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\begin{center}
{ \large \bf CONVOLUTION AND ADDITION THEOREMS \par FOR $q$-ORTHOGONAL POLYNOMIALS}
%{\Large \bf Convolution and addition theorems \par for $q$-orthogonal polynomials}
\end{center}
\begin{center}
{ \large \bf Erik~Koelink}
\end{center}

\vspace*{1cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
A short review of the use of quantum groups to addition
theorems for $q$-special functions is given. We discuss
a general convolution formula for Al-Salam and Chihara
polynomials arising from its quantum algebra as
transition coefficients. This contribution does not
contain new results.
\end{abstract}

\vspace*{1cm}

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\section{Introduction}

As is very well known there is a strong connection between
representation theory of groups and related structures, 
such as Lie groups, Lie algebras, finite groups, quantum 
groups, Hecke algebras, 
and all kinds of special functions, such as orthogonal
polynomials of hypergeometric type or basic hypergeometric type
in one or more variables, 
see e.g. Vilenkin and Klimyk 
\cite{VileK} and references therein. 
The theory of special functions has a very rich tradition
dating back to the great mathematicians of the 18th and 
19th century such as Euler, Gau\ss, Heine, Riemann, Laplace,
Legendre and others. The connection with
group theory is much more recent and started with
Gelfand and \v Sapiro in the 1950's who discovered
the relation between Jacobi polynomials and the
special linear group $SL(2,{\Bbb C})$. Since then
the literature on this relationship has expanded
explosively and many connection between group theory
and special functions, mainly of hypergeometric type,
have been established.

Very soon after the introduction of quantum groups 
by Jimbo, Drinfeld and Worono\-wicz it
became clear that these `groups' are strongly related
to $q$-special functions in a way that resembles the
relation discussed in the previous paragraph.
In \S 2 we discuss
a quantum group analogue of the Gelfand-\v Sapiro result
and use it to derive an addition formula for a
two-parameter family of Askey-Wilson polynomials. 
Classically, i.e. in the corresponding group case,
this would give the addition formula for 
the Legendre polynomials. This section is of a 
descriptive nature and takes all of its results
from \cite{KoelFIC}.

In \S 3 we discuss how a very general convolution
identity for the Al-Salam and Chihara polynomials
which also involves Askey-Wilson polynomials. It arises
from the interpretation of the Al-Salam and Chihara
polynomials as transition coefficients in the
representation space of the tensor product of two
positive discrete series representations of the
quantised universal enveloping algebra for
${\frak{su}}(1,1)$. Since we work with a tensor product
the Clebsch-Gordan coefficients will come into play.
As a special case we obtain 
the convolution property which originally led
Al-Salam and Chihara to the introduction of these
polynomials. The result discussed in \S 3 is a special
case of more general results obtained in collaboration
with Joris Van der Jeugt \cite{KoelJ} using an idea
of Granovskii and Zhedanov.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Addition theorems}

In this section we state some generalities on addition
theorems and we discuss some of the addition theorems
that can be derived using quantum groups without going
through (heavy) calculations. This section doesn't
contain any new results.
Explicit formulas with proofs and 
references to the literature for other approaches
to addition formulas for $q$-special functions
and addition formulas for specific sets of $q$-special
functions can be found in \cite{KoelFIC}, see also
\cite{KoelAAM}, \cite{KoelRCM}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Generalities on addition formulas}

The exponential function satisfies the addition
formula $e^{x+y}=e^xe^y$. More generally, the function
$f$ satisfies an {\sl algebraic addition theorem} if
$A(f(x),f(y),f(x+y))$ is identically zero for some
polynomial $A(x,y,z)$. By a theorem
of Weierstra\ss , a single valued analytic function $f(z)$ 
satisfying an algebraic addition theorem is either a rational 
function in $z$, or a rational function in $e^{\lambda z}$
for some $\lambda$, or an elliptic function, see Askey 
\cite[Lecture~4]{Aske} for more information and references.
The addition formula for Legendre polynomials $P_n(x)$ 
(Laplace, 1872) is the prototype of an addition theorem of a 
more general nature. It gives the explicit Fourier-cosine 
expansion of $P_n(\cos \phi \cos \psi + \sin \phi \sin \psi 
\cos \theta)$ in terms of $\cos m\theta$. The Fourier coefficients 
depending on $\phi$ and $\psi$ factorize as products of two 
associated Legendre functions. This addition formula was 
generalized by Gegenbauer (1875) to ultraspherical polynomials and
more generally by Koornwinder (1972) to Jacobi polynomials.
Addition theorems of this type exist for a large number of 
special functions of hypergeometric type. See
\cite[Lecture~4]{Aske} and in particular
\cite{Wats} for several addition formulas for Bessel functions.

