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%   Hypergeometric series related to the 9-j coefficient of su(1,1)
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\begin{document}
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\begin{center}
{\Large \bf Hypergeometric series related to the 
9-$j$ coefficient of $\su(1,1)$}\\[3cm]
{\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}

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\vspace{1 cm}
\noindent {\bf Abstract}\\
We study various expressions for the 9-$j$ coefficient of $\su(1,1)$, with an emphasis on multiple hypergeometric series. An orthogonal polynomial in two discrete variables is associated to the 9-$j$ coefficient. Some applications are considered, dealing with formulas in which the 9-$j$ coefficient appears as a connection or expansion coefficient, and discussing some transformation formulas for double and triple hypergeometric series of unit argument.

\vspace{1cm}
\noindent Keywords~: Hypergeometric series, Kamp\'e de F\'eriet
functions, 9-$j$ coefficient, discrete orthogonal polynomial, multiple hypergeometric series, transformation formulas.

\vspace{5mm}
\noindent AMS codes~: 33C50, 33C45, 33C20, 33C30, 33C25, 81R05. 

\newpage

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\section{Introduction}  \label{sec-intro}
\setcounter{equation}{0}

The interplay between representation theory of Lie algebras (or quantum groups) and special functions can be approached from very different points of view. Vilenkin and Klimyk~\cite{VK} have compiled the enormous amount of work done in this area, and summarized in three volumes. 

The Lie algebra (resp.\ Lie group) that occurs as a standard example in this field is $\su(2)$ (resp.\ $SU(2)$). Also from the physical point of view, this Lie algebra is mostly important since the irreducible finite-dimensional $\su(2)$ representations correspond to the multiplets in quantum theory of angular momentum. For this reason, the theory of coupling and recoupling of angular momenta (or in group theoretical terms~: the theory of decompositions of tensor products of representations) has become so important~\cite{Edmonds,BL,Var}.
In one approach, the relation with special functions stems from the fact that these coupling and recoupling coefficients for $\su(2)$ (Clebsch-Gordan or 3-$j$ coefficients; Racah or 6-$j$ coefficients) can be expressed in terms of hypergeometric series $_3F_2$ or $_4F_3$ of unit argument. The relation goes even deeper~: the Clebsch-Gordan coefficient can be expressed in terms of a Hahn polynomial, and the Racah coefficient in terms of a Racah polynomial (see e.g.~\cite{KS} for the Askey scheme of hypergeometric orthogonal polynomials). Many of the properties of Hahn and Racah polynomials (e.g.\ orthogonality and recurrence relations) can be deduced from the natural properties of Clebsch-Gordan and Racah coefficients. Moreover, many relations involving Hahn and Racah polynomials together with other special functions also have a natural explanation in this context, e.g. when these other special functions can be realized as basis vectors for an $\su(2)$ representation~\cite{VK}. 

In quantum theory of angular momentum, the next recoupling coefficient that plays an important role is the 9-$j$ coefficient for $\su(2)$. This 9-$j$ coefficient, introduced for the first time by Wigner~\cite{Wigner}, can easily be re-expressed in terms of sums over products of 3-$j$ or 6-$j$ coefficients. A simple expression for the 9-$j$ coefficient in terms of a hypergeometric series was lacking, however, and thus no direct relation to special functions was found for a long time. The simplest known algebraic form of the 9-$j$ coefficient is now a triple sum series due to 
Ali\v sauskas, Jucys and Bandzaitis~\cite{Ali,Jucys}. 
This triple sum series has been shown~\cite{RaoRajJMP} to
be a particular case of the extremely general triple hypergeometric
series studied by Lauricella~\cite{Lauricella}, Saran~\cite{Saran} and Srivastava~\cite{Sri64}. This formula does not exhibit any one of the 72 symmetries of the 9-$j$ coefficient which are obvious from other expressions. The lack of symmetry of this triple sum series was exploited in order to deduce new transformation and summation formulas for double hypergeometric series corresponding to stretched 9-$j$ 
coefficients~\cite{Vanderjeugt,Pitre,VPR}.

The main object of study in the present paper is the 9-$j$ coefficient for  $\su(1,1)$, and its relation to orthogonal polynomials and hypergeometric series expressions. The Lie algebra $\su(1,1)$ has a class of positive discrete series representations, and the tensor product of such representations decomposes again as a direct sum of positive discrete series representations~\cite{VK}. It is for these representations that we consider 3-$j$, 6-$j$ and 9-$j$ coefficients. The 3-$j$ and 6-$j$ coefficients for $\su(1,1)$ have already drawn much attention in the literature, see~\cite{VK} or~\cite{VdJ1} for references. Roughly speaking, expressions for the 3-$j$ and 6-$j$ coefficients for $\su(1,1)$ can be obtained from those of $\su(2)$ by analytic continuation. So at first sight one does not expect many new hypergeometric series results by going from $\su(2)$ to $\su(1,1)$. There is however a main difference~: representations of $\su(2)$ are labelled by a positive integer, and those of $\su(1,1)$ by a positive real number. As a consequence, in the $_3F_2$ series expression of an $\su(2)$ 3-$j$ coefficient, only integer parameters appear; whereas both integer and real parameters appear in the $_3F_2$ series expression of an $\su(1,1)$ 3-$j$ coefficient. Because of this, identifying a Hahn polynomial in the $\su(1,1)$ 3-$j$ coefficient is straightforward, whereas it appears to be very much at random in the $\su(2)$ case~\cite{VK}. The same holds for the identification of a Racah polynomial. By considering the 9-$j$ coefficient for $\su(1,1)$ rather than for $\su(2)$, we hoped to discover new properties; in particular to identify an orthogonal polynomial related to the 9-$j$ coefficient, and to find new relations for multiple hypergeometric series related to this coefficient.

