} (q^{2k_2};q)_{n_2} (q^{2k_1};q)_j}}
\over {\sqrt{ (q;q)_n(q;q)_{n_1}(q;q)_{n_2}(q;q)_j
(q^{2k_1+2k_2+2j};q)_n (q^{2k_2};q)_j
(q^{2k_1+2k_2+j-1};q)_j}}} ,
$$
and $Q_j$ a $q$-Hahn polynomial~:
\beq
Q_j(q^{-x};a,b,N;q) =
{}_3\vp_2 \left( {{q^{-j}, q^{-x},abq^{j+1}}\atop{
aq,\ q^{-N}}};q,q\right).
\eeq

In the tensor product space, 
\beq
v^{k_1;k_2}(x_1,x_2)=\sum_{n_1,n_2}
{S_{n_1}(\mu(x_1);q^{k_1}x_2,q^{k_1}/x_2|q)} 
{S_{n_2}(\mu(x_2);q^{k_2}s,q^{k_2}/s|q)}
 e^{(k_1)}_{n_1} \otimes e^{(k_2)}_{n_2}
\label{q-v1}
\eeq
is a generalized eigenvector of $\De(Y_sA)$ for the eigenvalue
$\la(x_1)$. In ``coupled'' form, 
\beq
v^{(k_1k_2)k}(x_1)=\sum_{n=0}^\infty
{S_{n}(\mu(x_1);q^{k}s,q^{k}/s|q)} e^{(k_1k_2)k}_n,
\label{q-v2}
\eeq
is a generalized eigenvector of $\De(Y_sA)$ for the same eigenvalue.
The relation between the two generalized eigenvectors is given
by~\cite{KV}~: 
\beq
v^{k_1;k_2}(x_1,x_2)=\sum_{j=0}^\infty C_j p_j(\mu(x_2);q^{k_1}x_1,
q^{k_1}/x_1, q^{k_2}s, q^{k_2}/s|q) v^{(k_1k_2)k_1+k_2+j}(x_1),
\label{q-vv}
\eeq
where $p_j$ is an Askey-Wilson polynomial and 
$$
C_j=\left( (q,q^{2k_1},q^{2k_2},q^{2k_1+2k_2+j-1};q)_j \right)^{-1/2}. 
$$
Thus, the $q$-analog of (\ref{ident}) reads,
\bea
&& \sum_{n_1+n_2=n+j} C^{k_1,k_2,k}_{n_1,n_2,n}(q) 
{S_{n_1}(\mu(x_1);q^{k_1}x_2,q^{k_1}/x_2|q)} 
{S_{n_2}(\mu(x_2);q^{k_2}s,q^{k_2}/s|q)} = \nn\\
&& \qquad {S_{n}(\mu(x_1);q^{k}s,q^{k}/s|q)}\ C_j\
p_j(\mu(x_2);q^{k_1}x_1, 
q^{k_1}/x_1, q^{k_2}s, q^{k_2}/s|q).
\label{sp}
\eea
This relation gives rise to a new convolution identity for the
Al-Salam--Chihara polynomials involving the $q$-Hahn polynomials and 
the Askey-Wilson polynomials~:
\bea
&&(q^{2k_1};q)_j \sum_{l=0}^{n+j}
q^{k_1(n-l)}
\left[ {{n+j}\atop l}\right]_{q}\
Q_j(q^{-l};q^{2k_1-1}, q^{2k_2-1},n+j;q)\nn\\
&&\quad\times
s_l(\mu(x_1);q^{k_1}x_2,q^{k_1}/x_2|q)\,
s_{n+j-l}(\mu(x_2);q^{k_2}s,q^{k_2}/s|q) = \nn\\
&& s_n(\mu(x_1);q^{k_1+k_2+j}s,q^{k_1+k_2+j}/s|q)\,
p_j(\mu(x_2);q^{k_1}x_1,q^{k_1}/x_1,q^{k_2}s,q^{k_2}/s|q).
\eea

Using similar techniques in the tensor
product of three representations, and the explicit expression of 
$U_q[su(1,1)]$ Racah coefficients in terms of $q$-Racah polynomials,
one can obtain a convolution identity for the Askey-Wilson polynomials.


\begin{thebibliography}{99}
\itemsep=-.2pc
\bibitem{VK}
N.Ja.\ Vilenkin and A.U.\ Klimyk, {\em Representation of Lie Groups and
Special Functions,} vols.~1, 2 and 3 (Kluwer Academic Publishers,
Dordrecht, 1991).
\bibitem{GZ}
Ya.I.\ Granovski\u\i\ and A.S.\ Zhedanov, ``New construction of
$3nj$-symbols,'' {\em J.\ Phys.\ A} {\bf 26} (1993) 4339--4344.
\bibitem{V}
J.~Van der Jeugt,``Coupling coefficients for Lie algebra
representations and addition formulas for special functions,''
University of Ghent preprint (1996). 
\bibitem{KV}
H.T.\ Koelink and J.\ Van der Jeugt, ``Convolutions for orthogonal
polynomials from Lie and quantum algebra representations,'' University
of Amsterdam preprint 96-11 (1996); see also {\tt q-alg/9607010}.
\bibitem{KS}
R.~Koekoek and R.F.~Swarttouw,
``The Askey-scheme of hypergeometric or\-tho\-go\-nal polynomials
and its $q$-analogue,'' Report 94-05, Technical University Delft (1994);
available from {\tt ftp.twi.tudelft.nl} in
directory /{\tt pub}/{\tt publications}/{\tt tech-reports}.
\bibitem{GR}
G.~Gasper and M.~Rahman, {\em Basic Hypergeometric Series,}
(Cambridge University Press, 1990).
\end{thebibliography}

\end{document}