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\begin{document}
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\begin{center}
{\Large \bf 
Transformation formula for a double Clausenian hypergeometric series, 
its $q$-analogue, and its invariance group}\\[1cm]
{\bf J.\ Van der Jeugt}\\[1cm] 
Department of Applied Mathematics and Computer Science,\\
University of Ghent,\\
Krijgslaan 281-S9, \\
B-9000 Gent, Belgium.\\[2mm]
E-mail : Joris.VanderJeugt@rug.ac.be.
\end{center}

\begin{abstract}
A transformation formula for a double basic hypergeometric series
of type $\Phi^{1:2;2}_{0:2;2}$ is derived. This transformation
yields a double series analogue of Sears' transformation for
a terminating ${}_3\Phi_2$ series.
In the limit $q\rightarrow 1$, the formula reduces to a
transformation for a terminating double Clausenian hypergeometric
series of unit argument (one of the proper Kamp\'e de F\'eriet series, $F^{1:2;2}_{0:2;2}(1,1)$).
This formula is a double series analogue of Whipple's
terminating ${}_3F_2$ transformation.
This transformation gives rise to a transformation group (the invariance
group) acting on the parameters of the double series.
The invariance group is examined and shown to be a subgroup of a 
double copy of the symmetries of the square.
\end{abstract}
\vskip 5mm

\noindent Keywords~: double hypergeometric series, transformation
formulae, transformation group, double basic hypergeometric series.

\noindent AMS Subject codes~: 33C70, 33D70, 33C80.

\section{Introduction}

For classical or basic hypergeometric series of a single variable,
transformation formulas (see~\cite[Appendix III]{GR})
play an important role.
For hypergeometric series of two or more variables, not many
transformation formulas are known, even though there are a number of
special reduction formulas. 
In the present paper we are dealing with a transformation formula
for a double Clausenian hypergeometric series and its basic analogue. 
The double Clausenian series~\cite{Karlsson1,Karlsson2}
are the ten proper Kamp\'e de F\'eriet
functions $F^{p:r;r}_{q:s;s}$ for which $q+s=2$ and $p+r=3$.
The function considered in this paper is
\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(a')_k(b')_k \over (c)_j(d)_j(c')_k(d')_k} 
{x^j y^k \over j!k!},
\label{deff}
\eeq
where $(a)_k$ is the classical Pochhammer symbol~\cite{Bailey,Slater}. 
The transformation formula discussed here is in the terminating case 
($e=-n$, $n$ being a nonnegative integer) and for unit argument ($x=y=1$),
with one relation between the parameters (namely $d+d'=1-n$).
We shall also be dealing with the basic analogue of~(\ref{deff}).
General basic double series were defined by Srivastava and 
Karlsson~\cite[p.\ 349]{Srivastava}, and 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 \la,\mu,\nu}\right] = \nn\\
&& \sum_{j,k=0}^\infty q^{{\la\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{phi}
\eea
Our notation for $q$-shifted factorials (and single basic series) is that
of Gasper and Rahman~\cite{GR}.
For double basic series such as (\ref{phi}), 
$\nu$ 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}, 
and play a role in the transformation formula given here.
In our case, the transformation formula is in the terminating case
($e=q^{-n}$), with again one relation between the remaining parameters
(namely $dd'=q^{1-n}$).

Double hypergeometric series, or their $q$-analogues, have received 
sporadic attention in the literature. The simplest double hypergeometric
series are the Appell series and the Horn series; more general double
series are the Kamp\'e de F\'eriet series~\cite{Appell,Kampe,Srivastava}
and the hypergeometric series in more variables~\cite{Lauricella}.

For Kamp\'e de F\'eriet series, some isolated reduction and
summation formulas have been published. 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 $F^{0:3;3}_{1:1;1}(1,1)$ series;
these were also studied and $q$-generalized by 
Srivastava~\cite{Sri}; see also~\cite{Sri2} for further results.
Two transformation formulas for series of the type 
$F^{1:2;2}_{1:1;1}(1,1)$ and $F^{0:3;3}_{1:1;1}(1,1)$ were derived
by Sighal~\cite{Sighal}, and their basic analogue was given
by Singh~\cite{Singh}.
Inspired by identities arising in the study
of the so-called 9-$j$ coefficient of angular momentum 
theory~\cite{paper1,Vanderjeugt}, a long list of new reduction
and summation formulas
for $F^{0:3;3}_{1:1;1}(1,1)$ series~\cite{Pitre}, for other
double series~\cite{jcam97}, and even for triple series~\cite{jcam00}
were derived.