In the present paper the emphasis is on special functions related to the 9-$j$ coefficient of $\su(1,1)$. We have reduced the representation theoretic results to an absolute minimum. For details of $\su(1,1)$ representations, see~\cite{VK}; for a detailed description of positive discrete series representations, their 3-$j$ and 6-$j$ coefficients, and relations with orthogonal polynomials, see~\cite{VdJ1,KV}. In section~2 of this paper we briefly mention the labelling of these representations, and recall the definition of 3-$j$, 6-$j$, and 9-$j$ coefficients. Then we devote a section on the triple sum series expression of the 9-$j$ coefficient of $\su(1,1)$, and its hypergeometric function notation. In section~4 we investigate the nature of a discrete orthogonal polynomial associated to the 9-$j$ coefficient. This problem was already considered by Suslov~\cite{Suslov}, who derived some basic properties of polynomials of two discrete variables. However, Suslov did not consider the degrees of such polynomials, nor the explicit weight function or the hypergeometric nature. Here, we describe these polynomials in more detail, determine their degree in two variables, and give the discrete orthogonality relation and weight function.

The paper ends with two applications. In section~5, we summarize some formulas in which the $\su(1,1)$ 9-$j$ coefficient appears as a connection coefficient, or more generally, as an expansion coefficient between products of orthogonal polynomials. Of particular interest is equation~(\ref{PPFPP}), where a Kamp\'e de F\'eriet function (associated to the 9-$j$ coefficient) appears in the expansion of the product of two Jacobi polynomials. In section~6, we discuss some results on transformation formulas of double and triple hypergeometric series, that follow as a result of considering symmetries of the $\su(1,1)$ 9-$j$ coefficient. All these formulas are for terminating hypergeometric series. 

Since many of the results in this paper follow either from arguments in representation theory or from manipulating complicated series expressions for the 9-$j$ coefficient, most of the proofs are outlined but not presented in much detail. Moreover, since the 9-$j$ coefficient depends already on 9 parameters, it can be anticipated that many of the formulas to follow are rather complicated and heavy in notation (a fact almost impossible to avoid when dealing with multiple hypergeometric series). By collecting some expressions in the Appendix, we hope the paper has become more readable.

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\section{Definitions}  \label{defs}
\setcounter{equation}{0}

The positive discrete series representations $\pi_k$ of $\su(1,1)$ are
unitary representations labelled by a positive real number $k>0$. The
representation space $\ell_2(\Z_+)$
is equipped with an orthonormal basis $e^k_n$
($n=0,1,2,\ldots$). In this paper, the important ingredients are the
decomposition of tensor products of such representations, and the
coefficients relating two orthogonal bases in the tensor products~\cite{VK,VdJ1,KV}. 

For the tensor product of two representations one has the decomposition
\beq
\pi_{k_1}\otimes \pi_{k_2} = \bigoplus_{j=0}^\infty \ \pi_{k_1+k_2+j}.
\eeq
There are two natural bases in this tensor product space~: the so-called uncoupled basis vectors $e^{k_1}_{n_1}\otimes e^{k_2}_{n_2}$ in $\pi_{k_1}\otimes \pi_{k_2}$, and the coupled basis vectors $e^{(k_1k_2)k}_n$ with $k=k_1+k_2+j$.
The coefficients relating these two basis vectors are known as the Clebsch-Gordan coefficients or 3-$j$ 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} .
\eeq
These coefficients are non-zero only if $k=k_1+k_2+j$, $j\in\Z_+$,
and $n_1+n_2=n+j$ with $n_1,n_2,n \in \Z_+$. 
An expression for the Clebsch-Gordan coefficient is given by~\cite{VK,VdJ1}~:
\bea
C^{k_1,k_2,k}_{n_1,n_2,n}&=& \sqrt{
{{(2k_1)_{n_1}(2k_2)_{n_2}(2k_1)_j}\over
{n!n_1!n_2!j!\, (2k_1+2k_2+2j)_n (2k_2)_j (2k_1+2k_2+j-1)_j}} } \nn \\
&&\times (n+j)!\,{}_3F_2\left[{-j,j+2k_1+2k_2-1,-n_1 \atop 2k_1,-n-j} \sc
 1 \right],
\eea
where we have used the common notation for Pochhammer symbols and for
generalized hypergeometric series~\cite{Bailey,Slater}. The above terminating ${}_3F_2$ series can be
written in terms of a Hahn polynomial, and the orthogonality of the
Clebsch-Gordan coefficients implies the orthogonality of Hahn
polynomials. 

In the tensor product of three representations $\pi_{k_1} \otimes
\pi_{k_2} \otimes \pi_{k_3}$ one considers two different bases
coupled in a different way, namely
\bea
e^{((k_1k_2)k_{12}k_3)k}_n &=& \sum_{n_1,n_2,n_{12},n_3} C^{k_1,k_2,k_{12}}_{n_1,n_2,n_{12}} C^{k_{12},k_3,k}_{n_{12},n_3,n}\ e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \otimes e^{k_3}_{n_3},\label{scheme1}\\
e^{(k_1(k_2k_3)k_{23})k}_n &=& \sum_{n_1,n_2,n_3,n_{23}} C^{k_2,k_3,k_{23}}_{n_2,n_3,n_{23}} C^{k_1,k_{23},k}_{n_1,n_{23},n}\ e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \otimes e^{k_3}_{n_3}\label{scheme2}
\eea
In the first coupling scheme~(\ref{scheme1}), $k_1$ and $k_2$ are first coupled to $k_{12}$, and then this is coupled with $k_3$ to $k$; in the second coupling scheme~(\ref{scheme2}), $k_2$ and $k_3$ are first coupled to $k_{23}$, and then this is coupled with $k_1$ to $k$.
The coefficients relating those
bases are the Racah coefficients or 6-$j$ coefficients~:
\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 .
\label{defU}
\eeq
In the previous formula the following constraints hold
\beq
\begin{array}{l}
k_{12}=k_1+k_2+j_{12},\\
k_{23}=k_2+k_3+j_{23}, \\
k=k_{12}+k_3+j =k_1+k_{23}+j',\\
j_{12},j,j_{23},j' \in \Z_+,\ {\rm and}\ j_{12}+j=j_{23}+j'.
\end{array}
\eeq
Thus the above sum is finite with $k_{23}$ going from $k_2+k_3$ up to
$k_2+k_3+j_{12}+j$ in steps of 1. 
The Racah coefficients have an expression in terms of a Saalsch\"utzian
${}_4F_3$ series~\cite{VK,VdJ1}~:
\bea
&&U^{k_1,k_2,k_{12}}_{k_3,k,k_{23}}=
\left(j+j_{12}\atop j_{23}\right)
{(2k_2)_{j_{12}}(2k_3)_j(2k_1+2k_2+2k_3+j+j_{12}-1)_{j_{23}}
\over (2k_3,2k_2+2k_3+j_{23}-1)_{j_{23}}
(2k_2+2k_3+2j_{23})_{j'} } \nn\\
&&\qquad\times \left({j'!(2k_1,2k_{23},2k_1+2k_{23}+j'-1)_{j'} \;
j_{23}!(2k_2,2k_3,2k_2+2k_3+j_{23}-1)_{j_{23}} \over
j!(2k_{12},2k_3,2k_{12}+2k_3+j-1)_j\;
j_{12}!(2k_1,2k_2,2k_1+2k_2+j_{12}-1)_{j_{12}} }
\right)^{1/2} \nn\\
&&\qquad\times {}_4F_3\left[
{2k_1+2k_2+j_{12}-1,2k_2+2k_3+j_{23}-1,-j_{12},-j_{23} \atop
2k_2,2k_1+2k_2+2k_3+j+j_{12}-1,-j-j_{12}}\sc 1\right],
\eea
The Racah coefficients can be rewritten in terms of the
Racah polynomials~\cite{KS} defined by
\beq
R_n(\la(x);a,b,c,d)= {}_4F_3\left[
{{-n, n+a+b+1,-x,x+c+d+1}\atop
{a+1,\ b+d+1,\ c+1}}\sc 1 \right],
\label{polRacah}
\eeq
where one of the lower parameters
equals $-N$, $N\in\Z_+$ and $0\leq n\leq N$. It is a polynomial
of degree $n$ in $\lambda(x)=x(x+c+d+1)$.
The orthogonality
relations for the Racah polynomials follow from the fact that the
Racah coefficients form a unitary (orthogonal) matrix.