For a more general definition of an addition formula
we follow Koornwinder \cite[\S 4]{KoorJLMS}.
An {\sl addition formula} for a family of special functions 
$\phi_\lambda$ expands $\phi_\lambda(\Lambda(x,y,t))$ ($\Lambda$ 
a fixed elementary function) explicitly and orthogonally in terms 
of certain special functions $\psi_\mu(t)$ with expansion 
coefficients of the form constant time $\phi_\lambda^\mu(x)
\,\phi_\lambda^\mu(y)$. The $\phi_\lambda^\mu$ are special 
functions {\sl associated with} $\phi_\lambda$. A further 
requirement is that $\psi_0(t)=1$ and $\phi_\lambda^0=
\phi_\lambda$. An important corollary of any addition formula
is the {\sl product 
formula} giving $\phi_\lambda(x) \phi_\lambda(y)$ as an integral 
of $\phi_\lambda$ using the orthogonality properties 
of $\psi_\mu(t)$.
The addition theorems for the Legendre, Gegenbauer and Jacobi
polynomials fit into this definition.

For special functions of basic hypergeometric type this
definition is no longer sufficiently general, although
the addition theorems for the spherical functions on
finite groups of Lie type, which are expressible in terms
of $q$-Krawtchouk polynomials, are still of this type,
see Stanton \cite{Stan}. However, for most of the 
addition theorems derived from the interpretation of
special functions on quantum groups this definition
no longer suffices. Then we use the following
variant that $\phi_\lambda(t)$, or even 
$\phi_\lambda(t)\psi^{(x,y)}_\nu(t)$, is expanded 
explicitly and orthogonally in terms of 
$\psi_\mu^{(x,y)}(t)$, with expansion coefficients 
of the form  $\xi_\lambda^\mu(x)
\,\eta_\lambda^\mu(y)$ for certain special functions
$\xi_\lambda^\mu$ and $\eta_\lambda^\mu$.
This type of addition formula also gives as a corollary the product 
formula for $\xi_\lambda^0\eta_\lambda^0(y)$. 
Of course, $\xi_\lambda^\mu$ and $\eta_\lambda^\mu(y)$ are
related to $\phi_\lambda$ in some sense. In the next
subsection we see this for the case that $\phi_\lambda$
is an Askey-Wilson polynomial and the $\xi_\lambda^\mu$ and 
$\eta_\lambda^\mu(y)$ are also Askey-Wilson polynomials
but with different parameters.

In general addition formulas are hard to prove from scratch.
Additional structure can provide a conceptual setting
for such formulas. For the Jacobi polynomials and
Bessel functions it is known that addition theorems 
follow from the representation
theory of specific Lie groups. For certain $q$-special functions
we can use the theory of quantum groups as sketched in the
next subsections.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Addition theorem from Askey-Wilson polynomials}

To understand the quantum group approach we recall that a 
quantum group $A_q(G)$ is a deformation 
(with deformation parameter $q$) of a function
algebra $A(G)$ on the
group $G$. For a compact group $G$ this would mean the 
polynomials on $G$ or the continuous functions on $G$,
see Chari and Pressley \cite{CharP}, Kassel \cite{Kass}
or \cite{KoelRCM} and references therein.
We assume that the comultiplication
$$
\Delta \colon A(G)\to A(G)\otimes A(G)
\cong A(G\times G),
\qquad \bigl( \Delta f\bigr) (g,h) = f(gh),
$$
is well-defined and we demand that
$\Delta \colon A_q(G)\to A_q(G)\otimes A_q(G)$
survives the deformation unchanged (in a suitable sense). 
In the deformation the representations of $G$
correspond to corepresentations of $A_q(G)$ and for a 
suitable basis of the representation space we get matrix elements
$t_{n,m}\in A_q(G)$ satisfying the cohomomorphism property
\begin{equation}
\Delta(t_{n,m}) = \sum_p t_{n,p}\otimes t_{p,m},
\label{vglDeltamatrixelt}
\end{equation}
For the undeformed case this is just
$t_{n,m}(gh) = \sum_p t_{n,p}(g) t_{p,m}(h)$, $g,h\in G$,
meaning that $t$ is a representation of $G$. Classically, this 
identity leads to addition formulas for special functions, 
see \cite{VileK} for numerous examples.

If the matrix elements can be expressed in terms of 
$q$-special functions we consider
(\ref{vglDeltamatrixelt}) as an (implicit) addition formula. 
Usually
$A_q(G)$ is a non-commutative algebra and thus we need to 
investigate the representations of $A_q(G)$ in order to 
convert (\ref{vglDeltamatrixelt}) 
into an addition formula involving $q$-special
functions in commuting variables. Usually, we normalise $n=m=0$ in
(\ref{vglDeltamatrixelt}) and $t_{0,0}$ is considered as 
an analogue of a spherical function, i.e. as a function which 
is left and right $K$-invariant for a certain subgroup 
$K\subset G$.

In some specific cases we can fill all the details of
this program in order to derive very general addition 
theorems for $q$-special functions. As an example we
discuss the addition formula for a two-parameter
family of Askey-Wilson polynomials arising from 
the interpretation of Askey-Wilson polynomials as
generalised matrix elements on the quantum $SU(2)$ group.

