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%     Transformation formulas for double hypergeometric series        %
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%     by S. Lievens and J. Van der Jeugt                              %
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{\Large \bf Transformation formulas for double hypergeometric series}\\[4mm]
{\Large \bf related to 9-$j$ coefficients and their basic analogues}\\[3cm]
{\bf S.\ Lievens and J.\ Van der Jeugt}\\[2mm]
Department of Applied Mathematics and Computer Science,\\
University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium.\\
E-mails~: Stijn.Lievens@rug.ac.be, Joris.VanderJeugt@rug.ac.be.
\end{center}

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\begin{abstract}
In a recent paper, Ali\v sauskas deduced different triple
sum expressions for the 9-$j$ coefficient of $su(2)$ and
$su_q(2)$. For a singly stretched 9-$j$ coefficient, these
reduce to different double sum series.
Using these distinct series, we deduce a set of new
transformation formulas for double
hypergeometric series of Kamp\'e de F\'eriet type and their
basic analogues. 
These transformation formulas are valid for rather
general parameters of the series, although a common feature
is that all the series appearing here are terminating.
It is also shown that the transformation formulas
deduced here generate a group of transformation formulas,
thus yielding an invariance group or symmetry group
of particular double series. 
\end{abstract}

\vspace{1cm}
\noindent
Running title~: Transformations for double series.\\[2mm]
PACS~: 02.20.+b, 02.30.+g, 03.65.Fd.

\vspace{1cm}

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\section{Introduction}

The Wigner 9-$j$ coefficients (or 9-$j$ symbols) arise as recoupling
coefficients in the coupling (tensor product) of four irreducible
representations of $su(2)$, and play an important role in the quantum
theory of angular momentum~\cite{Wigner,Edmonds,BiedenharnLouck}.
Although the relation between recoupling coefficients, such as the
3-$j$ coefficient and the 6-$j$ coefficient, and hypergeometric series
or (discrete) orthogonal polynomials of hypergeometric type 
is well understood~\cite{BiedenharnLouck, VK, RaoRajeswari, Varshalovich},
the \hbox{9-$j$} coefficient remains somewhat a mystery in this respect.
There are many known expressions for the 9-$j$ coefficient as a
multiple hypergeometric series. The most compact formula is the
so-called triple sum series, originally derived by Ali\v sauskas and
Jucys~\cite{AlisauskasJucys}, and rederived in~\cite{JucysBandzaitis}.
Whether a triple sum expression is really the best one can do for the
9-$j$ coefficient, is not known; specialists in the field still guess
that a double sum series might exist~\cite{Askey}.

The triple sum series of Ali\v sauskas and Jucys was recognized
as a special case~\cite{RaoRajeswariChiu, RaoRajeswari} of a triple hypergeometric series defined by Srivastava~\cite{Srivastava}.
It was used to speed up the numerical computation of 9-$j$
coefficients~\cite{RaoRajeswariChiu}, and to derive certain summation
and reduction formulas for hypergeometric series by using particular
classes of 9-$j$ coefficients~\cite{RaoVdj, VdjPitreRao, PitreVdj}.

Ali\v sauskas and Jucys's triple sum series was recently rederived
in two ways. In~\cite{Rosengren1}, Rosengren deduced the triple sum
series for 9-$j$ coefficients (of $su(1,1)$ rather than of $su(2)$)
based upon the use of coupling kernels; in~\cite{Rosengren2}, he
showed that the same formula can be deduced starting from the classical
expansion of the 9-$j$ coefficient in terms of 6-$j$ coefficients and
performing Dougall's summation formula~\cite{Slater} for a very 
well-poised ${}_4F_3(-1)$ series. In a recent paper~\cite{A}, 
Ali\v sauskas realized that this technique can be applied for several 
distinct expansions of the 9-$j$ coefficient in terms of 6-$j$ 
coefficients. Thus he obtained seven different triple sum formulas for
the 9-$j$ coefficient of $su(2)$. At the same time, he showed that
this technique has a basic analogue (or $q$-analogue), depending
upon a $q$-analogue of Dougall's summation formula~\cite{Alisauskas97}.
So he also obtained seven triple sum formulas for the 9-$j$
coefficient of $su_q(2)$, i.e.\ for the $q$-9-$j$ coefficients.

The study of these different triple sum formulas from the point of
view of multiple hypergeometric series would be interesting,
though rather tedious because of the complicated structure of 
the formulas. However, when considering the class of singly stretched
9-$j$ coefficients (i.e.\ one of the arguments in the 9-$j$
coefficient is the sum of two others), most of these triple sum
formulas reduce to double sum formulas which are less complicated 
and easier to handle. Ali\v sauskas actually wrote down these
double sum formulas~\cite[Section~IV.B]{A}, and used them to derive
certain rearrangement formulas of double sum series and their
basic analogues. 

In the present paper we shall show that the double sum formulas for
the singly stretched 9-$j$ coefficient actually give rise to a fairly
complete theory of transformation formulas for terminating
double hypergeometric series of Kamp\'e de F\'eriet type. This is
particularly interesting because until now not many transformation
formulas for multiple hypergeometric series are known, even though
transformation formulas for hypergeometric series of a single variable
play an important role~\cite{Slater,Bailey}.
The double hypergeometric series appearing in this context are
proper Kamp\'e de F\'eriet functions
$F^{p:r;r}_{q:s;s}$ with $q+s=2$ and $p+r=3$. Such functions
have been defined in~\cite{Appell, Kampe}, and studied by Srivastava
and Karlsson~\cite{SrivastavaKarlsson}, whose notation we follow. 
This notation is a rather straightforward extension of that for single
hypergeometric series, e.g.
\beq
F^{1:2;2}_{0:2;2}\left[ {e \atop \ }{:\atop :}
 {a,b \atop c,d} {;\atop ;} {a',b' \atop
c',d'} {;\atop ;} x,y\right] = \sum_{j,k=0}^\infty 
{(e)_{j+k}} {(a)_j(b)_j \over (c)_j(d)_j} {(a')_k(b')_k \over
(c')_k(d')_k} {x^j\over j!} {y^k \over k!},
\label{defF122022}
\eeq
and
\beq
F^{1:2;2}_{1:1;1}\left[ {e \atop d }{:\atop :}
 {a,b \atop c} {;\atop ;} {a',b' \atop
c'} {;\atop ;} x,y\right] = \sum_{j,k=0}^\infty 
{(e)_{j+k}\over (d)_{j+k}} 
{(a)_j(b)_j \over (c)_j} {(a')_k(b')_k \over
(c')_k} {x^j\over j!} {y^k \over k!}.
\label{defF122111}
\eeq
Herein, $(a)_k$ is the classical Pochhammer symbol~\cite{Bailey,Slater},
\beq
(a)_k = a(a+1)\cdots (a+k-1);
\eeq 
$a$, $b$, $\ldots$ are referred to as the parameters of the series,
and $x$, $y$ as the variables.
Observe that factors of the form $(d)_{j+k}$ or $(e)_{j+k}$ are
responsible for the fact that such double series cannot simply be
written as the product of two single hypergeometric series.
The Kamp\'e de F\'eriet series appearing in the context of double sums
related to the 9-$j$ coefficients are those of type $F^{1:2;2}_{0:2;2}$,
$F^{1:2;2}_{1:1;1}$ and $F^{0:3;3}_{1:1;1}$. 