Consider next the tensor product of four representations $\pi_{k_1} \otimes
\pi_{k_2} \otimes \pi_{k_3} \otimes \pi_{k_4}$, and the following two basis vectors~:
\beas
e^{ ((k_1k_2)k_{12}(k_3k_4)k_{34})k}_n &=& \sum_{{n_1,n_2,n_3,n_4 \atop n_{12},n_{34}}}
C^{k_1,k_2,k_{12}}_{n_1,n_2,n_{12}} C^{k_3,k_4,k_{34}}_{n_3,n_4,n_{34}} 
C^{k_{12},k_{34},k}_{n_{12},n_{34},n} \ e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \otimes e^{k_3}_{n_3} \otimes e^{k_4}_{n_4}, \\
e^{ ((k_1k_3)k_{13}(k_2k_4)k_{24})k}_n &=& \sum_{{n_1,n_2,n_3,n_4 \atop n_{13},n_{24}}}
C^{k_1,k_3,k_{13}}_{n_1,n_3,n_{13}} C^{k_2,k_4,k_{24}}_{n_2,n_4,n_{24}} 
C^{k_{13},k_{24},k}_{n_{13},n_{24},n} \ e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \otimes e^{k_3}_{n_3} \otimes e^{k_4}_{n_4}.
\eeas

\begin{defi}
The 9-$j$ coefficients for $\su(1,1)$ are the transition coefficients relating the above two bases~:
\beq
e^{ ((k_1k_2)k_{12}(k_3k_4)k_{34})k}_n = \sum_{k_{13}, k_{24}}
\nj{k_1}{k_2}{k_{12}}{k_3}{k_4}{k_{34}}{k_{13}}{k_{24}}{k} 
e^{ ((k_1k_3)k_{13}(k_2k_4)k_{24})k}_n.
\label{def9j}
\eeq
This 9-$j$ coefficient is a function of 4 real and 5 integer variables. It will be denoted by
\beq
J(\al,\be,\ga,\de;m,n,p,q,N)=
\nj{\al}{\be}{\al+\be+m}{\ga}{\de}{\ga+\de+n}{\al+
 \ga+p}{\be+\de+q}{\al+\be+\ga+\de+N}.
\eeq
Herein, $\al,\be,\ga,\de$ are positive real numbers and $m,n,p,q,N$ are
nonnegative integers such that $m+n\leq N$ and $p+q\leq N$. 
\end{defi}

From the definition, it follows that an expression for the 9-$j$ coefficient is given by
\bea
&&\nj{k_1}{k_2}{k_{12}}{k_3}{k_4}{k_{34}}{k_{13}}{k_{24}}{k} = \nn\\
&&\sum_{{n_1,n_2,n_3,n_4 \atop n_{12},n_{34},n_{13},n_{24}}} C^{k_1,k_2,k_{12}}_{n_1,n_2,n_{12}} C^{k_3,k_4,k_{34}}_{n_3,n_4,n_{34}} 
C^{k_{12},k_{34},k}_{n_{12},n_{34},n}
C^{k_1,k_3,k_{13}}_{n_1,n_3,n_{13}} C^{k_2,k_4,k_{24}}_{n_2,n_4,n_{24}} 
C^{k_{13},k_{24},k}_{n_{13},n_{24},n}.
\label{9j-3j}
\eea

\begin{prop}
The 9-$j$ coefficients satisfy the following orthogonality relation~:
\beq
\sum_{p=0}^{N}\sum_{q=0}^{N-p}
J(\al,\be,\ga,\de;m,n,p,q,N)J(\al,\be,\ga,\de;m',n',p,q,N)=
\delta_{m,m'}\delta_{n,n'}.
\label{orthj}
\eeq
\end{prop}
This relation follows directly from (\ref{def9j}) and the orthonormality of the basis vectors in (\ref{def9j}).

Just as the 3-$j$ coefficients are related to the Hahn polynomials, and the 6-$j$ coefficients to the Racah polynomials, one may ask the question which (discrete) orthogonal polynomial is related to the 9-$j$ coefficients. This is one of the problems considered in this paper. The orthogonality relation (\ref{orthj}) indicates that we are dealing with a polynomial in two variables. 
However, it is non-trivial to identify which of the quantities
$\al,\be,\ga,\de,m,n,p,q,N$ are corresponding to the variable(s) of the
polynomial, which to the degree(s) and which to the parameters. 
In the following sections we shall consider a number of alternative expressions for the 9-$j$ coefficient, that will finally lead to the identification of a polynomial.