In this paper, the main result is the derivation of a genuine 
transformation formula for a terminating $\Phi^{1:2;2}_{0:2;2}$
series. 
By a ``genuine transformation formula'' we mean a formula expressing
one $\Phi^{1:2;2}_{0:2;2}$ series (with certain parameter constraints) 
into another $\Phi^{1:2;2}_{0:2;2}$ series
(with different parameters and/or arguments). 
A formula expressing a $\Phi^{1:2;2}_{0:2;2}$
series (with parameter constraints) as a single basic hypergeometric series
of the form ${}_p\Phi_q$ will be called a ``reduction formula.'' 

For the case $q\rightarrow 1$, a transformation formula for a terminating
$F^{1:2;2}_{0:2;2}$ series of unit argument is obtained.
This transformation formula can be expressed by means of
a simple and elegant operation on the series parameters. In view of this,
it is worth studying the invariance group of $F^{1:2;2}_{0:2;2}$ series
transformations. This invariance group is shown to be of order 32, and
is described explicitly. 
It extends the known results about invariant groups of single
hypergeometric series transformations or their $q$-analogues~\cite{VR}.

\section{The transformation formula}

The main result of this paper is the following transformation formula~:
\begin{prop}
Let $n$ be a nonnegative integer, and $a,b,c,d,a',b',c'$ and $d'$ be
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{qtf}
\eea
\end{prop}

\noindent
{\bf Proof.}
Denote the lhs of (\ref{qtf}) by $L$; this can be written as
\beq
L=\sum_{k=0}^n {(q^{-n};q)_k(a';q)_k(b';q)_k \over (q;q)_k(c';q)_k(d';q)_k}
\left({c'd'q^n\over a'b'}\right)^k \;{}_3\Phi_2 \left[ 
{q^{-n+k}, a, b \atop c, d }{;\atop ;} q,{cdq^{n-k}\over ab} \right].
\label{tmp1}
\eeq
Applying Sears' formula~\cite[(III.13)]{GR} on the ${}_3\Phi_2$ yields
\beas
L&=& \sum_{k=0}^n {(q^{-n};q)_k(a';q)_k(b';q)_k \over (q;q)_k(c';q)_k(d';q)_k}
\left({c'd'q^n\over a'b'}\right)^k {(d/b;q)_{n-k}\over (d;q)_{n-k}}
\;{}_3\Phi_2 \left[ 
{q^{-n+k}, c/a, b \atop c, bq^{1-n+k}/d }{;\atop ;} q,q \right]\nn\\
&=&\sum_{j,k} {(q^{-n};q)_{j+k}(a';q)_k(b';q)_k \over (q;q)_k(c';q)_k(d';q)_k}
\left({c'd'q^n\over a'b'}\right)^k {(d/b;q)_{n-k}\over (d;q)_{n-k}}
{(c/a;q)_j(b;q)_j \over (q;q)_j(c;q)_j(bq^{1-n+k}/d;q)_j } q^j.
\eeas
Using some elementary identities for $q$-shifted factorials, 
and $dd'=q^{1-n}$, one can write
\[
{(d/b;q)_{n-k} \over (d';q)_k(d;q)_{n-k}(bq^{1-n+k}/d;q)_j} =
{(d'b;q)_n \over (d';q)_n} {b^{k-n} \over (d'b;q)_j(d'bq^j;q)_k}.
\]
Subsituting this in the last double sum gives
\beas
L&=&{(d'b;q)_n \over (d';q)_n}b^{-n}
\sum_{j,k} {(q^{-n};q)_{j+k}(a';q)_k(b';q)_k \over (q;q)_k(c';q)_k(d'bq^j;q)_k}
\left({b c'd'q^n\over a'b'}\right)^k 
{(c/a;q)_j(b;q)_j \over (q;q)_j(c;q)_j(d'b;q)_j } q^j \nn\\
&=&  {(d'b;q)_n \over (d';q)_n}b^{-n}
\sum_{j=0}^n {(q^{-n};q)_j(c/a;q)_j(b;q)_j \over (q;q)_j(c;q)_j(d'b;q)_j } q^j
\;{}_3\Phi_2\left[ {q^{-n+j}, a', b' \atop c', d'bq^j }
{;\atop ;} q, {b c'd'q^n\over a'b'}\right]
\eeas
Applying once more Sears' formula~\cite[(III.13)]{GR} on this ${}_3\Phi_2$ 
leads to
\beas
L&=&{(d'b;q)_n \over (d';q)_n}b^{-n}
\sum_{j=0}^n {(q^{-n};q)_j(c/a;q)_j(b;q)_j \over (q;q)_j(c;q)_j(d'b;q)_j } q^j
{(d'bq^j/b';q)_{n-j} \over (d'bq^j;q)_{n-j}}
\;{}_3\Phi_2\left[ {q^{-n+j}, c'/a', b' \atop c', b'q^{1-n}/d'b }
{;\atop ;} q, q\right]\nn\\
&=& {(d'b;q)_n \over (d';q)_n}b^{-n}
\sum_{j,k} {(q^{-n};q)_{j+k}(c/a;q)_j(b;q)_j \over (q;q)_j(c;q)_j(d'b;q)_j } q^j
{(d'bq^j/b';q)_{n-j} \over (d'bq^j;q)_{n-j}}
{(c'/a';q)_k(b';q)_k \over (q;q)_k(c';q)_k (b'q^{1-n}/d'b;q)_k }q^k\nn\\
&=& {(d'b/b';q)_n \over (d';q)_n}b^{-n}
\sum_{j,k} {(q^{-n};q)_{j+k}(c/a;q)_j(b;q)_j \over (q;q)_j(c;q)_j(d'b/b';q)_j } q^j
{(c'/a';q)_k(b';q)_k \over (q;q)_k(c';q)_k (b'd/b;q)_k }q^k,
\eeas
proving the proposition.\mybox