Convergence properties of such Kamp\'e de F\'eriet series have been 
considered in~\cite{Hai}. In this paper, however, all the series
dealt with are {\em terminating} series and hence there are no
convergence conditions. Note that the termination of Kamp\'e de 
F\'eriet series such as~(\ref{defF122022}) or~(\ref{defF122111})
can be assured in two ways~:
\begin{itemize}
\item a common numerator parameter equals a negative integer~: 
e.g.\ $e=-n$, with $n$ a positive integer, in~(\ref{defF122022}) 
or~(\ref{defF122111}) yields a terminating series irrespective of the
value of the other parameters;  
\item two separate numerator parameters are equal to negative
integers~: e.g.\ $a=-n$ and $a'=-m$ in~(\ref{defF122022}) 
or~(\ref{defF122111}), with $m$ and $n$ positive integers. 
\end{itemize}
In both cases the denominator parameters of the Kamp\'e de F\'eriet
series should not be negative integers. If some
of the denominator parameters are nevertheless negative integers,
then they should be smaller (or equal) than the parameters responsible
for the termination of the series. 

The transformation formulas deduced here from the double sum expressions
for 9-$j$ coefficients turn out to be of a quite general nature.
Apart from the parameter(s) responsible for the termination, the
remaining parameters of the series are completely general. 
Furthermore, the transformation formulas, together with trivial
permutation symmetries, are shown to generate a {\em symmetry group}
for the double hypergeometric series. In other words, we shall show
that for each of the double hypergeometric series considered here,
there exists a whole set of transformation formulas related to
a group action on the parameters of the double series.

In the paper of Ali\v sauskas~\cite{A}, the emphasis is on the
$q$-9-$j$ coefficients, i.e.\ the 9-$j$ coefficients of $su_q(2)$.
So it would be interesting to see if the transformation theory
developed here could be generalized to the basic analogue (i.e.\
the $q$-analogue). This is indeed the case. We shall give and prove
a set of new transformation formulas for basic double series.
For the notation related to $q$-series and single basic hypergeometric
series, we refer to the standard book of Gasper and Rahman~\cite{GR}.
The double series appearing in this context, however, are special
cases of general basic double series defined by Srivastava
and Karlsson~\cite[p.~349]{SrivastavaKarlsson}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The stretched 9-$j$ coefficient and double series}

Ali\v sauskas~\cite{A} considers the stretched 9-$j$ coefficient
denoted by
\beq
\left\{ 
\begin{array}{ccc} j_1 & j_2 & j_{12} \\ j_3 & j_4 & j_{34} \\
j_{13} & j_{24} & j_{12}+j_{34} \end{array} \right\},
\label{9j}
\eeq
which is a transformation coefficient connecting two different
ways in which four angular momenta $j_1$, $j_2$, $j_3$ and $j_4$
can be coupled. Since they stand for angular momenta, all the
arguments in~(\ref{9j}) are nonnegative integers or half-integers. 
In fact, in~\cite{A}, the $q$-analogues of such 9-$j$ coefficients
are considered, but here we first treat the classical case ($q=1$).

In~\cite[Section~IV.B]{A}, a list of double sum expressions is
determined for~(\ref{9j}). It is not difficult to rewrite these
in terms of double hypergeometric series of Kamp\'e de F\'eriet type.
For example, from~\cite[(4.3d)]{A} one deduces~:
\bea
&&\left\{ 
\begin{array}{ccc} j_1 & j_2 & j_{12} \\ j_3 & j_4 & j_{34} \\
j_{13} & j_{24} & j_{12}+j_{34} \end{array} \right\} = C \times 
\label{43d}\\
&& F^{1:2;2}_{0:2;2}\left[ {-j_1-j_2+j_{12} \atop \ }{:\atop :}
{-j_2-j_4+j_{24}, -j_2-j_4-j_{24}-1 \atop 
-2j_2, -j_2-j_4+j_{12}+j_{34}-j_{13}} {; \atop ;} \right. \nn\\
&& \qquad \left. 
{j_{13}-j_1+j_3+1, -j_1-j_3+j_{13} \atop -2j_1,j_4-j_1-j_{34}+j_{13}+1}
 {; \atop ;}  1,1 \right], \nn
\eea
where $C$ is some constant.
Similarly, \cite[(4.4b)]{A} yields~:
\bea
&&\left\{ 
\begin{array}{ccc} j_1 & j_2 & j_{12} \\ j_3 & j_4 & j_{34} \\
j_{13} & j_{24} & j_{12}+j_{34} \end{array} \right\} = C' \times 
\label{44b}\\
&& F^{1:2;2}_{1:1;1}\left[ {-j_1-j_2+j_{12} \atop 
1-j_1-j_{34}+j_{13}-j_2+j_{24} }
{:\atop :} {-j_2-j_4+j_{24}, 1+j_4-j_2+j_{24} \atop 
-2j_2} {; \atop ;} \right. \nn\\
&&\left. \qquad {j_{13}-j_1+j_3+1, -j_1-j_3+j_{13} \atop -2j_1 }
 {; \atop ;}  1,1 \right], \nn
\eea
where $C'$ is another constant. Upon equating the rhs's of~(\ref{43d})
and~(\ref{44b}), using the actual values of $C$ and $C'$, and relabelling
the parameters of the series, one finds~:
\begin{equation}
\label{trans0}
F^{1:2;2}_{0:2;2}\left[{-n\atop \phantom{}}{:\atop :} 
{a, b\atop c,d} {;\atop ;} {a', b'\atop c',d'} {;\atop ;}
1,1 \right] = {(d-a+n-1)!(d-1)! \over (d-a-1)!(d+n-1)!}
F^{1:2;2}_{1:1;1}\left[
 {-n\atop d'+a} {:\atop :} {a, c-b\atop c}
{;\atop ;} {a', b'\atop c'} {;\atop ;} 1,1 \right].
\end{equation}
Herein, the parameters satisfy $d+d' = 1-n$, so there are in total
eight free parameters (as there are in~(\ref{9j})).
Since all the parameters in~(\ref{9j}) are nonnegative integers or 
half-integers, the parameters in~(\ref{trans0}) in first instance
all correspond to integers. In particular, $-n$ corresponds to a
negative integer (due to triangular conditions satisfied by the
angular momentum coefficients). 
However, once the equation is rewritten in the form~(\ref{trans0}),
with ${(d-a+n-1)!(d-1)! \over (d-a-1)!(d+n-1)!}={(d-a)_n\over(d)_n}$,
it is obvious that this is a {\em rationial identity} in the remaining
parameters $a$, $b$, $c$, $d$, $a'$, $b'$, $c'$ and $d'$, once $-n$ is
a fixed negative integer. 
Therefore, (\ref{trans1}) holds for arbitrary parameters 
$a$, $b$, $c$, $d$, $a'$, $b'$, $c'$ and $d'$ (but still subject
to the constraint $d+d' = 1-n$).
As such, we have found a rather general transformation formula
between two terminating Kamp\'e de F\'eriet series.
This proves the first formula of the following theorem~:

\begin{theo}
Let $n$ be a nonnegative integer and $a$, $b$, $c$, $d$, $a'$, $b'$, 
$c'$ and $d'$ arbitrary parameters with $d+d'=1-n$. Then the
following transformation formulas hold~:
\begin{subeqnarray}
F^{1:2;2}_{0:2;2}\left[{-n\atop \phantom{}}{:\atop :} 
{a, b\atop c,d} {;\atop ;} {a', b'\atop c',d'} {;\atop ;}
1,1 \right] &=& 
{(d-a)_n\over (d)_n}F^{1:2;2}_{1:1;1}\left[
 {-n\atop d'+a} {:\atop :} {a, c-b\atop c}
{;\atop ;} {a', b'\atop c'} {;\atop ;} 1,1 \right] \slabel{trans1}\\
&=&
{(d-b+b')_n\over (d)_n} F^{1:2;2}_{0:2;2}\left[{-n\atop \phantom{}}
{:\atop :} {c-a, b\atop c,d'-b'+b} {;\atop ;}
{c'-a', b'\atop c',d-b+b'}{; \atop ;} 1,1\right], \nn\\
&& \slabel{F122022joris}
\end{subeqnarray}
and
\beq
F^{1:2;2}_{1:1;1}\left[{-n\atop d}{:\atop :} {a, b\atop c}
{;\atop ;}{a', b'\atop c'}{;\atop ;} 1,1\right]
= {(d-b-b')_n\over (d)_n} 
F^{1:2;2}_{1:1;1}\left[{-n\atop 1-n-d+b+b'}{:\atop :}
{c-a, b\atop c}{;\atop ;} {c'-a', b'\atop c'}
{; \atop ;} 1,1\right].
\label{F122111-n}
\eeq
\label{theo:1}
\end{theo}