For the rest of the paper, it will be useful to relabel $(\al,\be,\ga,\de)$ as follows 
\beq
(a,b,c,d)=(2\al-1,2\be-1,2\ga-1,2\de-1),
\label{relab}
\eeq
and use both notations $(a,b,c,d)$ and $(\al,\be,\ga,\de)$ simultaneously.

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\section{Some expressions for the 9-$j$ coefficient}  \label{expr}
\setcounter{equation}{0}

A first expression for the 9-$j$ coefficient was given in (\ref{9j-3j}). An alternative expression follows from explicit
recoupling theory, i.e.\ by using definitions (\ref{defU}) and
(\ref{def9j}). Then one finds
\beq
\nj{k_1}{k_2}{k_{12}}{k_3}{k_4}{k_{34}}{k_{13}}{k_{24}}{k} = \sum_{K}
(-1)^{K-k_{24}-k_{34}+k_4}\ U^{k_1,k_2,k_{12}}_{k_{34},k,K}\ 
U^{k_2,k_4,k_{24}}_{k_3,K,k_{34}} U^{k_1,k_3,k_{13}}_{k_{24},k,K}.
\label{singlesum}
\eeq
Herein, $K$ goes from $\max(k_{34}+k_2,k_{24}+k_3)$ upto $k-k_1$ in
steps of 1. Since every Racah coefficient in (\ref{singlesum}) is
proportional to a single hypergeometric series of ${}_4F_3$ type, the
above expression is 
proportional to a quartic hypergeometric series. For this series it is
difficult to identify a polynomial in (two) variables.

A third expression presented here is a triple sum series. The triple
sum series for the $\su(2)$ 9-$j$ coefficient was first obtained by
Ali\v sauskas and Jucys~\cite{Ali}, and later rederived by Jucys and Bandzaitis~\cite{Jucys}. Just as
the 3-$j$ (Clebsch-Gordan) and 6-$j$ (Racah) coefficients of $\su(1,1)$
can be obtained from those of $\su(2)$ by certain formal substitutions,
this can also be performed for the triple sum series expression for the
9-$j$ coefficient. Omitting the details of this derivation, we have
obtained the following (new) formula~:
\begin{prop}
Let $a,b,c,d\in (-1,\infty)$, $m,n,p,q,N\in\N$ with $m+n\leq N$ and 
$p+q\leq N$. Then
\bea
&&J(\al,\be,\ga,\de;m,n,p,q,N)=\label{triple}\\
&&C_1 \sum_{j=0}^q\sum_{k=0}^{\min(N-m-n,N-p-q)} \sum_{l=0}^{\min(p,m)} 
{(-c-n)_j(-b-q)_j(1+d+n)_j(-q)_j \over j!(1+d)_j}\nn\\
&&\times{(-1-c-d-n-N+m)_k(-1-a-c-p-N+q)_k(m+n-N)_k(p+q-N)_k \over
k!(-2-a-b-c-d-2N)_k }\nn\\
&&\times{(-b-m)_l(-m)_l(1+a+c+p)_l(-p)_l\over l! (1+a)_l}
{1\over (-c-N+m)_{j+k}(-b-m-q)_{j+l}(-N+q)_{k+l}}, \nn
\eea
where $C_1$ is given in the Appendix and the relabelling (\ref{relab}) 
has been used. 
\end{prop}
An equivalent expression was recently derived by Rosengren~\cite{Rosengren}.
This expression is a terminating triple hypergeometric series.
For generic parameters $a,b,c,d$ it is well defined, and since $k+l\leq
N-q$ the denominator $(-N+q)_{k+l}$ does not go to zero.
This triple sum series can be identified as a particular case of a general function defined in three variables by Srivastava~\cite{Sri67}, which is a unification of the triple hypergeometric functions of Lauricella-Saran~\cite{Saran} and the Srivastava functions~\cite{Sri64}. For the case of $\su(2)$, this identification was made by Srinivasa Rao and Rajeswari~\cite{RaoRajJMP,RaoRaj}. This general function takes the following form~:
\bea
&&F^{(3)} \left[ {(a)\atop (e)} {:: \atop ::} {(b) \atop (f)} \sc
{(b') \atop (f')} \sc {(b'') \atop (f'')} \cl  {(c) \atop (g)} \sc
{(c') \atop (g')} \sc {(c'') \atop (g'')} \sc x,y,z \right] \label{F3}\\
&&= \sum_{m,n,p} { ((a))_{m+n+p} \over ((e))_{m+n+p} }
{ ((b))_{m+n} ((b'))_{n+p} ((b''))_{p+m} \over
((f))_{m+n} ((f'))_{n+p} ((f''))_{p+m} }
{((c))_m ((c'))_n ((c''))_p \over ((g))_m ((g'))_n ((g''))_p }
{ x^m y^n z^p \over m!n!p!}. \nn
\eea
Herein, $(a)$ denotes a sequence of parameters $a_1,a_2,\ldots, a_A$,  and $((a))_m$ stands for $(a_1, \ldots, a_A)_m \equiv (a_1)_m \cdots (a_A)_m$. Observe that the 9-$j$ coefficient is thus equal to $C_1$ times
\beas
&&F^{(3)} \left[ {\hy\atop\hy} {:: \atop ::} {\hy\atop -c-N+m} \sc
{\hy\atop -N+q} \sc {\hy\atop -b-m-q} \cl  {-c-n,-b-q,1+d+n,-q \atop 1+d} \sc \right. \\
&&{-1-c-d-n-N+m,-1-a-c-p-N+q,m+n-N,p+q-N \atop -2-a-b-c-d-2N} \sc \\
&&\left. {-b-m,-m,1+a+c+p,-p \atop 1+a} \sc 1,1,1 \right].
\eeas
This triple sum is again rather complicated, and does not directly reveal
a polynomial expression in two variables. However, by considering first
a degenerate case, the polynomial nature will become clear.
Moreover, using symmetry properties of the 9-$j$ coefficient, some interesting transformation formulas for multiple hypergeometric series will be deduced.