This is not the only transformation that can be derived for the
given double series. Indeed, one finds~:
\begin{prop}
Let $n$ be a nonnegative integer, and $a,b,c,d,a',b',c'$ and $d'$ be
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 \\
&&{(cd/ab;q)_n \over (d;q)_n} \ 
\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{c/a,c/b \atop c,cd/ab} {; \atop ;} {a',b' \atop c',abd'/c}
{;\atop ;} {q;q^nd,qabc'/a'b'cd \atop 0,0,-1} \right].
\label{qtf2} \\
&&{(c'd'/a'b';q)_n \over (d';q)_n} \ 
\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{a,b \atop c,a'b'd/c'} {; \atop ;} {c'/a',c'/b' \atop c',c'd'/a'b'}
{;\atop ;} {q;qa'b'c/abc'd',q^nd' \atop 0,0,-1} \right].
\label{qtf3} \\
&&{(a'b'cd/abc';q)_n \over (d;q)_n}(a'b'/c')^{-n} \ 
\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{c/a,c/b \atop c,a'b'cd/abc'} {; \atop ;} {c'/a',c'/b' \atop c',abc'd'/a'b'c}
{;\atop ;} {q;qa'b'/c'd',qab/cd \atop 0,0,-1} \right].\nn\\
&& \label{qtf4} 
\eea
\end{prop}

\noindent
{\bf Proof.} Applying one of Sears' formulas, namely
\[
{}_3\Phi_2 \left[ {q^{-n},a,b \atop c,d} {;\atop ;} q, {cdq^n\over ab}
\right] = { (cd/ab;q)_n \over (d;q)_n }\
{}_3\Phi_2 \left[ {q^{-n},c/a,c/b \atop c,cd/ab} {;\atop ;} q, q^nd
\right],
\]
to the ${}_3\Phi_2$ in (\ref{tmp1}), and using some identities for
$q$-shifted factorials, yields (\ref{qtf2}). Formula (\ref{qtf3})
follows from (\ref{qtf2}) by replacing the primed and unprimed parameters.
Finally, applying (\ref{qtf3}) on the rhs of (\ref{qtf2}) yields
(\ref{qtf4}). \mybox

Some other formulas can be deduced from the given ones. For example, 
applying the transformation formula (\ref{qtf}) on the rhs of 
(\ref{qtf2}) gives again a $\Phi^{1:2;2}_{0:2;2}$ with arguments $(q,q)$.
Comparing this with the rhs of (\ref{qtf}) (and relabelling the
parameters) yields (under the same condition $dd'=q^{1-n}$)~:
\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;q,q \atop 0,0,0} \right]= \nn\\
&&{(cd/ab;q)_n \over (d;q)_n} (ab/c)^{n} \ 
\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{c/a,c/b \atop c,cd/ab} {; \atop ;} {a',b' \atop c',abd'/c}
{;\atop ;} {q;q,q \atop 0,0,0} \right].
\label{qtf0}
\eea


\section{Reduction and summation formula}

By further specialization of the parameters, the transformation formula
(\ref{qtf}) gives rise to two reduction formulas
and one summation formula.