For the quantum
$SU(2)$ group we have a generalisation of 
(\ref{vglDeltamatrixelt}) by varying the subgroup $K$.
Then all the matrix elements of the finite dimensional
irreducible representations, which can be labelled
by the spin $l\in{1\over 2}{\Bbb Z}_+$ such that the dimension is
$2l+1$, are explicitly known in terms of Askey-Wilson
polynomials \cite{AskeW}. This set of orthogonal polynomials
has four degrees of freedom (apart form the $q$) and these
correspond in this interpretation with $n$ and $m$, i.e.
discrete parameters corresponding to the labeling of the
vectors in the representation space, and two parameters,
say $\sigma$ and $\tau$,
labeling the `quantum subgroups' under which 
these matrix elements behave nicely for left and right
multiplication. This is due to Noumi and Mimachi following
Koornwinder's introduction of `quantum subgroups', 
see references in \cite{KoelAAM}, \cite{KoelRCM}. 
In this case, taking $n=m=0$ in the appropriate
analogue of (\ref{vglDeltamatrixelt}),
we get 
\begin{equation}
a^l_{0,0}(\tau,\sigma) = \sum_{p=-l}^l a^l_{0,p}(\tau,\mu)\otimes 
a^l_{p,0}(\mu,\sigma), \qquad l\in {\Bbb Z}_+,
\label{vglgeneralmatelts}
\end{equation}
as an identity in $A_q(SU(2))\otimes A_q(SU(2))$. In
general the (generalised) matrix elements $a^l_{n,m}(\tau,\sigma)$
are an explicitly known element of $A_q(SU(2))$ times
$p_{l-k}^{(|n-m|,|n+m|)}(\rho;s,t|q^2)$, with 
$n,m\in\{ -l,\ldots,l\}$, $k=\max( |n|,|m|)$, 
labeling the vectors,
$s,t\in\{ q^{\pm \sigma},q^{\pm\tau}\}$ with $\sigma$, $\tau$
labeling the quantum subgroups and $\rho\in A_q(SU(2))$ some
explicitly known quadratic element. This analogue of the
Jacobi polynomial is
$$
p_n^{(\alpha,\beta)}(x;s,t| q) = 
p_n(x;q^{1/2}t/s,q^{1/2+\alpha}s/t,
-q^{1/2}/(st), -stq^{1/2+\beta}| q),
$$
where the left hand side denotes a standard Askey-Wilson
polynomial. For $s=t=1$ this corresponds to Rahman's
continuous $q$-Jacobi polynomials, see \cite{AskeW}.

So (\ref{vglgeneralmatelts}) can be considered as the
first stage of an addition formula. To obtain the second
stage we use the representation theory of the non-commutative
algebra $A_q(SU(2))$. Its irreducible $\ast$-representations 
are either one-dimensional or infinite dimensional acting in 
$\ell^2({\Bbb Z}_+)$. Applying a one dimensional representation
on the first leg and an infinite dimensional representation
on the second leg of (\ref{vglgeneralmatelts}) we obtain
an operator identity in the space of bounded linear operators
of $\ell^2({\Bbb Z}_+)$. The spectral decompostion of the
operator corresponding to $\rho$, which is a five-term
recurrence operator, can be determined by splitting the
five-term recurrence in two three-term recurrences, which
each can be solved using orthogonal polynomials, 
namely Al-Salam and Carlitz polynomials and 
Al-Salam and Chihara polynomials, see \cite{KoelV}
for the explicit calculations. Having these (generalised)
eigenvectors at hand we can let the operator identity on
such an eigenvector and take inner product with the
appropriate eigenvectors for the right hand side. This leads to 
a very general formula, which we call an addition theorem
for a two-parameter family of Askey-Wilson polynomial, see 
\cite[Thm.~4.1]{KoelFIC}. However, in general this 
addition formula doesn't fit the general definition
of \S 2.1, since the functions $\psi_\mu^{(x,y)}(t)$
are not (yet) known to be orthogonal in a suitable sense.
In this case the $\psi_\mu^{(x,y)}(t)$ are explicitly
known in terms of very-well-poised ${}_8\varphi_7$-series,
see Gasper and Rahman \cite{GaspR}, which is related
to the non-symmetric Poisson kernel for the
Al-Salam and Chihara polynomials, see 
\cite{AskeRS}. It's an open problem to find suitable 
orthogonality relations for the functions $\psi_\mu^{(x,y)}(t)$
in this case.