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%     SECTION                                                         %
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\section{An orthogonal polynomial related to the 9-$j$ coefficient} \label{polyn}
\setcounter{equation}{0}

Recall the definition of a generalized Kamp\'e de F\'eriet series~\cite{Appell,Kampe,Srivastava}~:
\beq
F^{A:B;B'}_{C:D;D'} \left[ {(a) \atop (c)} \cl {(b) \atop (d)}\sc {(c)
\atop (d')} \sc x,y \right] = 
\sum_{k,l} {((a))_{k+l}((b))_k((b'))_l \over ((c))_{k+l}
((d))_k ((d'))_l} {x^k y^l \over k!l!}.
\eeq
As in the previous section, $(b)$ stands for $b_1,b_2,\ldots,b_B$, and $((b))_k$ is the product of a number of Pocchammer symbols $(b_1)_k(b_2)_k \ldots (b_B)_k$ (and similar for the other parameters). General convergence criteria for such series were studied by Srivastava and Daoust~\cite{Sriva}, and by H'ai et al.~\cite{Hai}. However, the series appearing here will
always be terminating, so there is no problem with convergence. 

Now, consider a so-called stretched 9-$j$ coefficient with $m+n=N$. In this degenerate case, the triple sum  series appearing in (\ref{triple}) reduces
to the following double series~:
\beq
F^{0:4;3}_{1:2;1}\left[ {\hy \atop -b-m-q}\cl {-b-m, -m, -p, a+c+p+1 \atop
-m-n+q, a+1}\sc {-q,1+d+n, -b-q \atop d+1}\sc 1,1\right].
\label{double1}
\eeq
Next, one can use the following transformation formula for generalized Kamp\'e de F\'eriet series (see \cite[Proposition~1]{VPR})~:
\bea
&&F^{0:4;3}_{1:2;1}\left[ {\hy \atop E}\cl{A,B,C,D \atop F,G}\sc{A',B',C'\atop
F'}\sc 1,1 \right] = 
\Gamma\left[ E,F\atop E-A',F+A' \right] \nn\\
&&\times F^{0:3;3}_{1:1;1}\left[ {\hy \atop E+F'-B'-C'}\cl{B,C,D \atop
G}\sc{A',F'-B',F'-C'\atop F'}\sc 1,1 \right] ,
\label{eq43}
\eea
provided that $A=E-A'$, $F=E+F'-A'-B'-C'$, and both sides converge
absolutely (in~(\ref{eq43}), the standard notation~\cite[(2.1.1.4)]{Slater} 
for a product of Gamma functions has been used).
Thus the series (\ref{double1}) becomes equal to
\beq
{(1+m+n-q)_q \over (1+b+m)_q}
F^{0:3;3}_{1:1;1}\left[ {\hy \atop -m-n}\cl{-m,-p,a+c+p+1 \atop
a+1}\sc{-n,-q,b+d+q+1\atop d+1}\sc 1,1 \right]
\label{eq44}
\eeq
Consequently, we have
\beq
J(\al,\be,\ga,\de;m,n,p,q,m+n) = C_2\ R_{m,n}(p,q;a,b,c,d,m+n),
\label{defC2}
\eeq
with $C_2$ a constant given in the Appendix (or, equivalently, $C_2$ is equal to the coefficient in (\ref{eq44}) multiplied by $C_1$ in which $N$ is specified to $N=m+n$) and
\bea
&&R_{m,n}(p,q;a,b,c,d,m+n) = \nn\\
&&F^{0:3;3}_{1:1;1} \left[{\hy\atop -m-n}\cl {a+c+p+1,-p,-m
\atop a+1}\sc {b+d+q+1,-q,-n \atop d+1}\sc 1,1\right].
\label{defR1}
\eea
This expression is very symmetric, and shows a resemblance with the Racah polynomial~(\ref{polRacah}). Moreover, it is a polynomial of degree
$m$ in $\la(p)=p(a+c+p+1)$ and of degree $n$ in $\mu(q)=q(q+b+d+1)$.
This is the case since in (\ref{defR1}) $p$ appears only in the Pochhammer symbols $(a+c+p+1)_k (-p)_k$, and $(a+c+p+1+j)(-p+j) = j^2+j(a+c+1)-\la(p)$.

The general case is much more complicated than the special degenerate case with $m+n=N$ treated so far. An ideal situation would be if the summation part in (\ref{triple}) could be rewritten such that it obviously becomes a polynomial in $\la(p)$ and $\mu(q)$. However, a general transformation theory of multiple hypergeometric series of the type (\ref{F3}) is missing, and hence we have been unable to perform such a transformation.
As a consequence, we have to work in an indirect way.

Consider again (\ref{triple}), and rewrite it in the following form~:
\beq
J(\al,\be,\ga,\de;m,n,p,q,N) = C_3\
 r_{m,n}(p,q;a,b,c,d,N),
\label{coefr}
\eeq
with $C_3$ given in the appendix. 
This defines $r$ as a coefficient that can be determined when $p$ and $q$ are nonnegative integers, but not when $p$ and $q$ are complex variables. 
We really wish to define a polynomial $R$ in the variables $(p,q)$, or rather in the variables $(\la(p),\mu(q))$. For
$m+n=N$, we already have such a proper polynomial definition, see
eq.~(\ref{defR1}). For the general case, we shall use a recurrence
relation that holds for the coefficient $r$ in (\ref{coefr}), and use
this recurrence relation 
as the defining relation for general polynomials $R$.
\begin{prop}
Let $a,b,c,d\in (-1,\infty)$, $m,n,p,q,N\in\N$ with $m+n\leq N$ and 
$p+q\leq N$. For the coefficient $r_{m,n}(p,q;a,b,c,d,N)$ the following recurrence relation holds~:
\bea
&&c(a+1)(N-m-n)r_{m,n}(p,q;a,b,c,d,N)=\nn\\
&&-c(a+1)(b+m+1)(c+d+1+N-m+n) r_{m+1,n}(p,q;a,b,c,d,N) \nn\\
&& +(a+b+2m+2)(d+n+1)(a+p+1)(c+p)(m+n+1) 
r_{m,n+1}(p,q;a+1,b,c-1,d,N) \nn\\
&&-(a+b+2m+2)(c+n+1)(c+d+n+1)p(a+c+p+1) \nn\\
&&\qquad\qquad\times r_{m,n}(p-1,q;a+1,b,c+1,d,N-1). 
\label{recur}
\eea
\end{prop}

