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%     Transformation and summation formulas for double                %
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%     J. Comp. Appl. Math. 83 (1997) 185-193.
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\begin{document}
\begin{center}
{\Large \bf 
Transformation and summation formulas for \\[4mm]
double hypergeometric series}\\[3cm]
{\bf J.\ Van der Jeugt\footnote{Research
Associate of the Fund for Scientific Research -- Flanders
(Belgium).}, S.~N.\ Pitre and K.~Srinivasa Rao\footnote{Permanent
Address~: Institute of Mathematical Sciences, CIT Campus, Madras
600113, India} }\\[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, Pitre.Sangita@rug.ac.be,
rao@imsc.ernet.in. 
\end{center}

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\vspace{1 cm}
\noindent {\bf Abstract}\\
A number of new transformation formulas for double hypergeometric
series are presented. The series appearing here are so-called Kamp\'e
de F\'eriet functions of type $F^{0:3;4}_{1:1;2}(1,1)$ and
$F^{1:2;2}_{0:2;2}(1,1)$. The transformation formulas relate such
double series to a single hypergeometric series of
${}_4F_3(1)$ type. By specializing certain parameters, a list of new
summation formulas for $F^{1:2;2}_{0:2;2}(1,1)$ series is obtained. The
origin of the results comes from studying symmetries of the 9-$j$
coefficient appearing in quantum theory of angular momentum.

\vspace{1cm}
\noindent Keywords~: Hypergeometric series, Kamp\'e de F\'eriet
functions, summation formulas.

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

\newpage

\section{Introduction}

For the classical hypergeometric series ${}_pF_q$ of a single variable,
the most impressive results are the various summation theorems due to
Gauss, Vandermonde, Saalsch\"utz, Watson, Dougall, Dixon, and Bailey,
amongst many others~\cite{Slater}. Just as important are the
transformation formulas expressing one hypergeometric series in terms
of a different one. Fewer results are known for hypergeometric series
of two or more variables. The simplest such double hypergeometric
series are the Appell series and the Horn series; very general double
series are the Kamp\'e de F\'eriet series and its generalizations,
see~\cite{Srivastava}. Hypergeometric series in more variables were
also introduced by Lauricella~\cite{Lauricella}. 

The series appearing in this paper are Kamp\'e de F\'eriet series,
i.e.\ rather general hypergeometric series of two variables. For such
series, some isolated summation formulas have been published. Recently,
a new approach has given rise to a number of transformation and
summation formulas~\cite{Pitre}, and in the present paper we extend and
continue this approach.

Kamp\'e de F\'eriet series~\cite{Kampe,Appell} are defined as
follows~\cite{Srivastava}~: 
\bea
&&F^{A:B;B'}_{C:D;D'} \left[ {(a) \atop (c)} \cl {(b) \atop (d)} \sc  {(b')
\atop (d')} \sc x,y \right] \nn\\
&&=\sum_{k,l=0}^\infty {\prod_{j=1}^A (a_j)_{k+l} \over \prod_{j=1}^C
(c_j)_{k+l}} {\prod_{j=1}^B (b_j)_{k} \over \prod_{j=1}^D
(d_j)_{k}} {\prod_{j=1}^{B'} ({b'}_j)_{l} \over \prod_{j=1}^{D'}
({d'}_j)_{l}} {x^k \over k!}{y^l \over l!}.
\eea
Herein, $(a)=(a_1,a_2,\ldots,a_A)$, and $(\alpha)_n$ is the classical
Pochhammer symbol. It is understood (here and hereafter) that no zeros
appear in the denominator. We shall be concerned with the case $B'=B$
and $D'=D$.