Taking a suitable limit $\mu\to\infty$
for the free parameter $\mu$ in the right hand side of
(\ref{vglgeneralmatelts})
we can derive an addition formula
for a two-parameter Askey-Wilson polynomial which
fits into the general definition of \S 2.1, 
see \cite[\S 5]{KoelFIC}, \cite[Lecture~6]{KoelRCM}.
Then the Askey-Wilson polynomials in the right hand side
of (\ref{vglgeneralmatelts}) go over in big $q$-Jacobi
polynomials, so the functions $\xi_\lambda^\mu$ and 
$\eta_\lambda^\mu$ of \S 2.1 are expressible in terms
of big $q$-Jacobi polynomials. 
For this case the functions $\psi_\mu^{(x,y)}(t)$ 
are $q$-analogues of the Laguerre polynomials which can
be obtained by letting $\beta\to\infty$ in
$p_n^{(\alpha,\beta)}(\cdot;s,t|q)$.
This addition formula can also be given a direct 
proof, cf. \cite[Lecture~6]{KoelRCM},
and this was presented in one of my lectures at IWOP'96.

If we let also $\sigma=\tau\to\infty$ we obtain the
addition formula for little $q$-Legendre polynomials.
In the definition of \S 2.1 we have $\phi_\lambda$ 
as little $q$-Legendre polynomials, 
$\xi_\lambda^\mu$ and  $\eta_\lambda^\mu$ are
expressed in terms of little $q$-ultraspherical
polynomials, and the $\psi_\mu^{(x,y)}(t)$ are
Wall polynomials. This addition formula is due
to Koornwinder \cite{KoorSIAM} and is the first
example of an addition formula derived from the relation
between quantum groups and $q$-special functions.

A decade before this general addition formula 
Rahman and Verma \cite{RahmV} proved an addition formula
for the continuous $q$-ultraspherical polynomials 
by analytic means. This addition formula has the same
structure as the addition formula for the 
ultraspherical (or Gegenbauer) polynomials.
For the case $\sigma=\tau=\mu=0$
the Rahman-Verma addition formula for the continuous
$q$-Legendre polynomials should be recovered from the
general formula described here. This is indeed the case,
see \cite[\S 5]{KoelFIC} and \cite[\S 4]{KoelSIAM} for
another derivation of the Rahman-Verma result starting
from (\ref{vglgeneralmatelts}) using only one-dimensional
representations of $A_q(SU(2))$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Addition formulas from quantum groups}

In the previous subsection we discussed the 
quantum group theoretic derivation of an addition
theorem for a two-parameter subclass of Askey-Wilson
polynomials. This method also works for other groups,
notably the quantum group of plane motions, and
analogues of the space $U(n)/U(n-1)$, see references
in \cite{KoelFIC}.

Instead of working with deformed function algebras
it is also possible to work with the dual object
(in a suitable sense) of deformed universal enveloping
algebras. In this approach (see also the next section) 
it's also possible to derive addition formulas for
special functions. In a certain sense this method is
more general since it needs less
structure. See the references to the papers
by Kalnins, Miller and coworkers, and by 
Floreanini and Vinet in \cite{KoelFIC}.

We may consider also (\ref{vglDeltamatrixelt}) or
(\ref{vglgeneralmatelts}) as addition formulas 
in non-commuting variables. Even at this level
these formulas turn out to be useful. See e.g.
Floris \cite{Flor}, where non-negativity of
linearisation coefficients for so-called disk
polynomials involving little $q$-Jacobi polynomials
is proved from a non-commutative addition formula.
Its proof is similar to the proof in \cite{KoorJLMS}.
This proof also can be used to prove non-negativity
of the linearisation coefficients for 
$p_l^{(0,0)}(\cdot;s,s|q)$ from (\ref{vglgeneralmatelts}).
See \cite[\S 8.3]{KoelAAM}, \cite[\S 7]{KoorCM} for
different proofs of this fact and \cite[\S 5]{KoelJ},
cf. Remark \ref{remthmgeneralconvols}(iv),
for an explicit formula for the linearisation
coefficients for the more general case 
$p_l^{(0,0)}(\cdot;s,t|q)$.
See Koornwinder \cite{KoorFIC} for more information
on $q$-special functions in non-commuting variables.

Addition theorems for $q$-Krawtchouk polynomials
for three specific choices of the parameter
have been derived by Stanton \cite{Stan} using its
interpretation as spherical functions on certain
finite groups of Lie type. This interpretation
can be generalised to an interpretation of 
$q$-Krawtchouk polynomials on the Hecke
algebra of type $B$, see \cite{KoelHA}, as explained
in my other lecture at IWOP'96. It's my hope that
this may lead to an addition formula for 
$q$-Krawtchouk polynomials.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convolution theorems}

The results in this section have been obtained in collaboration
with Joris Van der Jeugt (Universiteit Gent, Belgium). Full details
for this section as well as references and the discussion
for the case of the continuous Hahn, Hahn, Meixner-Pollaczek, 
Meixner, Laguerre and Hermite polynomials 
and the Askey-Wilson polynomials can be found in \cite{KoelJ}.
For the orthogonal polynomials of hypergeometric type
this follows using the Lie algebra ${\frak{su}}(1,1)$,
whereas for the Al-Salam and Chihara and the Askey-Wilson
polynomials we have to use the quantised universal
enveloping algebra $U_q({\frak{su}}(1,1))$.