\noindent {\bf Proof.} 
The transformation formula~(\ref{F122022joris}) was deduced 
recently in a different context~\cite{Vdj}. This equation
can now also be seen in the context of the stretched 9-$j$ coefficient.
In fact, it corresponds to a symmetry of this 9-$j$ coefficient
(namely a transposition of the first and second column in~(\ref{9j})),
re-expressed by means of~(\ref{43d}).
Finally, applying~(\ref{trans1}) to the rhs of~(\ref{F122022joris}) 
and equating the resulting expression with the rhs of~(\ref{trans1}) 
yields~(\ref{F122111-n}) (after appropriate relabelling of the
parameters).  \mybox

Observe that in this section all Kamp\'e de F\'eriet series are
terminating because a common numerator parameter equals a negative
integer. In the following section we shall consider some transformation
formulas, also deduced from the stretched 9-$j$ coefficient, 
for Kamp\'e de F\'eriet series that are terminating because of
the appearance of two negative integers as separate numerator parameters.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Kamp\'e de F\'eriet series with two negative integers as
parameter}

Though the transformation formulas with a single common numerator
parameter as a negative integer (i.e.\ Theorem~\ref{theo:1}) are
new, there do exist some transformation formulas for Kamp\'e de 
F\'eriet series with two separate numerator parameters as
negative integers. One of these formulas is given by 
Singh~\cite{Singh}, and reads~:
\begin{eqnarray}
\lefteqn{
F^{0:3;3}_{1:1;1}\left[ {\phantom{}\atop d}{:\atop :}
{-n, a, b\atop c}{;\atop ;} {-m, a', b'\atop c'}
{;\atop ;} 1,1\right] = {(c-a)_n(c'-a')_m\over(c)_n(c')_m} }\nonumber \\
& & {}\times F^{0:3;3}_{1:1;1}\left[ {\phantom{}\atop d}
{:\atop :} {-n, a, b'\atop 1+a-c-n}{; \atop ;}
{-m, a', b\atop 1+a'-c'-m}{;\atop ;}1,1\right], 
\label{singh}
\end{eqnarray}
where $n$ and $m$ are nonnegative integers and $b+b'=d$. 
This, and some other transformation formulas of similar type, can be found
in or deduced from~\cite[Appendix~C]{A}.

Let us first consider some transformation formulas that express
a Kamp\'e de F\'eriet series of type $F^{1:2;2}_{0:2;2}$ into
a series of a different type~:
\begin{theo}
Let $m$ and $n$ be nonnegative integers, and $a$, $b$, $c$, $d$,
$a'$, $b'$ and $c'$ be arbitrary parameters with $c+c'=1+d$, then
\begin{subeqnarray}
\lefteqn{ 
F^{1:2;2}_{0:2;2} \left[{d\atop \phantom{}}{:\atop :}
{-n, a\atop b,c}{;\atop ;} {-m, a'\atop b',c'} {;\atop ;} 1,1\right]
} \nn\\
&=& 
{(b-a)_n(1-c)_m\over (b)_n (c')_m}  
F^{0:3;3}_{1:1;1}
\left[ {\phantom{}\atop c-m}{:\atop :} {-n, a, -d+c-m\atop -n+a-b+1}
{;\atop ;} {-m, b'-a', d\atop b'} {;\atop ;} 1,1  \right]
 \slabel{trans2} 
\\ &=& 
{(1-c)_m\over (c')_m}
F^{1:2;2}_{1:1;1}\left[ {d\atop c-m}{:\atop :}
{-n, a\atop b}{;\atop ;} {-m, b'-a'\atop b'}{;\atop ;} 1,1
\right] .
\slabel{trans3}
\end{subeqnarray}
\label{theo:mixed}
\end{theo}

The proof of~(\ref{trans2}) follows by comparing equations (4.3c) 
and (4.3e) of~\cite{A}, making appropriate relabellings, and using the
same rational expression argument as in the proof of 
Theorem~\ref{theo:1}. In a similar way, (\ref{trans3}) follows
from (4.3b) and (4.4c) of~\cite{A}. 

It is worth mentioning that transformation formulas~(\ref{trans1}) 
and~(\ref{trans3}) are formally equivalent (after rewriting the 
Pochhammer symbols in terms of Gamma functions and using the constraint
 $1-c = c'-d$ to eliminate $c$ from the Gamma functions in~(\ref{trans3})).

We can now present three results, giving transformation formulas
for Kamp\'e de F\'eriet series of a particular type into 
series of the same type, for each of the types
$F^{1:2;2}_{0:2;2}$, $F^{1:2;2}_{1:1;1}$ and $F^{0:3;3}_{1:1;1}$. 

\begin{theo} \label{theo:T1}
Let $n$ and $m$ be nonnegative integers and $a$, $b$, $c$, $a'$, 
$b'$, $c'$ and $d$ be arbitrary parameters with $c+c' = d+1$, then
\begin{subeqnarray}
&&F^{1:2;2}_{0:2;2}\left[{d\atop \phantom{}}{:\atop :}
{-n, a\atop b,c}{;\atop ;} {-m, a'\atop b',c'}{;\atop ;},1,1\right] = 
{(-1)^m(d)_n(b-a)_n(a')_m\over (b)_n(c')_m(b')_m(c)_{n-m}}  \slabel{Tmn1} \\
& & {}\times  F^{1:2;2}_{0:2;2}\left[{-m-c'+1\atop \phantom{}}{:\atop :}
{-n, -n-b+1\atop -n+a-b+1, -n-d+1}{; \atop ;}
{-m, -a'+b'\atop -m-a'+1, -m+n+c}{;\atop ;} 1,1\right] \nn \\
&&={(-1)^m(d)_m(b-a)_n(a')_m\over (b)_n(b')_m(c')_m} \nn \\ 
&& \times F^{1:2;2}_{0:2;2}\left[{-m-c'+1\atop \phantom{}}{:\atop :}
{-n, a\atop -n+a-b+1, c}{;\atop ;} 
{-m, 1-m-b'\atop 1-m-a', 1-m-d} {;\atop ;} 1,1\right].
\slabel{Tmn1bis} 
\end{subeqnarray}
\end{theo}

\noindent {\bf Proof.} The first formula, (\ref{Tmn1}), follows by
comparing expressions (4.3a) and (4.3d) of~\cite{A}, and using
the rational expression argument. The second formula is derived 
using~(\ref{trans2}) and Singh's formula~(\ref{singh}). \mybox


In~(\ref{Tmn1}) the difference $n-m$ might be negative, and then 
$  c_{n-m} = {(-1)^{m-n}/ (1-c)_{m-n}}$,
which is the natural extension of the Pochhammer symbol.