Convergence criteria for the above series were
studied by Srivastava and Daoust~\cite{Sriva}, and by Hai {\em et
al}~\cite{Hai}. For us, the case $x=y=1$, $A=0$ and
$C=1$ is of particular importance. 
In this situation, the series converges absolutely provided
\beq
\Re(c+\sum_{j=1}^D d_j -\sum_{j=1}^B b_j)>0 \qquad\hbox{and}\qquad
\Re(c+\sum_{j=1}^{D'} {d'}_j -\sum_{j=1}^{B'} {b'}_j)>0 .
\eeq

For double hypergeometric series of unit argument, there are
comparatively fewer
transformation and summation formulas available in the literature.
Jain~\cite{Jain} obtained a summation formula for a particular
$F^{0:3;3}_{1:1;1}(1,1)$ series, and Carlitz~\cite{Carlitz} for a
certain $F^{1:2;2}_{0:2;2}(1,1)$ series; these were also studied and
$q$-generalized by Srivastava~\cite{Sri}.  

Inspired by identities arising in the study of the so-called 9-$j$
coefficient of angular momentum theory~\cite{Vanderjeugt}, 
a long list of new summation
formulas for $F^{0:3;3}_{1:1;1}(1,1)$ series (including the one due to
Jain) was obtained~\cite{Pitre}. Further investigation of this 9-$j$
coefficient~\cite{thesis} 
has led us to more transformation and summation formulas,
in particular for $F^{0:3;4}_{1:1;2}(1,1)$ and $F^{1:2;2}_{0:2;2}(1,1)$
series. These results are presented in this paper. Of particular
interest is the list of summation formulas for $F^{1:2;2}_{0:2;2}(1,1)$
series (including the one obtained by Carlitz) given in Section~5. 

\section{A general transformation formula}

First, a transformation formula relating a $F^{0:3;p+3}_{1:1;p+1}(1,1)$
series to a $F^{0:3;p+2}_{1:1;p}(1,1)$ series is derived. The proof is
along the same lines as the proof of~\cite[equation~(4)]{Pitre}.

\begin{prop}
Let $f_{p+3}=d-a$, $g_{p+1}=d+e-a-b-c$, and
\beq
\Re(a+\sum_{j=1}^p g_j - \sum_{j=1}^{p+2} f_j)>0, \qquad
\Re(f_{p+3})>0, \qquad \Re(g_{p+1})>0.
\eeq
Then 
\bea
&&F^{0:3;p+3}_{1:1;p+1} \left[ {\hy \atop d} \cl  {a,b,c \atop e} \sc
{f_1, f_2, \ldots, f_{p+3} \atop g_1, g_2,\ldots, g_{p+1} } \sc 1,1\right]
= \Gamma\left[ d,d+e-a-b-c \atop d-a, d+e-b-c \right] \nn\\
&&\times
F^{0:3;p+2}_{1:1;p} \left[ {\hy \atop d+e-b-c} \cl {a,e-b,e-c \atop e} \sc
{f_1, f_2, \ldots, f_{p+2} \atop g_1, g_2,\ldots, g_{p} }\sc 1,1\right].
\label{gentrf}
\eea
\end{prop}
Herein, $\Gamma$ is the classical gamma function with the
convention~\cite[(2.1.1.4)]{Slater}
\[
\Gamma\left[{a_1,a_2,\ldots \atop b_1,b_2,\ldots}\right] =
{\Gamma(a_1)\Gamma(a_2)\ldots \over
\Gamma(b_1)\Gamma(b_2)\ldots}.
\]