\subsection{Al-Salam and Chihara polynomials}

The convolution of two sequences $(a_n)_{n=0}^\infty$ and
$(b_n)_{n=0}^\infty$ is the sequence $(c_n)_{n=0}^\infty$
with $c_n=\sum_{k+l=n} a_kb_l$. This arises for instance
from multiplying the formal power series 
$f(z)=\sum_{n=0}^\infty a_nz^n$ by 
$g(z)=\sum_{n=0}^\infty b_nz^n$ and then the $c_n$'s are
the coefficients in the resulting power series.
In case $a_n=p_n(x)$, $b_n=r_n(y)$ for a certain set of
orthogonal polynomials, we have $c_n=Q_n(x,y)$ for some
polynomial in two variables of total degree $n$.
Al-Salam and Chihara \cite{AlSaC}, see also \cite[\S 8]{AlSa},
posed and solved the question for which orthogonal polynomials
$p_n$ and $r_n$ the polynomial $Q_n(x,y)$ is orthogonal as
a polynomial in $x$. This led to the introduction of the
Al-Salam and Chihara polynomials, which was later identified
as a two-parameter subset of the Askey-Wilson polynomials.

To put this convolution property in a wider perspective
let us introduce the orthogonality measures $\mu$ and $\nu$ 
for $p_n$ and $r_n$. Then the two-variable polynomial
$R_{n,m}(x,y) = p_n(x)r_m(y)$ is orthogonal with 
respect to the product measure of $\mu$ and $\nu$. From 
the convolution property
$Q_n(x,y) = \sum_{k=0}^n R_{k,n-k}(x,y)$ we get
$$
\int\hspace*{-0.3cm}\int Q_n(x,y)Q_m(x,y)\, 
d\mu(x) d\nu(y) = 0, \qquad n\not=m.
$$
Now a natural question arises: is $Q_n(x,y)$ part of a 
set of two-variable orthogonal (with respect to the product
measure of $\mu$ and $\nu$) polynomials $P_{k,l}(x,y)$?
If so, can we calculate the connection coefficients
between $R_{n,m}(x,y)$ and $P_{k,l}(x,y)$ in general?
In the rest of this section we indicate how the
quantised universal enveloping algebra for ${\frak{su}}(1,1)$ 
can be used to obtain a formula which answers this question
for the Al-Salam and Chihara polynomials.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Positive discrete series representations}

{}From now on $q$ is assumed to satisfy $0<q<1$.
Let $U_q({\frak{su}}(1,1))$ be the complex unital
associative $\ast$-algebra generated by $A$, $B$, $C$, 
$D$ subject to the relations
$$
AD=1=DA, \quad AB=qBA,\quad AC=q^{-1}CA,\quad
BC-CB = {{A^2-D^2}\over{q-q^{-1}}}.
$$
The $\ast$-structure on $U_q({\frak{su}}(1,1))$ is given by
$A^\ast=A$, $B^\ast=-C$, $D^\ast=D$.

The positive discrete series representations $\pi_k$ of 
are $\ast$-representations of $U_q({\frak{su}}(1,1))$
labelled by $k>0$. They act in $\ell^2({\Bbb Z}_+)$
and the action of the generators is given by
\begin{eqnarray*}
\pi_k(A)\, e^k_n &=&q^{k+n}\, e^k_n, 
\qquad \pi_k(D)\, e^k_n =q^{-k-n} \\
\pi_k(C)\, e^k_n &=& q^{1/2-k-n}
{{\sqrt{(1-q^{2n})(1-q^{4k+2n-2})}}\over{q-q^{-1}}}
\, e^k_{n-1}, \\
\pi_k(B) \, e^k_n &=&
q^{-1/2-k-n}
{{\sqrt{(1-q^{2n+2})(1-q^{4k+2n})}}\over{q^{-1}-q}}
\, e^k_{n+1}. 
\end{eqnarray*}
Note that $\pi_k(D)$ is an unbounded operator, but that
$\pi_k(A)$, $\pi_k(B)$ and $\pi_k(C)$ are bounded. We consider
only bounded operators.