Using the above two formulas and~(\ref{trans3}) yields~:

\begin{theo} \label{theo:T2}
Let $n$ and $m$ be nonnegative integers and let $a$, $b$, $a'$, $b'$, 
$c$ and $d$ be arbitrary parameters, then
\begin{subeqnarray}
&& F^{1:2;2}_{1:1;1}\left[{c\atop d}{:\atop :} {-n, a\atop b}
{;\atop ;} {-m, a'\atop b'}{; \atop ;}1,1\right]
= {(c)_{n+m}(b-a)_n(b'-a')_m\over (d)_{n+m}(b)_n(b')_m }  \nonumber \\
& & {}\times F^{1:2;2}_{1:1;1}\left[{d-c\atop -n-m-c+1}{:\atop :}
{-n, -n-b+1\atop -n+a-b+1}{;\atop ;} 
{-m, -m-b'+1\atop -m+a'-b'+1}{;\atop ;}1,1\right]
\slabel{Tmn2} \\
&&= {(b-a)_n(b'-a')_m\over (b)_n(b')_m } 
 F^{1:2;2}_{1:1;1}\left[{d-c\atop d}{:\atop :}
{-n, a\atop -n+a-b+1}{;\atop ;} {-m, a'\atop -m+a'-b'+1}
{;\atop ;}1,1\right]. \slabel{Tmn2bis}
\end{subeqnarray}
\end{theo}

As a third and final result, we give the transformation formulas
for Kamp\'e de F\'eriet series of type $F^{0:3;3}_{1:1;1}$.
The first formula follows from~(\ref{Tmn1}) and~(\ref{trans2}); the
second is just Singh's formula~(\ref{singh}).

\begin{theo} \label{theo:T3}
Let $n$ and $m$ be nonnegative integers and let $a$, $b$, $c$, 
$a'$, $b'$, $c'$ and $d$ be arbitrary parameters such that 
$b+b'=d$, then
\begin{subeqnarray}
&& F^{0:3;3}_{1:1;1}\left[ {\phantom{}\atop d}{:\atop :}
{-n, a, b\atop c}{;\atop ;} {-m, a', b'\atop c'}{;\atop ;}1,1\right] = 
{(b')_{n+m}(a)_n(c'-a')_m\over (d)_{n+m}(c)_n(c')_m} \nn \\
& & {}\times F^{0:3;3}_{1:1;1}\left[ {\phantom{}\atop -n-m+1-b'}{:\atop :}
 {-n, c-a, 1-n-m-d \atop -n-a+1}{;\atop ;}
{-m, -c'-m+1, b\atop -m+a'-c'+1}{;\atop ;}1,1\right] \slabel{Tmn3} \\
&& = {(c-a)_n(c'-a')_m\over(c)_n(c')_m}
F^{0:3;3}_{1:1;1}\left[ {\phantom{}\atop d}{:\atop :}
 {-n, a, b'\atop 1+a-c-n}{;\atop ;}
{-m, a', b\atop 1+a'-c'-m}{;\atop ;}1,1\right]. 
\slabel{Tmn3bis}
\end{subeqnarray}
\end{theo}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symmetry groups of terminating Kamp\'e de F\'eriet series}

In the previous sections we have determined transformation formulas
between (terminating) Kamp\'e de F\'eriet series of the same type. 
It is known that transformation formulas of hypergeometric series of 
a single variable can give rise to a transformation group~\cite{VdjRao}. 
This transformation group, known as the symmetry group or invariance
group of the series, arises as a finite group acting on the 
parameters of the series. The existing transformation formulas are
then expressed as the invariance of a certain hypergeometric series
under the action of group elements on its parameters.
For single hypergeometric series (and basic series), this idea
has been expanded in~\cite{VdjRao}. 

So it would be interesting to see whether there are any invariance
groups behind the transformation formulas for double hypergeometric 
series, as the ones we are dealing with in this paper.
One such invariance group for a double series has recently been
discussed~\cite{Vdj}. This concerns the invariance group related
to the transformation formula~(\ref{F122022joris}). Observe
that~(\ref{F122022joris}) gives a transformation between two
series of the type $F^{1:2;2}_{0:2;2}$. Apart from this transformation,
there are also trivial transformations for
\beq
F^{1:2;2}_{0:2;2}\left[{-n\atop \phantom{}}{:\atop :} 
{a, b\atop c,d} {;\atop ;} {a', b'\atop c',d'} {;\atop ;}
1,1 \right]\qquad\hbox{with }d+d'=1-n,
\label{series1}
\eeq
namely the transposition of $a$ and $b$, or the transposition
of $a'$ and $b'$, or the exchange of all primed with the corresponding
unprimed parameters. It was shown in~\cite{Vdj} that superposing
such trivial transformations with~(\ref{F122022joris}) gives rise to
a set of 64 transformations for the series $F^{1:2;2}_{0:2;2}$ (with
one common numerator parameter equal to $-n$). These 64 
transformations correspond to a group $G$ of order~64, that we shall
briefly describe because it also plays a role in other transformations
considered in this paper.

First, consider the permutation group $S_8$ acting on $(x_1,x_2,x_3,
x_4,x_1',x_2',x_3',x_4')$, and its subgroup $D_8\times D_8'$. Herein,
$D_8$ stands for the dihedral group~\cite{Hamermesh} (sometimes
denoted by $D_4$) consisting of the 8 symmetries of the square (i.e.\
those permutations of $x_1, \ldots, x_4$ that preserve the square
whose sides are labelled by $x_1, \ldots, x_4$). Similarly, $D_8'$
is the same dihedral group but acting on the primed labels 
$x_1', \ldots, x_4'$. The group $D_8\times D_8'$ consists of 64
elements; superposing on this group the interchange of primed and
unprimed elements yields a group of order 128, denoted by 
$S_2\times(D_8\times D_8')$. This is the invariance group of 
two squares whose sides are labelled as follows~: 
\vskip 2mm
\[
\vbox{
\unitlength=1mm
%\special{em:linewidth 0.4pt}
%\linethickness{0.4pt}
\begin{picture}(57.00,152.00)
\put(10.00,150.00){\line(1,0){15.00}}
\put(25.00,150.00){\line(0,-1){15.00}}
\put(25.00,135.00){\line(-1,0){15.00}}
\put(10.00,135.00){\line(0,1){15.00}}
\put(10.00,150.00){\line(0,0){0.00}}
\put(40.00,150.00){\line(1,0){15.00}}
\put(55.00,150.00){\line(0,-1){15.00}}
\put(55.00,135.00){\line(-1,0){15.00}}
\put(40.00,135.00){\line(0,1){15.00}}
\put(18.00,152.00){\makebox(0,0)[cc]{$x_1$}}
\put(48.00,153.00){\makebox(0,0)[cc]{$x_1'$}}
\put(8.00,143.00){\makebox(0,0)[cc]{$x_2$}}
\put(38.00,143.00){\makebox(0,0)[cc]{$x_2'$}}
\put(18.00,133.00){\makebox(0,0)[cc]{$x_3$}}
\put(48.00,133.00){\makebox(0,0)[cc]{$x_3'$}}
\put(27.00,143.00){\makebox(0,0)[cc]{$x_4$}}
\put(58.00,143.00){\makebox(0,0)[cc]{$x_4'$}}
\end{picture}
\vskip -135mm
}
\]
The group $G$ now consists of those 64 elements of 
$S_2\times(D_8\times D_8')$ that preserve the
constraint
\beq
x_1+x_3+x_1'+x_3'-x_2-x_4-x_2'-x_4'=0,
\label{constraint}
\eeq
i.e.\ those elements that map $X=x_1+x_3+x_1'+x_3'-x_2-x_4-x_2'-x_4'$ 
into $\pm X$ by permuting the indices.
The following proposition~\cite{Vdj} then describes the invariance group
generated by the transformation~(\ref{F122022joris})~:

\begin{prop}
Let $x_i$, $x_i'$ ($i=1,\ldots,4$) be arbitrary parameters such that
$x_1+x_3+x_1'+x_3'=x_2+x_4+x_2'+x_4'$ and let $n$ be a 
nonnegative integer. Then the expression
\[
f_1(x)={\ts ({1-n\over 2}+x_2-x_2')_n }\ 
F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }
{:\atop :} {x_2+x_3,x_1+x_2 \atop \sum_i x_i ,{1-n\over 2}+x_2-x_2'} 
{;\atop;}  
{x_2'+x_3',x_1'+x_2' \atop \sum_i x_i' ,{1-n\over 2}+x_2'-x_2}
{;\atop ;} 1,1\right]
\]
is (upto a sign) invariant under the action of $G$.
The action of an element
$g$ of $G$ is by permuting the indices of $x_1, \ldots ,x_4'$, and we
can write
\[
f_1(g\cdot x) = \epsilon^n f_1(x),
\]
where $\epsilon = \pm 1$ is determined by $g(X)=\epsilon X$.
\end{prop}