\noindent {\em Proof}.
{}From the beta function integral representation
\[
B(x,y)=\Gamma\left[ x,y \atop x+y \right]=\int_0^1
t^{x-1}(1-t)^{y-1}dt, \qquad\Re(x)>0, \Re(y)>0,
\]
one deduces
\beq
{(x)_k(y)_l\over (x+y)_{k+l} } = \Gamma\left[ x+y\atop
x,y\right] \int_0^1 t^{x+k-1}(1-t)^{y+l-1}dt.
\label{beta}
\eeq
Assume that $\Re(a)>0$, and apply the above formula to
$(a)_k(f_{p+3})_l/(d)_{k+l}$ with $f_{p+3}=d-a$. Using the Pochhammer
symbol convention $(b,c)_k=(b)_k(c)_k$, the lhs of (\ref{gentrf}) becomes
\beq
\Gamma\left[d\atop a,d-a\right] \int_0^1 \sum_{k,l}
{(b,c)_k(f_1,\ldots,f_{p+2})_l \over (e)_k k! (g_1,\ldots,g_{p+1})_l l!}
t^{a+k-1}(1-t)^{d-a+l-1}dt. 
\label{tmp1}
\eeq
In this last expression, we use Euler's identity (see
Ref.~\cite{Slater}, eq.\ (1.3.15))~:
\beq
\sum_k{(b,c)_k t^k\over (e)_k k!}={\ }_2F_1\left[{b,c \atop
e}\sc t\right] =
(1-t)^{e-b-c}{\ }_2F_1\left[{e-b,e-c\atop e}\sc t\right].
\eeq
We obtain
\beq
\Gamma\left[d\atop a,d-a\right] \int_0^1 \sum_{k,l}
{(e-b,e-c)_k(f_1,\ldots,f_{p+2})_l \over (e)_k k! (g_1,\ldots,g_{p+1})_l l!}
t^{a+k-1}(1-t)^{d+e-a-b-c+l-1}dt. 
\eeq
For the last integral, we apply again~(\ref{beta}). This gives 
\beq
\Gamma\left[d,d+e-a-b-c\atop d-a,d+e-b-c\right]
\sum_{k,l=0}^\infty { (a,e-b,e-c)_k (f_1,\ldots,f_{p+2},d+e-a-b-c)_l \over
(e)_k (g_1,\ldots g_{p+1})_l (d+e-b-c)_{k+l} k!l!},
\label{tmp2}
\eeq
leading to the rhs of (\ref{gentrf}) since $g_{p+1}=d+e-a-b-c$. The absolute
convergence conditions are necessary for the interchange of summation
and integration; the extra condition $\Re(a)>0$ used to apply the beta
function integral disappears by analytic continuation. \mybox

\section{Transformation formulas for $F^{0:3;4}_{1:1;2}$}

In a previous paper~\cite{Pitre}, three transformation formulas for
double series of the type $F^{0:3;3}_{1:1;1}(1,1)$ were obtained (see
also~\cite{Karlsson2} for~(\ref{12})).
We recall these formulas here, in a notation more appropriate for the
present paper~:

\begin{itemize}
\item
For $\Re(d+e-a-b-c)>0$, $\Re(d-a)>0$ and $\Re(d+e-b-c-b'-c')>0$,
\bea
&&F^{0:3;3}_{1:1;1}\left[ {\hy\atop d}\cl {a,b,c \atop e}\sc {d-a,b',c'
\atop d+e-a-b-c}\sc 1,1\right] \nn\\
&=& \Gamma\left[{d,d+e-a-b-c,d+e-b-c-b'-c' \atop d-a, d+e-b-c-b',
d+e-b-c-c'} \right] \nn\\
&&\times{}_4F_3\left[{a,e-b,e-c, d+e-b-c-b'-c' \atop e,
d+e-b-c-b', d+e-b-c-c'}\sc 1\right].
\label{10}
\eea
\item
For $n$ a nonnegative integer and $\Re(e'-a'-c'-n)>0$,
\bea
&&F^{0:3;3}_{1:1;1}\left[ {\hy\atop 1+a+b-e-n}\cl {-n,a,b \atop e}\sc
{1+a+b-e,a',b' \atop e'}\sc 1,1\right] \nn\\
&=& {(e-a,e-b)_n\over(e,e-a-b)_n}
\Gamma\left[{e',e'-a'-b' \atop e'-a',e'-b'} \right] \nn\\
&& \times
{}_4F_3\left[{-n,a',b',1-e-n \atop
1+a-e-n,1+b-e-n,1+a'+b'-e'}\sc 1\right]. 
\label{11}
\eea
\item
For a nonnegative integer $n$ and $\Re(e-a-c-n)>0$,
\bea
&&F^{0:3;3}_{1:1;1}\left[ {\hy\atop d}\cl {a,b,d+n \atop e}\sc
{d-a,b',-n \atop e'}\sc 1,1\right] \nn\\
&=& {(a)_n\over(d)_n}
\Gamma\left[{e,e-a-b-n \atop e-b,e-a-n} \right] 
{}_4F_3\left[{-n,d-a,e'-b',e-a-b-n \atop e',1-a-n,e-a-n}\sc 1\right].
\label{12}
\eea
\end{itemize}