Recall that $U_q({\frak{su}}(1,1))$ is actually a Hopf algebra,
so that we can define the tensor product of two representations
by use of the comultiplication.
The tensor product of two positive discrete series representation
decomposes as for the Lie algebra ${\frak{su}}(1,1)$ in 
positive discrete series representations;
$\pi_{k_1}\otimes \pi_{k_2} \cong
\bigoplus_{j=0}^\infty \pi_{k_1+k_2+j}$.
So there exists a unitary matrix mapping
the orthogonal basis $e_{n_1}^{k_1}\otimes e_{n_2}^{k_2}$ onto
$e_n^{k_1+k_2+j}$ intertwining the action of 
$U_q({\frak{su}}(1,1))$.
The matrix elements of this unitary mapping are
the Clebsch-Gordan coefficients;
\begin{equation}
e_n^k = \sum_{n_1,n_2=0}^\infty C^{k_1,k_2,k}_{n_1,n_2, n}\,
e_{n_1}^{k_1}\otimes e_{n_2}^{k_2},
\label{vglCGC}
\end{equation}
where $k=k_1+k_2+j$ for $j\in {\Bbb Z}_+$. The sum is finite;
the Clebsch-Gordan coefficients are non-zero only for 
$n_1+n_2=n+j$ as follows from considering the action of the
group-like element $A$ on both sides of (\ref{vglCGC}).
The Clebsch-Gordan coefficients are normalised
by $\langle e_0^k,e_0^{k_1}\otimes e_j^{k_2}\rangle >0$.

These results can be found in Burban and Klimyk \cite{BurbK}
and Kalnins, Manocha and Miller \cite{KalnMM}. See
Chari and Pressley \cite{CharP} and Kassel \cite{Kass} 
for general information on
quantised universal enveloping algebras.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spectral analysis of a self-adjoint element}

We now define
$$
Y_s = q^{1/2}B-q^{-1/2}C + {{s^{-1}+s}\over{q^{-1}-q}}(A-D)
\in U_q({\frak{su}}(1,1)).
$$
Then $Y_sA$ is a self-adjoint element in 
$U_q({\frak{su}}(1,1))$ for
$s\in{\Bbb R}\backslash\{ 0\}$, or
$s\in{\Bbb T}$. We want to calculate the action of $Y_sA$
in $\pi_{k_1}\otimes \pi_{k_2}$ and in 
$\bigoplus_{j=0}^\infty \pi_{k_1+k_2+j}$, and then match the
results.
 
In order to do so we first find the spectrum of $\pi_k(Y_sA)$.
We need the orthonormal Al-Salam and Chihara polynomials 
$S_n(x;a,b|q)$, which satisfy
the three-term recurrence relation
$$
2x\, S_n(x) = a_{n+1}\, S_{n+1}(x) + q^n(a+b)\, S_n(x) +
a_n\, S_{n-1}(x),
$$
$a_n = \sqrt{ (1-abq^{n-1})(1-q^n)}$. Let $dm(\cdot;a,b|q)$
denote the corresponding normalised orthogonality measure,
see \cite{AskeW} with $c=d=0$. Let $M_g$ denote multiplication
by the function $g$, i.e. $M_gf(x)= g(x)f(x)$. We use 
the notation $\mu(x)=(x+x^{-1})/2=\mu(x^{-1})$ for $x\not= 0$.

\begin{prp} \label{propspectrumYsApik}
Let $\Lambda\colon \ell^2({\Bbb Z}_+) \to 
L^2({\Bbb R},dm(\cdot;q^{2k}s,q^{2k}/s|q^2))$
be the unitary mapping defined by
$\Lambda \colon e^k_n \mapsto S_n (\cdot;q^{2k}s,q^{2k}/s|q^2)$,
then $\Lambda$ intertwines $\pi_k(Y_sA)$ acting in
$\ell^2({\Bbb Z}_+)$
with $2(M_x-\mu(s))/(q^{-1}-q)$.
\end{prp}

Since $\Delta(Y_sA)=A^2\otimes Y_sA +Y_sA\otimes 1$ we can 
also calculate the spectrum of $Y_sA$ in 
$\pi_{k_1}\otimes \pi_{k_2}$ using Proposition 
\ref{propspectrumYsApik}.

\begin{prp} \label{propspectrumYsAintensorprod}
Define $\Upsilon\colon \ell^2({\Bbb Z}_+)\otimes\ell^2({\Bbb Z}_+)
\to L^2({\Bbb R}^2,dm(x_1,x_2))$,
where
$$
dm(x_1,x_2) = dm(x_1;q^{2k_1}w_2,q^{2k_1}/w_2|q^2)\,
dm(x_2;q^{2k_2}s,q^{2k_2}/s|q^2), \qquad x_2=\mu(w_2),
$$
by
$$
\Upsilon \colon
e^{k_1}_{n_1} \otimes e^{k_2}_{n_2}
\mapsto S_{n_1} (x_1;q^{2k_1}w_2,q^{2k_1}/w_2|q^2)\,
S_{n_2}(x_2;q^{2k_2}s,q^{2k_2}/s|q^2)
$$
then $\Upsilon$ is a unitary mapping intertwining
$\pi_{k_1}\otimes\pi_{k_2}(\Delta(Y_sA))$
with $2(M_{x_1}-\mu(s))/(q^{-1}-q)$ in $L^2(dm(x_1,x_2))$.
\end{prp}

The proof of both propositions consists of comparing 
three-term recurrence relations arising from the operator under 
consideration with the three-term
recurrence relations for the Al-Salam and Chihara polynomials.
For Proposition \ref{propspectrumYsAintensorprod} we use the
result of Proposition \ref{propspectrumYsApik} on the second
leg of the tensor product. See \cite{KoelJ} for details.