When determining the invariance group of the series
\beq
F^{1:2;2}_{1:1;1}\left[{-n\atop d}{:\atop :} {a, b\atop c}
{;\atop ;}{a', b'\atop c'}{;\atop ;} 1,1\right]
\label{series2}
\eeq
the following relabelling is appropriate~:
\bea
&& a=x_2+x_3,\qquad b=x_1+x_2,\qquad c=\sum_i x_i, \nn\\
&& a'=x_2'+x_3',\qquad b'=x_1'+x_2',\qquad c'=\sum_i x_i',
\qquad d={ 1-n\over  2}+x_2+x_2'.
\eea
Here again, $-n$ is a negative integer and $x_1,\ldots, x_4'$
are arbitrary parameters satisfying~(\ref{constraint}). 
Using this relabelling in~(\ref{series2}), 
the transformation~(\ref{F122111-n})
corresponds (apart from a factor) to the permutation
$g_1=(x_1\ x_2)(x_3\ x_4)(x_1'\ x_2')(x_3'\ x_4')$.
The trivial transposition of $a$ and $b$ in~(\ref{series2})
corresponds to the permutation $g_2=(x_1\ x_3)$. 
And the interchange of primed and unprimed parameters 
in~(\ref{series2}) corresponds to the permutation
$g_3=(x_1\ x_1')(x_2\ x_2')(x_3\ x_3')(x_4\ x_4')$.
It is now easy to see that the elements $g_1$, $g_2$ and
$g_3$ generate the group $G$ described earlier. Thus
we have the following result~:

\begin{prop}
Let $x_i$, $x_i'$ ($i=1,\ldots,4$) be arbitrary parameters such that
$x_1+x_3+x_1'+x_3'=x_2+x_4+x_2'+x_4'$ and let $n$ be a 
nonnegative integer. Then the expression
\[
f_2(x)= {\ts ({1-n\over 2}+x_2+x_2')_n} \ 
F^{1:2;2}_{1:1;1}\left[ {-n \atop { 1-n\over  2}+x_2+x_2' }
{:\atop :} {x_2+x_3,x_1+x_2 \atop \sum_i x_i } 
{;\atop;}  
{x_2'+x_3',x_1'+x_2' \atop \sum_i x_i' }
{;\atop ;} 1,1\right]
\]
is (upto a sign) invariant under the action of $G$, i.e.\
$f_2(g\cdot x) = \epsilon^n f_2(x)$,
where $\epsilon = \pm 1$ is determined by $g(X)=\epsilon X$.
\end{prop}

So the invariance groups of~(\ref{series1}) and~(\ref{series2})
are the same~: both series have 64 symmetries.  Moreover, the two 
non-trivial transformations~(\ref{F122022joris}) and~(\ref{F122111-n}) 
both correspond to the same element, namely $g_1$, of $G$.\\[5mm]

Now we shall show that also the transformations with two
numerator parameters $-n$ and $-m$ being negative integers
give rise to an interesting symmetry group. 
It will be convenient to first describe the group, and then
show that under a certain relabelling of the parameters it is indeed
the symmetry group of the transformations given in
Theorems~\ref{theo:T1}, \ref{theo:T2} and~\ref{theo:T3}.

Consider a prism with an equiangular triangle as basis and 
edges orthogonal to this basis. The sides of the triangles are
labelled by $x_1$, $x_2$, $x_3$ and $x_1'$, $x_2'$, $x_3'$;
the three edges are labelled by $x_1''$, $x_2''$, $x_3''$.
For convenience we shall also label the basis triangle by $n$ and the
opposite triangle by $m$~:
\vskip 2mm
\[
\vbox{
\unitlength=1mm
%\special{em:linewidth 0.4pt}
%\linethickness{0.4pt}
\begin{picture}(62.00,140.00)
\put(30.00,130.00){\line(1,0){30.00}}
\put(60.00,130.00){\line(-3,2){15.00}}
\put(45.00,140.00){\line(-3,-2){15.00}}
\put(30.00,130.00){\line(0,-1){30.00}}
\put(30.00,100.00){\line(1,0){30.00}}
\put(60.00,100.00){\line(0,1){30.00}}
\put(30.00,100.00){\line(3,2){15.00}}
\put(45.00,110.00){\line(3,-2){15.00}}
\put(45.00,110.00){\line(0,1){30.00}}
\put(42.00,97.00){\makebox(0,0)[cc]{$x_1$}}
\put(37.00,107.00){\makebox(0,0)[cc]{$x_2$}}
\put(52.00,107.00){\makebox(0,0)[cc]{$x_3$}}
\put(39.00,128.00){\makebox(0,0)[cc]{$x_1'$}}
\put(36.00,137.00){\makebox(0,0)[cc]{$x_2'$}}
\put(54.00,137.00){\makebox(0,0)[cc]{$x_3'$}}
\put(47.00,120.00){\makebox(0,0)[cc]{$x_1''$}}
\put(63.00,116.00){\makebox(0,0)[cc]{$x_2''$}}
\put(27.00,116.00){\makebox(0,0)[cc]{$x_3''$}}
\put(45.00,104.00){\makebox(0,0)[cc]{$n$}}
\put(48.00,134.00){\makebox(0,0)[cc]{$m$}}
\end{picture}
\vskip -95mm
}
\]
The symmetry group $H$ of this prism is generated by four planes
of symmetry~: the three planes of symmetry through an edge $x_i''$
($i=1,2,3$) and the plane of symmetry parallel with the basis.
Let $r_i$ ($i=1,2,3$) denote the reflection about a plane of
symmetry through an edge $x_i''$, and let $r_0$ denote the reflection
about the plane of symmetry that is parallel with the basis.
These four reflections map the prism into itself, and they generate
the symmetry group of the prism. This symmetry group $H$ is a group
of order~12, and it is easy to verify that it is isomorphic to 
the dihedral group $D_{12}$ (i.e.\ the symmetries of the hexagon).
The generating reflections correspond to permutations of $x_1,
x_2,\ldots, x_3''$ (and possibly an interchange of $n$ and $m$)~:
\beas
r_1 & : & (x_2\ x_3)(x_2'\ x_3')(x_2''\ x_3''),\\
r_2 & : & (x_1\ x_3)(x_1'\ x_3')(x_1''\ x_3''),\\
r_3 & : & (x_1\ x_2)(x_1'\ x_2')(x_1''\ x_2''),\\
r_0 & : & (x_1\ x_1')(x_2\ x_2')(x_3\ x_3')(n\ m).
\eeas
It turns out that the transformations given in
Theorems~\ref{theo:T1}, \ref{theo:T2} and~\ref{theo:T3}
all have the same symmetry group, described by $H$.
Thus we can state the following~:
\begin{prop}
Let $m$ and $n$ be nonnegative integers, and let $x_i$, $x_i'$, $x_i''$
($i=1,2,3$) be arbitrary parameters such that $\sum_{i=1}^3 x_i=0$,
$\sum_{i=1}^3 x_i'=0$, $\sum_{i=1}^3 x_i''=0$. Then the following
expressions
\bea
&&g_1(x)= {\ts ({2(1-n)\over 3}-x_1)_n ({2-2n+m\over 3}-x_2'')_n
({2(1-m)\over 3}-x_1')_m ({2-2m+n\over 3}-x_3'')_m }
\label{case1}\\
&\times& 
F^{1:2;2}_{0:2;2}\left[ {{ 1-n-m\over 3}+x_1'' \atop \ }
{:\atop :} { -n, { 1-n\over 3}+x_2 \atop 
 { 2(1-n)\over 3}-x_1 ,{ 2-2n+m\over 3}-x_2'' } 
{;\atop;} { -m, { 1-m\over 3}+x_3' \atop 
 { 2(1-m)\over 3}-x_1',{ 2-2m+n\over 3}-x_3'' }
{;\atop ;} 1,1\right], \nn \\[3mm]
&&g_2(x)= {\ts ({2(1-n-m)\over 3}+x_2'')_{n+m} ({2(1-n)\over 3}-x_3)_n
({2(1-m)\over 3}-x_3')_m } \label{case2} \\
&\times& 
F^{1:2;2}_{1:1;1}\left[ {{ 1-n-m\over 3}-x_3'' \atop 
 { 2(1-n-m)\over 3}+x_2'' }
{:\atop :} { -n, { 1-n\over 3}+x_2 \atop 
 { 2(1-n)\over 3}-x_3  } 
{;\atop;} { -m, { 1-m\over 3}+x_2' \atop 
 { 2(1-m)\over 3}-x_3' }
{;\atop ;} 1,1\right], \nn \\[3mm]
&&g_3(x)= {\ts ({2(1-n-m)\over 3}-x_1'')_{n+m} ({2(1-n)\over 3}+x_2)_n
({2(1-m)\over 3}+x_3')_m } \label{case3} \\
&\times& 
F^{0:3;3}_{1:1;1}\left[ {\ \atop { 2(1-n-m)\over 3}-x_1'' }
{:\atop :} { -n, { 1-n\over 3}-x_1, { 1-n-m\over  3}+x_2'' \atop 
 { 2(1-n)\over 3}+x_2  } 
{;\atop ;} { -m, { 1-m\over 3}-x_1', { 1-n-m\over  3}+x_3'' \atop 
 { 2(1-m)\over 3}+x_3'  } 
{;\atop ;} 1,1\right], \nn
\eea
are (upto a sign) invariant under the action of $H$, the symmetries
of the prism, i.e.\ $g_1(h\cdot x)=(-1)^{l_0(n+m)} g_1(x)$,
$g_2(h\cdot x)=(-1)^{l(n+m)} g_2(x)$ and
$g_3(h\cdot x)=(-1)^{l_0(n+m)} g_3(x)$,
where $l$ is the number of reflections $r_1$, $r_2$, $r_3$ in the expression
of $h$ and $l_0$ is the number of reflections $r_0$, $r_1$, $r_2$, $r_3$ 
in the expression of $h$.
\end{prop}