Next we shall present three transformation formulas for
$F^{0:3;4}_{1:1;2}(1,1)$ series. For each of these, the conditions of
Proposition~1 are satisfied, and they reduce to a
$F^{0:3;3}_{1:1;1}(1,1)$ series. Herein, the parameters are such that
(\ref{10}), (\ref{11}) or (\ref{12}) are applicable. Thus we obtain the
following results~:

\begin{coro}

Let $\Re(f)>0$, $\Re(f-c)>0$ and $\Re(e+f-a-b-c)>0$. Then
\bea
&&F^{0:3;4}_{1:1;2}\left[ {\hy\atop e}\cl {a,b,c \atop f}\sc {e-a,e-b,c',d'
\atop e+f-a-b-c,c+c'+d'}\sc 1,1\right] \nn\\
&=& \Gamma\left[{f,e+f-a-b-c \atop f-c, e+f-a-b} \right] 
{}_4F_3\left[{e-a,e-b,c+c',c+d' \atop e,
e+f-a-b, c+c'+d'}\sc 1\right].
\label{13}
\eea

Let $\Re(f-a-b-n)>0$ and $n$ be a nonnegative integer, then
\bea
&&F^{0:3;4}_{1:1;2}\left[ {\hy\atop e}\cl {a,b,e+n \atop f}\sc {-n,e-a,b',c'
\atop f-a-b-n,1+b+b'+c'-f}\sc 1,1\right] \nn\\
&=& {(e-a,1+b+b'-f,1+b+c'-f)_n\over (e,1+a+b-f,1+b+b'+c'-f)_n}
\Gamma\left[{f,f-a-b \atop f-a, f-b} \right] \nn\\
&&\times{}_4F_3\left[{-n,a,f-e-n,f-b-b'-c'-n \atop
1+a-e-n,f-b-b'-n,f-b-c'-n}\sc 1\right]. 
\eea

Let $\Re(f-a-b-n)>0$ and $n$ be a nonnegative integer, then
\bea
&&F^{0:3;4}_{1:1;2}\left[ {\hy\atop e}\cl {a,b,e+n \atop f}\sc {-n,e-a,e-b,c'
\atop f-a-b-n,f'} \sc 1,1\right] \nn\\
&=& {(1+e-f)_n\over(1+a+b-f)_n} \Gamma\left[{f,f-a-b \atop f-a, f-b} \right] 
{}_4F_3\left[{-n,e-a,e-b,f'-c' \atop e,1+e-f,f'}\sc 1\right].
\label{15}
\eea
\end{coro}

By making further specializations, e.g.\ $c'=e-c$ in (\ref{13}), we can
obtain transformation formulas reducing double series into a
${}_3F_2(1)$. 

\section{Transformation formulas for $F^{1:2;2}_{0:2;2}$}

Consider the Kamp\'e de F\'eriet series
\beq
F^{1:2;2}_{0:2;2}\left[ {e \atop \hy}\cl {a,b \atop
c,d}\sc {a',b' \atop c',d'}\sc 1,1\right]=
\sum_{k,l=0}^\infty {(e)_{k+l}(a,b)_k (a',b')_l \over
(c,d)_k(c',d')_l\; k!l!}.
\eeq
This series is terminating if one of the following conditions is
satisfied~: 
\begin{itemize}
\item $e$ is a negative integer,
\item $a$ or $b$, and $a'$ or $b'$ are negative integers.
\end{itemize}
It is also convergent when it is terminating in one summation and
convergent in the other, i.e.\ if
\begin{itemize}
\item $a$ is a negative integer and $\Re(c'+d'-a'-b'-e+a)>0$; or
\item $a'$ is a negative integer and $\Re(c+d-a-b-e+a')>0$.
\end{itemize}