It's important to observe that $\Upsilon
e^{k_1}_{n_1} \otimes e^{k_2}_{n_2}$ gives a complete set of
orthonormal polynomials in $L^2({\Bbb R}^2,dm(x_1,x_2))$. 
Here we use that the Al-Salam and Chihara polynomial is
symmetric in its parameters $a$ and $b$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A general convolution}

{}From the Clebsch-Gordan decomposition we see that $e_n^k$
is also an orthonormal basis of the representation 
space. So $\Upsilon\, e_n^k$ gives another orthonormal
basis in $L^2({\Bbb R}^2,dm(x_1,x_2))$.

\begin{prp} \label{propUpsilononenk}
Let $k=k_1+k_2+j$ for $j\in{\Bbb Z}_+$, and $x_1=\mu(w_1)$ then
\begin{eqnarray*}
\bigl( \Upsilon\, e^k_n\bigr)(x_1,x_2) &=&
S_n(x_1;q^{2k}s,q^{2k}/s|q^2)
\, \bigl( \Upsilon\, e^k_0\bigr)(x_1,x_2), \\
\bigl( \Upsilon\, e^k_0\bigr)(x_1,x_2) &=&C\,
p_j(x_2;q^{2k_1}w_1, q^{2k_1}/w_1,q^{2k_2}s,q^{2k_2}/s|q^2),\\
C^{-1}
&=& \sqrt{ (q^2,q^{4k_1},q^{4k_2},q^{4k_1+4k_2+2j-2};q^2)_j}.
\end{eqnarray*}
\end{prp}

\begin{prf}
For the first part of Proposition \ref{propUpsilononenk}
we use that $\Upsilon$ intertwines 
$\pi_{k_1}\otimes\pi_{k_2}(\Delta(Y_sA))$
with a certain multiplication operator, leading to
a three-term recurrence relation for $\Upsilon\, e^k_n$
with respect to $n$. The second part is then proved by
identifying orthogonality measures as follows. 
Since $\Upsilon$ is unitary we have the orthogonality relations
$\delta_{nm}\delta_{kl} = 
\langle \Upsilon e^k_n,\Upsilon e^l_m\rangle=$
$$
\int S_n(x_1;q^{2k}s,q^{2k}/s|q^2)S_m(x_1;q^{2l}s,q^{2l}/s|q^2)
\int \Upsilon e^k_0(x_1,x_2) \Upsilon e^l_0(x_1,x_2)\, dm(x_1,x_2),
$$
by the first part of the proposition. Since the moment problem
for the Al-Salam and Chihara polynomials is determined we 
see that for $k=l$ we get 
$$
\int \Upsilon e^k_0(x_1,x_2) \Upsilon e^l_0(x_1,x_2)\, dm(x_1,x_2)
= \delta_{kl}\, dm(x_1;q^{2k}s,q^{2k}/s|q^2)
$$
as measures with respect to functions in the variable $x_1$.
It also holds for $k\not= l$, since then
$\int \Upsilon e^k_0(x_1,x_2) \Upsilon e^l_0(x_1,x_2)\, 
dm(x_1,x_2)=0$ because integrated against any polynomial it gives
zero. Knowing that $\Upsilon e^k_0(x_1,x_2)$ is a polynomial
of degree $j$, $k=k_1+k_2+j$, and the explicit expression for 
the orthogonality measures involved we find the second part
of the proposition.
\end{prf}

We can now apply $\Upsilon$ to both sides of 
(\ref{vglCGC}). Apart from the explicit value of the
Clebsch-Gordan coefficient we have a complete 
description of the resulting identity in 
$L^2({\Bbb R}^2,dm(x_1,x_2))$ due to 
Propositions \ref{propspectrumYsAintensorprod}
and \ref{propUpsilononenk}. The explicit value of 
the Clebsch-Gordan coefficients can be found in e.g.
\cite{KalnMM}, but it can also be obtained from 
the identity arising from the application of 
$\Upsilon$ to both sides of 
(\ref{vglCGC}). Specialising $x_1$ and $x_2$ 
suitably and observing that the Clebsch-Gordan coefficients
are independent of $s$, we obtain a generating function
from which the value for the Clebsch-Gordan coefficients
can be determined;
$$
C^{k_1,k_2,k_1+k_2+j}_{n_1,n_2,n} =
C\ Q_j(q^{-2n_1};q^{4k_1-2},q^{4k_2-2},n+j;q^2)
$$
with $n_1+n_2=n+j$. The constant is explicitly known and
$Q_j$ is the $q$-Hahn polynomial;
$$
Q_n(q^{-x};a,b,N;q) =
{}_3\varphi_2 \left( {{q^{-n}, q^{-x},abq^{n+1}}\atop{
aq,\ q^{-N}}};q,q\right).
$$
Plugging this information back into the formula we obtain
the following theorem.