\noindent {\bf Proof.}
Consider~(\ref{case1}). Equation~(\ref{Tmn1}) of Theorem~\ref{theo:T1}
expresses that $g_1(h_1\cdot x)= g_1(x)$, with 
$h_1=(x_1\ x_3\ x_2)(x_1'\ x_3'\ x_2')(x_1''\ x_3''\ x_2'')$.
Similarly, equation~(\ref{Tmn1bis}) of Theorem~\ref{theo:T1}
expresses that $g_1(h_2\cdot x)= (-1)^{m+n}g_1(x)$, with 
$h_2=(x_1\ x_3)(x_1'\ x_3')(x_1''\ x_3'')$. Apart from the
two transformations given in Theorem~\ref{theo:T1}, there
is of course also the trivial transformation interchanging
$-n,a,b,c$ with $-m,a',b',c'$; this expresses that
$g_1(h_3\cdot x)= g_1(x)$ with 
$h_3=(x_1\ x_1')(x_2\ x_3')(x_3\ x_2')(x_2''\ x_3'')(n\ m)$.
It is now easy to verify that $h_1$, $h_2$ and $h_3$ generate
$H$, i.e.\ the same group as generated by $r_i$ ($i=0,1,2,3$).
Thus the symmetry statement for~(\ref{case1}) follows.
The remaining cases~(\ref{case2}) and~(\ref{case3}) follow
in a similar way from Theorems~\ref{theo:T2} and~\ref{theo:T3}.
\mybox

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic analogues of some transformation formulas}

In this section we shall be dealing with the basic analogues
(or $q$-analogues) of some of the transformation formulas
for double hypergeometric series considered in sections~II and~III.
For a general introduction and background to basic
hypergeometric series, see~\cite{GR}, whose notation we
follow~: thus $q$ is a parameter with $|q|<1$; $(a;q)_n$ is
the $q$-shifted factorial; $(a,b,c;q)_n$ stands for $(a;q)_n
(b;q)_n(c;q)_n$; ${}_{p+1}\Phi_p$ is the common notation
for a basic hypergeometric series in one variable; etc.

The double basic hypergeometric series appearing in the present context
is a special case of general double basic series 
defined by Srivastava and Karlsson~\cite[p.\ 349]{SrivastavaKarlsson}.
So we use their notation to define the series
\bea
&&\Phi^{1:2;2}_{0:2;2}\left[ {e \atop \ }{:\atop :} {a,b \atop c,d} 
{;\atop ;} {a',b' \atop c',d'}{;\atop ;} 
{q;x,y \atop \lambda,\mu,\nu}\right] = \nn\\
&& \sum_{j,k=0}^\infty q^{{\lambda\over 2}j(j-1)
+{\mu\over 2} k(k-1)+\nu jk}
(e;q)_{j+k}{(a;q)_j(b;q)_j\over (c;q)_j(d;q)_j}
{(a';q)_k(b';q)_k \over (c';q)_k(d';q)_k} 
{x^j\over (q;q)_j} {y^k \over (q;q)_k};
\label{phidouble}
\eea
the definition of $\Phi^{1:2;2}_{1:1;1}$ and $\Phi^{0:3;3}_{1:1;1}$
is completely analogous.
For double basic series such as (\ref{phidouble}), 
$\nu$ is usually taken to be 0, in which case this is a straightforward
double series 
analogue of the basic series ${}_3\Phi_2$. However, also the cases
with $\nu=+1$ or $\nu=-1$ appear in the literature~\cite{Denis,Singh}, 
and will play a role in the transformation formulas given here.

The main purpose of this section is to show that the different
expressions of $q$-9-$j$ coefficients of~\cite{A},
in the singly stretched case, give rise
to new transformation formulas for double basic hypergeometric 
series of the type $\Phi^{1:2;2}_{0:2;2}$,
$\Phi^{1:2;2}_{1:1;1}$ and $\Phi^{0:3;3}_{1:1;1}$.
Ali\v sauskas actually realized that his expressions gave rise
to ``rearrangement formulas of double sums'' (see~\cite[Appendix~C]{A}),
but he did not write them as transformation formulas of
series of the type~(\ref{phidouble}). Furthermore, he did
not recognize that some of these formulas allow for a set
of very general parameters.

In this section we shall discuss some of the $q$-analogues of
theorems given in sections~II and~III. Rather than derive these
$q$-analogues from the different double series expressions
of Ali\v sauskas~\cite{A}, a direct proof is given. It turns out
that the direct proofs of such transformation formulas are
fairly easy, and all rely on the same technique.