In this section we derive a transformation formula relating a
$F^{1:2;2}_{0:2;2}(1,1)$ series to a $F^{0:3;3}_{1:1;1}(1,1)$ series.
Then (\ref{10})--(\ref{12}) can be used to find formulas expressing a
$F^{1:2;2}_{0:2;2}(1,1)$ series into a ${}_4F_3(1)$ series.

\begin{prop}
Let $n$ be a nonnegative integer. If $\Re(e'-a'-b'-n)>0$ then
\bea
&&F^{0:3;3}_{1:1;1}\left[ {\hy\atop d}\cl {-n,a,b\atop e}\sc {d+n,a',b'\atop
e'} \sc 1,1\right] = {(e-a,b)_n\over(d,e)_n} \nn\\
&&\times F^{1:2;2}_{0:2;2}\left[ {d-b\atop \hy}\cl {-n,1-e-n\atop
1+a-e-n,1-b-n}\sc {a',b'\atop e',d-b} \sc 1,1\right].
\label{17}
\eea
\end{prop}

\noindent {\em Proof}.
The lhs of (\ref{17}) can be written as
\beq
\sum_{l=0}^\infty {(d+n,a',b')_l \over l! (d,e')_l} 
{}_3F_2\left[ {-n,a,b \atop d+l ,e}\sc 1\right].
\label{18}
\eeq
For the terminating ${}_3F_2$, one can apply a transformation from
Whipple's list between a $Fp(0;4,5)$ and a $Fp(1;2,4)$~\cite[Ch.
3]{Bailey}, see also~\cite[equation~(XVI)]{Rao}~:
\beq
{}_3F_2\left[ {-n,a,b \atop d+l ,e}\sc 1\right]= {(b,e-a)_n\over(d+l,e)_n}
{}_3F_2\left[ {-n,d+l-b,1-e-n \atop 1+a-e-n ,1-b-n}\sc 1\right].
\eeq
Writing $(d+n)_l$ as $(d)_l(d+l)_n/(d)_n$, (\ref{18}) reduces to
\beq
\sum_{l=0}^\infty {(a',b')_l \over l! (e')_l} 
{(b,e-a)_n\over(d,e)_n}
{}_3F_2\left[ {-n,d+l-b,1-e-n \atop 1+a-e-n ,1-b-n}\sc 1\right],
\eeq
and using $(d-b+l)_k=(d-b)_{k+l}/(d-b)_l$ this can be written in the
terms of the $F^{1:2;2}_{0:2;2}$ of (\ref{17}).
\mybox

\begin{coro}

For $\Re(1+e-a'-b'-e-n)>0$~: 
\bea
&&F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,d}\sc
{a',b' \atop 1+e-c,e}\sc 1,1\right] =
{(d-a)_n\over(d)_n}
\Gamma\left[{1+e-c,1+e-c-a'-b' \atop 1+e-c-a',1+e-c-b'} \right] \nn\\
&&\times
{\ }_4F_3\left[{-n,a,c-e+a',c-e+b' \atop c,1+a-d-n,c-e+a'+b'}\sc 1\right].
\label{21}
\eea

For $\Re(c'-a'-b'-n)>0$~:
\bea
&&F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,e}\sc
{a',b' \atop c',e}\sc 1,1\right] =
{(c-a)_n\over(c)_n}
\Gamma\left[{c',c'-a'-b' \atop c'-a',c'-b'} \right] \nn\\
&&\times
{\ }_4F_3\left[{-n,a,a',b' \atop e,1+a-c-n,1+a'-c'+b'}\sc 1\right].
\label{22}
\eea

For $\Re(b+c-a-e+c'-a'-1)>0$~:
\bea
&&F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop b,c}\sc
{a',1+a+e-n-b-c \atop c',e}\sc 1,1\right] =\nn\\
&&{(a,b+c-a-e)_n\over(b,c)_n}
\Gamma\left[{c',b+c-a-e+c'-1-a' \atop c'-a',b+c-a-e+c'-1} \right] \nn\\
&&\times {\ }_4F_3\left[{-n,b-a,c-a,b+c-a-e+c'-1-a' \atop 
1-a-n,b+c-a-e,b+c-a-e+c'-1}\sc 1\right].
\label{23}
\eea
\end{coro}