To prove this relation, one can use the triple sum series expressions
for the three terms in the rhs and simplify in order to get the lhs;
this is a very difficult step however. Another way to proceed is to
start from one of the classical 4-term recurrences for the $\su(2)$
9-$j$ coefficient (see~\cite[eqs.~(2)-(4), p.~346]{Var}), and perform the
formal substitutions that transform $\su(2)$ coupling coefficients into
$\su(1,1)$ coupling coefficients. 

If we define the excess of
$r_{m,n}(p,q;a,b,c,d,N)$ as $N-m-n$, the $r's$ on the rhs of
(\ref{recur}) all have the same excess $N-m-n-1$.

We are now in a position to define the general polynomial $R_{m,n}(p,q;a,b,c,d,N)$.
\begin{defi}
Let $a,b,c,d\in (-1,\infty)$, $m,n,N\in\N$ with $m+n\leq N$ and 
let $p$ and $q$ be arbitrary variables. 
For $m+n=N$, let $R_{m,n}(p,q;a,b,c,d,N)$ be defined by (\ref{defR1});
and for $m+n<N$, let $R_{m,n}$ be defined through the recurrence relation
\bea
&&c(a+1)(N-m-n)R_{m,n}(p,q;a,b,c,d,N)=\nn\\
&&-c(a+1)(b+m+1)(c+d+1+N-m+n) R_{m+1,n}(p,q;a,b,c,d,N) \nn\\
&& +(a+b+2m+2)(d+n+1)(a+p+1)(c+p)(m+n+1) 
R_{m,n+1}(p,q;a+1,b,c-1,d,N) \nn\\
&&-(a+b+2m+2)(c+n+1)(c+d+n+1)p(a+c+p+1) \nn\\
&&\qquad\qquad\times R_{m,n}(p-1,q;a+1,b,c+1,d,N-1). 
\label{Recur}
\eea
\end{defi}

\begin{prop}
$R_{m,n}(p,q;a,b,c,d,N)$ is a polynomial of degree $N-n$
in $\la(p)$ and of degree $N-m$ in $\mu(q)$. 
\end{prop}
For $m+n=N$ (i.e.\ excess 0) the statement has been proved already.
For $m+n=N-1$ the
three $R$'s in the rhs of (\ref{Recur}) have excess 0, thus they are polynomials respectively of degree $(m+1,n)$, $(m,n+1)$ and $(m,n)$ in
$(\la(p),\mu(q))$. The three factors in front of these $R$'s are
respectively of the form $c_1$, $c_2 \la(p)+c_3$, and $c_4 \la(p)$,
where all $c_i$'s are constants in which no $p$ or $q$ appear. The second
factor is indeed of the form $c_2 \la(p)+c_3$ because $(a+p+1)(c+p) =
\la(p) + (a+1)c$. Combining this shows that the rhs is then a
polynomial of degree $(m+1,n+1)=(N-n,N-m)$ in $(\la(p),\mu(q))$. The
general statement then follows by induction.

We are now in a position to state the orthogonality of the two-variable polynomial related to the 9-$j$ coefficient of $\su(1,1)$~:
\begin{theo}
Let $a,b,c,d\in (-1,\infty)$, $m,n,N\in\N$ with $m+n\leq N$ and let $p$ and $q$ be arbitrary variables. The polynomial
\beq
R_{m,n}(p,q)\equiv R_{m,n}(p,q;a,b,c,d,N)
\eeq
which is of degree $(N-n,N-m)$ in the variables $(\la(p),\mu(q))$ satisfies the orthogonality relation
\beq
\sum_{p=0}^N\sum_{q=0}^{N-p} w(p,q)
R_{m,n}(p,q)R_{m',n'}(p,q) =\delta_{m,m'}\delta_{n,n'} h_{m,n}.
\label{orthRR}
\eeq
Herein, the weight function $w(p,q)$ and the norm $h_{m,n}$ are given in the appendix, see equations~(\ref{w}) and~(\ref{h}).
\label{theo4}
\end{theo}
The proof of (\ref{orthRR}) follows directly from (\ref{orthj}), using (\ref{coefr}) and (\ref{C3}). The merit of this theorem is really the recognition of the polynomial part in 
the 9-$j$ coefficient, see~(\ref{coefr}). In this equation, the corresponding coefficient $C_3$ is then split up into a part depending only upon $(p,q)$, yielding the weight function $w(p,q)$, and into a part depending only upon $(m,n)$, yielding the constant $h_{m,n}$; of course, $a$, $b$, $c$, $d$ and $N$ appear as parameters in all these quantities.

Note that the orthogonality is on a triangular grid for $(p,q)$.
The weight function $w(p,q)$ is defined for integer $p$ and $q$ only, just as the (one-variable) weight function for the Hahn or Racah polynomial.
 