We know of two genuine transformation formulas for double basic 
hypergeometric series that have appeared in the literature (by
a genuine transformation formula, we mean a formula expressing
a basic double series of a particular type into another
series of the same type). One of these was given by 
Singh~\cite{Singh}~:
if $m$ and $n$ are nonnegative integers, and $a,b,c,a',b',c'$ and $d$
arbitrary parameters with $bb'=d$, then
\bea
&&\Phi^{0:3;3}_{1:1;1}\left[ {\ \atop d }{:\atop :}
{q^{-n},a,b \atop c}{;\atop ;} {q^{-m},a',b' \atop c'}{;\atop ;}
{q;cdq^n/ab,c'dq^m/a'b'\atop 0,0,1} \right] = 
\nn \\
&&{(c/a;q)_n (c'/a';q)_m \over (c;q)_n(c';q)_m} \ 
\Phi^{0:3;3}_{1:1;1}\left[ {\ \atop d }{:\atop :} 
{q^{-n},a,b' \atop q^{1-n}a/c} {; \atop ;} 
{q^{-m},a',b \atop q^{1-m}a'/c'}
{;\atop ;} {q;q,q \atop 0,0,0} \right].
\label{qTmn3bis}
\eea
The other was the topic of a recent paper~\cite{Vdj}~:
if $n$ is a nonnegative integer, and $a,b,c,d,a',b',c'$ and $d'$ are
arbitrary parameters with $dd'=q^{1-n}$, then
\bea
&&\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :}
{a,b \atop c,d}{;\atop ;} {a',b' \atop c',d'}{;\atop ;}
{q;cdq^n/ab,c'd'q^n/a'b'\atop 0,0,-1} \right] = 
\nn \\
&&{(d'b/b';q)_n \over (d';q)_n} b^{-n} \ 
\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{c/a,b \atop c,d'b/b'} {; \atop ;} {c'/a',b' \atop c',db'/b}
{;\atop ;} {q;q,q \atop 0,0,0} \right].
\label{qF122022joris}
\eea
Observe that (\ref{qTmn3bis}) is the basic analogue of~(\ref{Tmn3bis}),
and~(\ref{qF122022joris}) is the basic analogue of~(\ref{F122022joris}).

We shall now indicate how such formulas, and others, can
be derived directly. First, we shall deduce two basic analogues
of~(\ref{trans1}), namely~: if $dd'=q^{1-n}$ then
\beq
\Phi^{1:2;2}_{0:2;2}\left[
{q^{-n} \atop \ }{:\atop :}{a,b\atop c,d}{;\atop;}
{a',b'\atop c',d'}{;\atop;}
{q; cdq^n/ab, y\atop 0,0,-1}
\right] = {(d/a;q)_n\over(d;q)_n}  
\Phi^{1:2;2}_{1:1;1}\left[
{q^{-n}\atop ad'}{:\atop :}{a, c/b \atop c}{;\atop ;}
{a', b'\atop c'}
{;\atop ;} {q; q, ay\atop 0,0,0} 
\right],   \label{q1trans1}
\eeq
and
\beq
\Phi^{1:2;2}_{0:2;2}\left[
{q^{-n}\atop \ }{:\atop :}{a,b\atop c,d}{;\atop;}
{a',b'\atop c',d'}{;\atop;}
{q; q, y\atop 0,0,0}
\right] = {a^n(d/a;q)_n\over(d;q)_n}  
\Phi^{1:2;2}_{1:1;1}\left[
{q^{-n}\atop ad'}{:\atop :}{a, c/b \atop c}{;\atop ;}
{a', b'\atop c'}
{;\atop ;} {q; bq/d, y\atop 0,0,0} 
\right].   \label{q2trans1}
\eeq
Herein, as usual, $n$ is a nonnegative integer, $a,b,c,d,a',b',c'$ and $d$
are arbitrary parameters (subject to $dd'=q^{1-n}$), and $y$ is
an arbitrary variable.

For a proof, expand the lhs $L$ of~(\ref{q1trans1}) into a 
double series~:
\beas
L&=& \sum_{j,k} { (q^{-n};q)_{j+k} (a,b;q)_j(a',b';q)_k \over
(q,c,d;q)_j (q,c',d';q)_k} \left({cdq^n\over ab}\right)^j y^k q^{-jk}\\
&=& \sum_k { (q^{-n},a',b';q)_k \over (q,c',d';q)_k} y^k \ 
{}_3\Phi_2\left[ {q^{-n+k},a,b \atop c,d }{;\atop ;} 
q, cdq^{n-k}/ab \right].
\eeas
Now apply Sears' transformation formula~\cite[(III.13)]{GR}, and
expand again~:
\bea
L&=& \sum_k { (q^{-n},a',b';q)_k \over (q,c',d';q)_k} y^k
{(d/a;q)_{n-k} \over (d;q)_{n-k} } \ 
{}_3\Phi_2\left[ {q^{-n+k},a,c/b \atop c,aq^{1-n+k}/d }{;\atop ;} 
q, q \right] \nn \\
&=& \sum_{j,k} { (q^{-n};q)_{j+k} (a,c/b;q)_j (a',b';q)_k \over
(q,c,aq^{1-n+k}/d;q)_j (q,c',d';q)_k } y^k q^j 
{(d/a;q)_{n-k} \over (d;q)_{n-k} }. \label{tmp1}
\eea
Using $d'=q^{1-n}/d$, and elementary properties of $q$-shifted
factorials, there comes
\[
{1\over (aq^{1-n+k}/d;q)_j (d';q)_k }  
{(d/a;q)_{n-k} \over (d;q)_{n-k} }=
{a^k \over (aq^{1-n}/d;q)_{j+k} } {(d/a;q)_{n} \over (d;q)_{n} }.
\]
Plugging this in~(\ref{tmp1}) yields the rhs of~(\ref{q1trans1}).
The proof of~(\ref{q2trans1}) is completely analogous.

Now we can give the basic analogue of~(\ref{F122111-n})~:
\begin{prop}
Let $n$ be a nonnegative integer and $a,b,c,a',b',c'$ and $d$ be
arbitrary parameters, then
\begin{eqnarray}
&&\Phi^{1:2;2}_{1:1;1}\left[
{q^{-n}\atop d}{:\atop :}{a, b\atop c}{;\atop ;}{a', b'\atop c'}
{;\atop ;} {q; q, dc'q^n/a'b'\atop 0,0,0}
\right] = \nn\\
&& {b^n(d/bb';q)_n\over (d;q)_n}  \ 
\Phi^{1:2;2}_{1:1;1}\left[
{q^{-n}\atop q^{1-n}bb'/d}{:\atop :}{c/a, b\atop c}{;\atop ;}{c'/a', b'\atop c'}{;\atop ;}
{q; b'aq/d, q\atop 0,0,0} \right].  \label{qF122111-n}
\end{eqnarray}
\end{prop}

\noindent
{\bf Proof.}
This is now straightforward~: apply~(\ref{q1trans1}) to the
lhs of~(\ref{qF122022joris}) and~(\ref{q2trans1}) to the
rhs of~(\ref{qF122022joris}). Comparing these expressions 
yields~(\ref{qF122111-n}). \mybox

With this, we have given basic analogues of all transformation
formulas of section~II. 
Also for the transformation formulas with two separate
numerator parameters as negative integers, given in section~III,
the basic analogues can be deduced.
The proof of such formulas uses similar steps as
illustrated in the proof of~(\ref{q1trans1})~:
\begin{itemize}
\item[(a)]
rewrite the double sum as a single sum over a term 
containing a ${}_3\Phi_2$ series;
\item[(b)]
perform one of Sears' transformation formulas on the
${}_3\Phi_2$ and rewrite the result as a double sum;
\item[(c)]
make certain simplifications, using the constraint (if
present) between the parameters;
\item[(d)]
if necessary, repeat (a), (b) and (c) on the double sum
obtained so far, and finally rewrite it in the standard
notation of a double basic hypergeometric series.
\end{itemize}

Detailed proofs of the remaining formulas in this section
will not be given, since they all follow the above technique.
In fact, we will not even give the basic analogues of
all of the formulas of section~III, but just list
those corresponding to the transformation formulas of 
Theorems~\ref{theo:T1}, \ref{theo:T2} and~\ref{theo:T3}.