\begin{thm} \label{thmgeneralconvols}
Let $p_n(x;a,b,c,d|q)$ be Askey-Wilson polynomials
as in {\rm \cite{AskeW}} and let the 
Al-Salam and Chihara polynomials $s_n(x;a,b|q)=p_n(x;a,b,0,0|q)$.
With $x_i=(w_i+w_i^{-1})/2$ {\rm ($i=1,2$)}, $n,j\in{\Bbb Z}_+$, 
$k_1,k_2>0$ we have
\begin{eqnarray*}
&& (q^{4k_1};q^2)_j \sum_{l=0}^{n+j}
q^{2k_1(n-l)}
\left[ {{n+j}\atop l}\right]_{q^2}\
Q_j(q^{-2l};q^{4k_1-2}, q^{4k_2-2},n+j;q^2)\\
&&\qquad\quad\times
s_l(x_1;q^{2k_1}w_2,q^{2k_1}/w_2|q^2)\,
s_{n+j-l}(x_2;q^{2k_2}s,q^{2k_2}/s|q^2) =
\\ && s_n(x_1;q^{2k_1+2k_2+2j}s,q^{2k_1+2k_2+2j}/s|q^2)\,
p_j(x_2;q^{2k_1}w_1,q^{2k_1}/w_1,q^{2k_2}s,q^{2k_2}/s|q^2).
\end{eqnarray*}
\end{thm}

\begin{rmk} \label{remthmgeneralconvols} {\rm 
(i) Theorem \ref{thmgeneralconvols} is a connection
coefficient formula for orthogonal polynomials in two
variables, orthogonal for the same measure $dm(x_1,x_2)$
of Proposition \ref{propspectrumYsAintensorprod}, where the
connection coefficients are given by the
$q$-Hahn polynomials. The dual connection coefficient problem
follows from the orthogonality for the Clebsch-Gordan coefficients
or, equivalently, from the orthogonality
relations for the dual $q$-Hahn polynomials.

(ii) The case $j=0$ gives a simple convolution
property for the Al-Salam and Chihara polynomials, since
the $q$-Hahn and the Askey-Wilson polynomial reduce to $1$.
This
was the motivation for Al-Salam and Chihara \cite{AlSaC} to
introduce the Al-Salam and Chihara polynomials as the
most general set of orthogonal polynomials still
satisfying a convolution property.
The case $n=0$ is also of interest, since then the
$q$-Hahn polynomial can be evaluated and the Al-Salam and
Chihara polynomial on the right hand side reduces to $1$.
In both cases we have a free parameter in the sum.

(iii) Using Theorem \ref{thmgeneralconvols} we can also
work in the three-fold tensor product representation of
positive discrete series representations. Instead
of Clebsch-Gordan coefficients we have to work with
Racah coefficients. We obtain 
a similar formula with the $q$-Hahn and the Al-Salam and Chihara
polynomials replaced by the $q$-Racah and Askey-Wilson 
polynomials, see \cite[\S 4]{KoelJ}. The resulting formula
has the same structure as Theorem \ref{thmgeneralconvols}
due to the Wigner-Eckhart theorem, and it contains 
Theorem \ref{thmgeneralconvols} as a special case. 
A special case of this general convolution property 
has been obtained by Dunkl \cite{DunkSIAM} by
working on the general linear group over a finite field
with $q$ elements. (Here $q$ is a prime power.) 
This is the ``$q$-analogue'' (in Stanton's
\cite{StanAKS} terminology) of Dunkl's \cite{DunkPJM} proof of 
a special case of \cite[Thm.~3.13]{KoelJ} using the
representation theory of the symmetric group, which is
the Weyl group of the general linear group over a finite field
with $q$ elements. 

(iv) Using the the result of the previous remark and
the interpretation of a two-parameter family of Askey-Wilson
polynomials on the quantum $SU(2)$ group, see \S 2, we
can find the explicit linearisation coefficients for 
these polynomials as the product of two terminating
balanced ${}_4\varphi_3$-series, see \cite[\S 5]{KoelJ}.
}
\end{rmk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{0.3cm}
\underline{ACKNOWLEDGEMENTS} \quad
I thank Paco Marcell\'an and Renato \'Alvarez-Nodarse for the
invitation to give lectures at the 1996 International Workshop on
Orthogonal Polynomials in Mathematical Physics. The research for
this paper is supported by the Netherlands Organization for
Scientific Research (NWO) under project number 610.06.100.


\vspace*{0.3cm}

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\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Your adress %%%%%%%%%%%%%%%%%%%%%%%
{\sc
\begin{flushleft}
{Erik~Koelink} \\
{Vakgroep Wiskunde, Universiteit van Amsterdam, 
Plantage Muidergracht 24,
1018 TV Amsterdam, the Netherlands} \\
{{\sf E-mail:} {\tt koelink@@wins.uva.nl}}
\end{flushleft}
}
\end{document}

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