Choosing $b'=abq^{1-n}/cd$, the rhs of (\ref{qtf}) reduces to a 
$\Phi^{1:1;2}_{0:1;2}$ series, which can be (partially) summed by 
means of the $q$-Vandermonde sum~\cite[(II.6)]{GR}. Thus 
we find~:
\bea
&&\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{a,b \atop c,d} {; \atop ;} {a',abq^{1-n}/cd \atop c',q^{1-n}/d}
{;\atop ;} {q;{cdq^n/ab},{cc'q^n/aba'}\atop 0,0,-1}\right] = \nn\\
&&{(c/a;q)_n(c/b;q)_n \over (q^{1-n}/d;q)_n(c;q)_n}
{}_4\Phi_3\left[ {q^{-n}, c'/a', abq^{1-n}/cd, q^{1-n}/c \atop 
c', a q^{1-n}/c, b q^{1-n}/c }{;\atop ;} q, q\right].
\eea
Taking $c'=q^{1-n}/c$, this further reduces to
\bea
&&\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} {a,b \atop c,d}
{; \atop ;} {a',abq^{1-n}/cd \atop q^{1-n}/c,q^{1-n}/d}
{;\atop ;} {q;{cdq^n/ab},{q/aba'}\atop 0,0,-1} \right] = \nn\\
&&{(c/a;q)_n(c/b;q)_n \over (q^{1-n}/d;q)_n(c;q)_n}
{}_3\Phi_2\left[ {q^{-n}, q^{1-n}/ca', abq^{1-n}/cd \atop 
a q^{1-n}/c, b q^{1-n}/c }{;\atop ;} q, q\right].
\eea
Finally, when $a'=1/a$, this ${}_3\Phi_2$ reduces to a ${}_2\Phi_1$
which can be summed, and leads to~:
\beq
\Phi^{1:2;2}_{0:2;2}\left[ {q^{-n} \atop \ }{:\atop :} 
{a,b \atop c,d} {; \atop ;}
{1/a,abq^{1-n}/cd \atop q^{1-n}/c,q^{1-n}/d}{;\atop ;}
{q;{cdq^n/ab},{q/b} \atop 0,0,-1} \right] = 
a^n {(c/a;q)_n(d/a;q)_n \over (c;q)_n(d;q)_n}.
\eeq

\section{The transformation formula when $q\rightarrow 1$ and its
invariance group}

Under the limit $q\rightarrow 1$, and relabelling of the
parameters, (\ref{qtf}) becomes
a transformation formula for a $F^{1:2;2}_{0:2;2}$ series~:
\begin{prop}
Let $n$ be a nonnegative integer, and $a,b,c,d,a',b',c'$ and $d'$ be
arbitrary parameters with $d+d'=1-n$. Then
\beq
F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }{:\atop :} {a,b \atop c,d} 
{;\atop ;} {a',b' \atop c',d'}{;\atop ;} 1,1\right] = 
 {(d-b+b')_n\over (d)_n}
F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }{:\atop :} {c-a,b \atop c,d'-b'+b}
{;\atop ;} {c'-a',b' \atop c',d-b+b'}{;\atop ;} 1,1\right].
\label{tf}
\eeq
\end{prop}

Other transformation formulas follows from (\ref{qtf2})-(\ref{qtf4}).
Alternatively, one can use
the symmetry between the numerator parameters, and apply
Proposition~3 more than once, to deduce (under the same condition
that $d+d'=1-n$)~:
\bea
&& F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }{:\atop :} {a,b \atop c,d} 
{;\atop ;} {a',b' \atop c',d'}{;\atop ;} 1,1\right]  \nn\\
&=& {(c+d-a-b)_n\over (d)_n}
F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }{:\atop :} {c-a,c-b \atop c,c+d-a-b}
{;\atop ;} {a',b' \atop c',d'-c+a+b}{;\atop ;}1,1\right] \\
&=& {(d-c'+a'+b')_n\over (d)_n}
F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }{:\atop :} {a,b \atop c,d-c'+a'+b'}
{;\atop ;} {c'-a',c'-b' \atop c',c'+d'-a'-b'}{;\atop ;}1,1\right] \\
&=&{(c+d-a-b+a'+b'-c')_n\over (d)_n} \nn\\
&&\times 
F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }{:\atop :} {c-a,c-b \atop 
c,c+d-a-b+a'+b'-c'} {;\atop ;} {c'-a',c'-b' \atop 
c',d'-c+a+b-a'-b'+c'}{;\atop ;}1,1\right].
\eea
Note that the denominator parameters $c$ and $c'$ are left unchanged,
and that the sum of the remaining denominator parameters is always $1-n$.