Here are the basic analogues of~(\ref{Tmn1}) and~(\ref{Tmn1bis}),
given in Theorem~\ref{theo:T1}. The first, (\ref{Tmn1}),
has two basic analogues, namely
\begin{eqnarray}
&&\Phi^{1:2;2}_{0:2;2}\left[
{ d\atop\phantom{}}{:\atop :}{q^{-n},a\atop b,c}{;\atop;}
{q^{-m},a'\atop b',c'}{;\atop;}
{q; bcq^n/ad, b'c'q^m/a'd\atop 0,0,-1}
\right]  \nonumber \\ 
&& = {(-1)^m(d;q)_n(b/a;q)_n(a';q)_m (b'/a'c)^m(c/d)^nq^{{m+1\choose2} - mn} \over
(b;q)_n(c';q)_m(b';q)_m(c;q)_{n-m}  }  
\nonumber \\
& &  {}\times 
\Phi^{1:2;2}_{0:2;2}
\left[
{ q^{1-m}/c'\atop\phantom{}}{:\atop :}
{q^{-n},q^{1-n}/b\atop aq^{1-n}/b, q^{1-n}/d}{;\atop;}
{q^{-m},b'/a'\atop q^{1-m}/a',cq^{n-m}}{;\atop;}
{q; aq^{m+1}/c, dq^{n+1}/b'\atop 0,0,-1}
\right],   \label{q1Tmn1}
\end{eqnarray}
and
\begin{eqnarray}
&& \Phi^{1:2;2}_{0:2;2}\left[
{ d\atop\phantom{}}{:\atop :}{q^{-n},a\atop b,c}{;\atop;}
{q^{-m},a'\atop b',c'}{;\atop;}
{q; q, q\atop 0,0,0}
\right] = 
 {(-1)^m(d;q)_n(b/a;q)_n(a';q)_m (d/c)^ma^nq^{{m+1\choose2}} \over
(b;q)_n(c';q)_m(b';q)_m(c;q)_{n-m}  }   
\nonumber \\
& &  {}\times 
\Phi^{1:2;2}_{0:2;2}
\left[
{ q^{1-m}/c'\atop\phantom{}}{:\atop :}
{q^{-n},q^{1-n}/b\atop aq^{1-n}/b, q^{1-n}/d}{;\atop;}
{q^{-m},b'/a'\atop q^{1-m}/a',cq^{n-m}}{;\atop;}
{q; q, q\atop 0,0,0}
\right],   \label{q2Tmn1}
\end{eqnarray}
where in both formulas $cc'=qd$.
The basic analogue of~(\ref{Tmn1bis}) is
\begin{eqnarray}
&& \Phi^{1:2;2}_{0:2;2}\left[
{ {d\atop\phantom{}}}{:\atop :}{q^{-n},a\atop b,c}{;\atop;}{q^{-m},a'\atop b',c'}{;\atop;}
{q; cbq^n/ad, y\atop 0,0,-1}
\right] = {(-1)^m(b/a;q)_n(a',d;q)_m q^{- {m+1\choose2}}y^m\over (b;q)_n(b',c';q)_m } 
\nonumber \\
& & {}\times
\Phi^{1:2;2}_{0:2;2}\left[
{ {q^{1-m}/c'\atop\phantom{}}}{:\atop :}{q^{-n},a\atop c,aq^{1-n}/b}
{;\atop;}{q^{-m},q^{1-m}/b'\atop q^{1-m}/a',q^{1-m}/d}{;\atop;}
{q; q, b'q^{m+2}/a'cy\atop 0,0,0}
\right], \label{qTmn1bis}
\end{eqnarray}
where again $cc'=qd$.
 
The basic analogues of the formulas in Theorem~\ref{theo:T2} are
given by~:
\begin{eqnarray}
&& \Phi^{1:2;2}_{1:1;1}\left[
{c\atop d}{:\atop :}{q^{-n}, a \atop b}{;\atop ;}
{q^{-m}, a'\atop b'}
{;\atop ;} {q; bdq^n/ac, q\atop 0,0,0} 
\right]  =  {(b/a;q)_n(b'/a';q)_m(c;q)_{n+m}(a')^m(d/c)^n\over
  (b;q)_n(b';q)_m(d;q)_{n+m} }   \nonumber \\
& & {}\times 
\Phi^{1:2;2}_{1:1;1}\left[
{d/c\atop q^{1-n-m}/c}{:\atop :}{q^{-n}, q^{1-n}/b \atop aq^{1-n}/b}{;\atop ;}
{q^{-m}, q^{1-m}/b'\atop a'q^{1-m}/b'}
{;\atop ;} {q; aq^{1-m}/d, q\atop 0,0,0} 
\right]    \label{qTmn2}
\end{eqnarray}
and
\begin{eqnarray}
&& \Phi^{1:2;2}_{1:1;1}\left[
{c\atop d}{:\atop :}{q^{-n},a \atop b}{;\atop ;}{q^{-m}, a'\atop b'}
{;\atop ;}
{q; dbq^n/ac, q\atop 0,0,0}
\right] = { (a')^m(b/a;q)_n(b'/a';q)_m\over (b;q)_n(b';q)_m }  
\nonumber \\
& & {}\times
\Phi^{1:2;2}_{1:1;1}\left[
{d/c\atop d}{:\atop :}{q^{-n},a \atop aq^{1-n}/b}{;\atop ;}
{q^{-m}, a'\atop a'q^{1-m}/b'}
{;\atop ;} {q; q, cq/b'\atop 0,0,0} 
\right].   \label{qTmn2bis}
\end{eqnarray}

Finally, the basic analogues of the transformation 
formulas~(\ref{Tmn3}) and~(\ref{Tmn3bis}) of Theorem~\ref{theo:T3}
are given by
\begin{eqnarray}
&&\Phi^{0:3;3}_{1:1;1}\left[{\phantom{}\atop d}{:\atop :}
{q^{-n}, a, b\atop c} {;\atop ;}
{q^{-m}, a', b'\atop c'}{;\atop ;}{q; q, q\atop 0, 0, 0}\right] 
={(a')^mb^n(b';q)_{n+m}(a';q)_n(c'/a';q)_m\over (d;q)_{n+m}(c;q)_n(c';q)_m}
\nonumber \\
& & {}\times 
\Phi^{0:3;3}_{1:1;1}\left[{\phantom{}\atop q^{1-n-m}/b'}{:\atop :}
{q^{-n}, c/a, q^{1-n-m}/d\atop q^{1-n}/a}{;\atop ;}
{q^{-m}, q^{1-m}/c', b\atop q^{1-m}a'/c'}{;\atop ;}
{q; q, q\atop 0, 0, 0}\right]  \label{qTmn3}
\end{eqnarray}
and~(\ref{qTmn3bis}), where $bb'=d$ in both formulas.

This completes the list of $q$-analogues of the transformation
formulas of Kamp\'e de F\'eriet series with two nonnegative
integers as parameters, as given in
Theorems~\ref{theo:T1}, \ref{theo:T2} and~\ref{theo:T3}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}

Using the different double sum expressions of Ali\v sauskas~\cite{A}
for a singly stretched 9-$j$ coefficient of $su(2)$ or $su_q(2)$,
we have deduced a set of new transformation formulas for double
hypergeometric series of Kamp\'e de F\'eriet type and their
basic analogues. 
An important observation is that these transformation
formulas are valid for quite general parameters, even though
the original 9-$j$ coefficients assume only nonnegative integer or
half-integer values as arguments.
The transformation formulas given here for double
hypergeometric series of Kamp\'e de F\'eriet type 
are all terminating, which means that either a common
numerator parameter, or else two
separate numerator parameters are negative integers.

The transformation formulas seem to inherit some of the
symmetries of the 9-$j$ coefficient. In particular, we
have shown that the given transformation formulas for a
double series of a particular type generate a symmetry
group, acting on the parameters of the series. These
symmetry groups are explicitly determined and described
as subgroups of permutation groups, or as symmetry groups of
some geometric object.

In the case of basic double hypergeometric series,
corresponding to different expressions of 9-$j$ coefficients
of $su_q(2)$, the relevant series is a double $q$-series
as defined in~\cite{SrivastavaKarlsson}.
Also for these series, the transformation formulas are listed,
and we have shown that an independent proof of such
transformations is easy.

%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}