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\section{First application~: connection coefficients}
\label{appl1}
\setcounter{equation}{0}

Just as the $\su(1,1)$ Clebsch-Gordan and Racah coefficients have an interpretation as connection coefficients, see~\cite{VdJ1,KV}, the 9-$j$ coefficients have a similar property. This relation was already studied by Granovskii and Zhadanov~\cite{GZ}, and can be written in the following way~:
\bea
&&S_m^{(\al,\be)}(x,y) S^{(\ga,\de)}_n(z,u) 
S^{(\al+\be+m,\ga+\de+n)}_{N-m-n}(x+y,z+u) =\label{SSS}\\
&&\sum_{p=0}^{N} \sum_{q=0}^{N-p} J(\al,\be,\ga,\de;m,n,p,q,N)
S^{(\al,\ga)}_p(x,z) S^{(\be,\de)}_q(y,u) 
S^{(\al+\ga+p,\be+\de+q)}_{N-p-q}(x+z,y+u). \nn
\eea
This is a polynomial relation in $x,y,z,u$. 
The definition of $S$ given in~\cite{GZ} may, by the relabelling
(\ref{relab}), be stated as follows,
\beas
&&S^{(\al,\be)}_m(x,y)=\left( (b+1)_m \over
(a+1)_m(a+b+m+1)_m\; m! \right)^{1/2} 
x^m {\ }_2F_1\left({-m-a,-m\atop b+1};{-y\over x}\right)   \\
&&\qquad=(-1)^m
\left(m!\over (a+1)_m(b+1)_m(a+b+m+1)_m\right)^{1/2} 
(x+y)^m P_m^{(a,b)}\Bigl({y-x\over y+x}\Bigr),
\eeas
in terms of a Jacobi polynomial. The same relation~(\ref{SSS}) is also more generally valid~\cite{KV} for $S$ given in terms of a continuous Hahn polynomial~:
\bea
S^{(\al,\be)}_m(x,y)&=&\left({m!(2\al+2\be+2m-1)\Gamma(2\al+2\be+m-1)\over
\Gamma(2\al+m) \Gamma(2\be+m)} \right)^{1/2} \nn\\
&&\times p_m(x;\al,\be-i(x+y),\al,\be+i(x+y)),
\eea
where~\cite{KS}
\beq
p_m(x;a,b,c,d)=i^m {(a+c)_m(a+d)_m \over m!}\ _{3}F_2 \left[
{-n,n+a+b+c+d-1,a+ix \atop a+c, a+d} \sc 1 \right].
\eeq

Putting $z+u=-x-y$ in (\ref{SSS}) leads to a simplification if one
takes $S$ in terms of the Jacobi polynomials. 
With $r=(z-x)/(z+x)$ and $s=(x+2y+z)/(x+z)$, the equation becomes
$(m+n\leq N$)
\bea
&&\left(s-r\over 2\right)^{N} P_m^{(a,b)} \Bigl({2-r-s\over r-s}\Bigr)
P_n^{(c,d)} \Bigl({2+r+s\over s-r}\Bigr) = \nn\\
&& \qquad\qquad\qquad
\sum_{p=0}^N\sum_{q=0}^{N-p} C_4\;J(\al,\be,\ga,\de;m,n,p,q,N)\;
 P_p^{(a,c)}(r) P_q^{(b,d)}(s), 
\label{PPJPP}
\eea
where $C_4$ is given in the Appendix. Since in
the case $N=m+n$ the 9-$j$ coefficient takes the simple form 
given by (\ref{defC2}) and (\ref{defR1}), 
equation (\ref{PPJPP}) then becomes~:
\bea
&&\left(s-r\over 2\right)^{m+n} P_m^{(a,b)} \Bigl({2-r-s\over r-s}\Bigr)
P_n^{(c,d)} \Bigl({2+r+s\over s-r}\Bigr) =  \sum_{{p,q \atop p+q\leq m+n}}
C_5\; P_p^{(a,c)}(r) P_q^{(b,d)}(s)\nn\\
&&\qquad\times\ F^{0:3;3}_{1:1;1} \left[{\hy\atop -m-n}\cl {a+c+p+1,-p,-m
\atop a+1}\sc {b+d+q+1,-q,-n \atop d+1}\sc 1,1\right].
\label{PPFPP}
\eea
Again, $C_5$ is a coefficient given in the Appendix.

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\section{Second application~: transformation formulas}
\label{appl2}
\setcounter{equation}{0}

Since the $\su(1,1)$ Clebsch-Gordan coefficients satisfy the 
symmetry relation
\beq
C^{k_1,k_2,k}_{n_1,n_2,n} = (-1)^{k-k_1-k_2}\ C^{k_2,k_1,k}_{n_2,n_1,n},
\eeq
a number of symmetries can be deduced for the 9-$j$ coefficient from the expression (\ref{9j-3j}). For example, we have
\beq
J(\al,\be,\ga,\de;m,n,p,q,N) = J(\de,\ga,\be,\al,n,m,q,p,N),
\label{sym1}
\eeq
and many other symmetries can be written down. Though such a symmetry is obvious from (\ref{9j-3j}), it is hardly evident from the triple sum series expression (\ref{triple}). On the contrary, expressing the symmetry gives rise to a transformation formula for the triple sum series. Not all symmetries yield nice transformation formulas, but some do. For example, from the above symmetry (\ref{sym1}), after relabelling of the 9 parameters, the following transformation formula can be deduced~:

\begin{prop}
Let $m,n,p \in \N$ and $a_1,b_1,c_1,a_2,b_2,c_2$ be arbitrary parameters. Let 
\[
\begin{array}{ll}
a_3=a_1-b_2-n,\qquad & a_4=a_1+a_2+b_1-b_2-c_1-p, \\
b_3=b_1-c_2-p,       & b_4=b_1+b_2+c_1-c_2-a_1-m, \\
c_3=c_1-a_2-m,       & c_4=c_1+c_2+a_1-a_2-b_1-n.
\end{array}
\]
Then
\beas
&&F^{(3)} \left[ {\hy\atop\hy} {:: \atop ::} 
{\hy\atop a_1-n} \sc {\hy\atop b_1-p} \sc {\hy\atop c_1-m} \cl 
{-m,a_1,a_2,a_3 \atop a_4} \sc {-n,b_1,b_2,b_3 \atop b_4} \sc 
{-p,c_1,c_2,c_3 \atop c_4} \sc 1,1,1 \right] \\
&& = {(1-a_1)_p (1-b_1)_m (1-c_1)_n \over (1-a_1)_n (1-b_1)_p (1-c_1)_m}
\times F^{(3)} \left[ {\hy\atop\hy} {:: \atop ::} 
{\hy\atop b_1-m} \sc {\hy\atop c_1-n} \sc {\hy\atop a_1-p} \cl \right.\\
&& \left. {-m,a_1,a_4-a_2,a_4-a_3 \atop a_4} \sc {-n,b_1,b_4-b_2,b_4-b_3 \atop b_4} \sc {-p,c_1,c_4-c_2,c_4-c_3 \atop c_4} \sc 1,1,1 \right].
\eeas  
\end{prop}