\noindent {\em Proof}. Put, respectively, 
$e'=d+e+n-a-b$, $e=1+a+b-d-n$ and $a=d-a'$, in equation (\ref{17}) and
apply respectively (\ref{10}), (\ref{11}) and (\ref{12}). \mybox


\begin{prop}
For $n$ a nonnegative integer there holds
\bea
&&F^{1:2;2}_{0:2;2}\left[ {-n\atop\hy}\cl {a,b \atop c,d}\sc
{a',1+a+b-c-d-n \atop c',1-d-n}\sc 1,1\right] = \nn\\
&&{(c+d-a-b,c'-a')_n\over(d,c')_n}
{\ }_4F_3\left[{-n,c-a,c-b,1-c'-n \atop c,c+d-a-b,1+a'-c'-n}\sc 1\right].
\label{24}
\eea
\end{prop}

\noindent {\em Proof}.
Denoting the lhs of (\ref{24}) by $L$, it can be written as
\beq
L=\sum_{l=0}^n {(-n,a',1+a+b-c-d-n)_l \over l!(c',1-d-n)_l} 
{\ }_3F_2 \left[{-n+l,a,b \atop c,d}\sc 1\right].
\label{25}
\eeq
For the ${}_3F_2$,
applying a transformation from Whipple's list~\cite[Ch. 3]{Bailey} 
between $Fp(0;4,5)$ and $Fp(1;2,3)$, leads to~:
\bea
&&{\ }_3F_2 \left[{-n+l,a,b \atop c,d}\sc 1\right]=
{(1-b+l-n)_{n-l}\over(c,d)_{n-l}}\;
\Gamma\left[{1+a-d,1+a-c \atop
1+a-b,1+a+b-c-d-n+l}\right] \nn\\
&&\qquad \times{}_3F_2 \left[{1-b,c-b,d-b \atop 1+a-b,1-b-n+l}\sc 1 \right].
\label{27}
\eea
Substituting (\ref{27}) in (\ref{25}) yields, after some simplifications,
\bea
L&=&{(1-b-n)_n\over(c,d)_n} \Gamma\left[{1-c+a,1-d+a\atop 1+a-b,1+a+b-c-d-n
}\right]\nn\\
&& \times F^{0:3;3}_{1:1;1} \left[{\hy\atop 1-b-n} \cl {-n,a',1-c-n \atop
c'}\sc {1-b,c-b,d-b \atop 1+a-b}\sc 1,1\right].
\eea
Next, we apply (\ref{12}), and obtain
\beq
L={(1+a-c-n,c-b)_n\over(c,d)_n} 
{\ }_4F_3\left[{-n,1-c-n,1+a+b-c-d-n,c'-a' \atop
c',1+a-c-n,1+b-c-n}\sc 1\right]. 
\eeq
Finally, performing a reversal of series on this $_4F_3$ leads to (\ref{24}).
\mybox

\section{Some summation formulas}

Here some limiting cases of the above transformation formulas are
considered. In this section $m$ and $n$ always denote nonnegative integers.
Putting $f'=c'$ in (\ref{15}) leads directly to~:
\beq
F^{0:3;3}_{1:1;1}\left[ {\hy\atop e}\cl {a,b,e+n \atop f}\sc {-n,e-a,e-b
\atop f-a-b-n}\sc 1,1\right]
= {(1+e-f)_n\over(1+a+b-f)_n} \Gamma\left[{f,f-a-b \atop f-a, f-b} \right] ,
\eeq
where $\Re(f-a-b-n)>0$.