An interesting special case occurs when $a_2=b_2=c_2=0$; then we get the following summation formula~:
\begin{coro}
For $m,n,p \in \N$ and $a,b,c$ arbitrary parameters, we have
\beas
&&F^{(3)} \left[ {\hy\atop\hy} {:: \atop ::} 
{\hy\atop b-m} \sc {\hy\atop c-n} \sc {\hy\atop a-p} \cl 
{-m,a,b-c+n-p \atop \hy} \sc {-n,b,c-a+p-m \atop \hy} \sc \right. \\
&& \left. {-p,c,a-b+m-n \atop \hy} \sc 1,1,1 \right] 
= {(1-a)_n (1-b)_p (1-c)_m \over (1-a)_p (1-b)_m (1-c)_n}.
\eeas
\end{coro}

Similarly, when considering symmetries for the 9-$j$ coefficient in the degenerate case $m+n=N$, one obtains certain transformation formulas for the Kamp\'e de F\'eriet series of unit argument. Again, after proper relabelling one can deduce~:

\begin{prop}
Let $m,n\in\N$, and let $a,b,e,a',b',e',d$ be arbitrary parameters with $d=a+a'$. Then the following transformation formulas hold~:
\beas
&& F^{0:3;3}_{1:1;1}\left[ {\hy \atop d}\cl{a,b,-m \atop
e}\sc{a',b',-n\atop e'}\sc 1,1 \right] \\
&&= {(e-b)_m (e'-b')_n \over (e)_m (e')_n} 
F^{0:3;3}_{1:1;1}\left[ {\hy \atop d}\cl{a',b,-m \atop
b-e-m+1}\sc{a,b',-n\atop b'-e'-n+1}\sc 1,1 \right] \\
&&= (-1)^n {(a')_{m+n} (b')_n \over (d)_{m+n} (e')_n} 
F^{0:3;3}_{1:1;1}\left[ {\hy \atop 1-a'-m-n}\cl{a,e-b,-m \atop
e}\sc{1-d-m-n,1-e'-n,-n\atop 1-b'-n}\sc 1,1 \right] .
\eeas
\end{prop}
Applying the above transformation formulas recursively, one obtains in total a set of six such Kamp\'e de F\'eriet series which are transformed into one another.

The above transformation formulas are for terminating multiple hypergeometric series of unit argument. This has the advantage that they can easily be verified using a symbolic computation package.

\section*{Appendix}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

Here, we have collected some of the complicated expressions in order not to overload the notation in earlier formulas. The weight function $w(p,q)$ and the constant $h_{m,n}$ of Theorem~\ref{theo4} are given by
\bea
&&w(p,q)\equiv w(p,q;a,b,c,d,N)={(-1)^{p+q} (-N)_{p+q} \over
p!q!(a+b+c+d+N+3)_{p+q}} {(a+1)_p(d+1)_q \over (c+1)_p(b+1)_q} \nn\\
&&\times {(a+c+1)_p(a+c+2)_{2p} \over (a+c+1)_{2p}(a+c+N+2)_{p-q}}
{(b+d+1)_q(b+d+2)_{2q} \over (b+d+1)_{2q}(b+d+N+2)_{q-p}},
\label{w}
\eea
and
\bea
&&h_{m,n} \equiv h_{m,n}(a,b,c,d,N)= {(-1)^{m+n} m!n! (-N)_{m+n} \over
(m+n)!^2 (a+b+c+d+N+3)_{m+n} } \nn\\
&&\times {(a+c+2,b+d+2,a+b+2,c+d+2)_N \over (a+1,b+1)_m (c+1,d+1)_n }
\nn\\ 
&&\times {(a+b+1)_{2m}(a+b+N+2)_{m-n} \over (a+b+1)_m(a+b+2)_{2m}} 
{(c+d+1)_{2n}(c+d+N+2)_{n-m} \over (c+d+1)_n(c+d+2)_{2n}}.
\label{h}
\eea
The coefficients $C_1$, $C_2$, and $C_3$ are defined in terms of this weight function $w$ and the norm $h_{m,n}$, 
\bea
C_1&=&{(-1)^{N+m+n+p+q}(N-q)! \over (m+n)!(N-m-n)!}
(b+m+1)_q(c+n+1)_{N-m-n}\nn\\
&&\times (a+b+c+d+3+N+m+n)_{N-m-n} \sqrt{w(p,q)\over
h_{m,n}} . \\
C_2&=&(-1)^{p+q} \sqrt{w(p,q;a,b,c,d,m+n)\over h_{m,n}(a,b,c,d,m+n)}.\\
C_3&=&(-1)^{p+q} \sqrt{w(p,q;a,b,c,d,N)\over h_{m,n}(a,b,c,d,N)}. \label{C3}
\eea
The remaining coefficients are
\bea
C_4&=&(-1)^{m+p}\Bigl(
{p!q!(N-m-n)!(a+1)_m (b+1)_m (c+1)_n(d+1)_n  \over
 m!n!(N-p-q)!(a+1)_p (b+1)_q (c+1)_p(d+1)_q }\nn\\
&&\times{ (a+b+c+d+N+3)_{m+n}(a+b+m+1)_m (c+d+n+1)_n   \over (a+b+c+d+N+3)_{p+q} (a+c+p+1)_p (b+d+q+1)_q }    \nn\\                                             
&&\times{(a+b+2m+2)_{N-m-n} (c+d+2n+2)_{N-m-n}\over
         (a+c+2p+2)_{N-p-q} (b+d+2q+2)_{N-p-q}}
 \Bigr)^{1/2}; \\
C_5&=&{(-1)^{m+q}(m+n)! (a+1)_m(b+1)_m(c+1)_n(d+1)_n \over
m!n!(m+n-p-q)!(a+c+p+1)_p (b+d+q+1)_q (b+1)_q (c+1)_p } \nn\\
&&\times {(a+b+c+d+m+n+p+q+3)_{m+n-p-q} \over
(a+c+2p+2)_{m+n-p-q} (b+d+2q+2)_{m+n-p-q} }.
\eea


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\end{document}