Substituting $b'=1+a+e-n-c-d$ and $a'=d+n-1$ in (\ref{21}), the $_4F_3(1)$
reduces to a $_2F_1(1)$, leading to~
\bea
&&F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,d}\sc
{d+n-1,1+a+e-n-c-d \atop 1+e-c,e}\sc 1,1\right] = \nn\\
&&\qquad {(d-e,c+d-e-1)_n\over(d,c)_n}
\Gamma\left[{1+e-c,1-a \atop d-a,2+e-c-d} \right],
\eea
where $\Re(1+c-a-e-n)>0$. This can be further specialized to a double series
terminating in both variables~:
\bea
&&F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,1+a+e-c+m-n}\sc
{a+e+m-c,-m \atop 1+e-c,e}\sc 1,1\right] = \nn\\
&&\qquad (-1)^m{(a)_m(c-e-m,c-a-m)_n\over(1+e-c)_m(c,c-a-e-m)_n}.
\eea

Consider next (\ref{22}) under the extra conditions $a'=1+a-c-n$ and
$c'=2+b'-c-n$. There comes
\beq
F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,e}\sc
{1+a-c-n,b' \atop 2+b'-c-n,e}\sc 1,1\right] =
{(c-a,e-b')_n\over(c,e)_n}
\Gamma\left[{2+b'-c-n,1-a \atop 1+b'-a,2-c-n} \right] ,
\eeq
where $\Re(1-a-n)>0$. For both sides terminating, i.e.\ $b'=-m$, this
reduces to Carlitz's identity~\cite{Carlitz}~:
\beq
F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,e}\sc
{a',-m \atop c',e}\sc 1,1\right] =
{(e)_{m+n}(c-a)_n(c'-a')_m\over(c,e)_n(c',e)_m},
\eeq
where the parameters must satisfy $a'=1+a-c-n$ and $a=1+a'-c'-m$.

Consider again (\ref{22}) but now with $a'=1+a-c-n$ and $c'=1+e-c$; one gets
\bea
&&F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,e}\sc
{1+a-c-n,b' \atop 1+e-c,e}\sc 1,1\right] =\nn\\
&&\qquad {(c-a,e-b')_n\over(c,e)_n}
\Gamma\left[{1+e-c,e-a-b' \atop 1+e-c-b',e-a} \right],
\eea
for $\Re(e-a-b')>0$. For $b'=-m$ both sides are terminating, and one
obtains 
\beq
F^{1:2;2}_{0:2;2}\left[ {e\atop\hy}\cl {a,-n \atop c,e}\sc
{1+a-c-n,-m \atop 1+e-c,e}\sc 1,1\right] =
{(e)_{m+n}(c-a)_n(e-a)_m\over(c,e)_n(1+e-c,e)_m}.
\eeq

Finally, an interesting formula is obtained from (\ref{24}) by choosing
$a'=-a$ and $c'=1-c-n$~:
\beq
F^{1:2;2}_{0:2;2}\left[ {-n\atop\hy}\cl {a,b \atop c,d}\sc
{-a,1+a+b-c-d-n \atop 1-c-n,1-d-n}\sc 1,1\right] = 
{(c-a,d-a)_n\over(c,d)_n}.
\eeq

\section{Comments}

The transformation formulas presented here were originally obtained in
the context of a study of the
series expressions for symmetries of the 9-$j$ angular momentum recoupling
coefficient. In particular, the five distinct types
of doubly stretched 9-$j$ coefficients have been considered. The
original method was presented in~\cite{paper1}, and a complete
classification of the series related to one of the five types of doubly
stretched 9-$j$ coefficients was given in~\cite{Vanderjeugt}. The
classification of the remaining types was completed in~\cite{thesis}.

For single series appearing in this framework, no new summation or
transformation formulas were obtained~: they turned out to be one of the
classical results (Vandermonde's , Saalsch\"utz's or Karlsson-Minton's
summation theorem; Thomae's or Weber-Erd\'elyi's $_3F_2$
transformation formula; Whipple's or Bailey's
Saalsch\"utzian $_4F_3$ transformation formula).
The results concerning double series have been published
in~\cite{Pitre} for the series of type $F^{0:3;3}_{1:1;1}$, and are
presented in the present paper for the remaining ones. Thus the study
of double hypergeometric series arising in the framework of 9-$j$
coefficients has now been completed. 
What remains to be considered are the triple series appearing in this
context. Here, the analysis is more difficult. We hope to report on
this in the future.

\section*{Acknowledgements}

This research was partly supported by the E.E.C. (contract No.
CI1*-CT92-0101).

%\newpage

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