Transformation formulas for classical hypergeometric series
can often be seen as a transformation acting upon the parameters
of the series (since the argument itself is usually 1). 
Together with trivial transformations, deduced from numerator or
denominator permutations, such transformation formulas give rise
to a group of transformations -- the invariance group.
For example, the invariance group of Thomae's ${}_3F_2$ transformation
is known to be the symmetric group 
$S_5$~\cite{Hardy,BLS}; that of Bailey's
transformation for terminating Saalsch\"utzian ${}_4F_3$ series is
the symmetric group $S_6$~\cite{BLS};
and that of Whipple's transformations for terminating ${}_3F_2$ series
is a 72 element subgroup of $S_6$~\cite{Rao,VR}. 
Also for some basic series transformations
the invariance group has been determined, and has lead to other
groups such as the dihedral group $D_{12}$ or a group of signed
permutations $WB_5$~\cite{VR}.

Also in the present case, the transformation formula (\ref{tf}) has an
appealing invariance group. 
Rather than dealing with the invariance group of the basic series
generated by (\ref{qtf}), we shall be dealing only with the
classical case generated by (\ref{tf}). This is mainly to avoid
some notational complexities regarding the arguments $x$, $y$, $\lambda$,
$\mu$ and $\nu$ arising only in the basic series.

The elements of the invariance group
act on the parameters $a,b,\ldots,d',-n$ of the double series (\ref{tf}).
The invariance group is generated by three elements~: the interchange
of $a$ and $b$, the interchange of $a'$ and $b'$, and finally the
action corresponding to (\ref{tf})~:
\[
\left[ \begin{array}{ccccc}
-n&a&b&a'&b'\\ &c&d&c'&d' \end{array} \right]
\rightarrow
\left[ \begin{array}{ccccc}
-n&c-a&b&c'-a'&b'\\ &c&d'-b'+b&c'&d-b+b' \end{array} \right].
\]
Note that the interchanges of $c\leftrightarrow d$ and 
$c'\leftrightarrow d'$ are not
considered as actions of the invariance group, even though they are
obviously valid transformations of the double series. The reason is
that the original condition $d+d'=1-n$ should remain valid for the
transformed parameters in the position of $d$ and $d'$.

In order to determine the invariance group, it is useful to make a 
relabelling of the parameters of the double series. There are in total
9 parameters, with one condition $d+d'=1-n$. The relabelling that will
illuminate the invariance group is also in terms of 9 parameters $x_i$
($i=0,\ldots,8$) satisfying one constraint, and is given by~:
\[
\begin{array}{ll}
a=x_2+x_3 & a'=x_6+x_7\\
b=x_1+x_2 & b'=x_5+x_6\\
c=x_1+x_2+x_3+x_4 & c'=x_5+x_6+x_7+x_8\\
d=x_0-x_4-x_8-2x_6 & d'=x_0-x_4-x_8-2x_2
\end{array}
1-n= 2x_0-\sum_{i=1}^8 x_i,
\]
satisfying the constraint
\[
x_1+x_3+x_5+x_7=x_2+x_4+x_6+x_8.
\]
Thus the first transformation, the interchange of $a$ and $b$, is
equivalent to the transposition $x_1 \leftrightarrow x_3$, i.e.\ it
corresponds to the permutation $(1\;3)$ in cycle notation.
Similarly, the second transformation, the interchange of $a'$ and $b'$, is
equivalent to the transposition $x_5 \leftrightarrow x_7$, i.e.\ 
the permutation $(5\;7)$ in cycle notation.
The third transformation, given by (\ref{tf}), is now equivalent
with $x_1 \leftrightarrow x_2$, $x_3 \leftrightarrow x_4$, 
$x_5 \leftrightarrow x_6$, $x_7 \leftrightarrow x_8$, or 
$(1\;2)(3\;4)(5\;6)(7\;8)$. Since $x_0$ is left unchanged by the three
transformations, the resulting invariance group is a subgroup of $S_8$,
acting on $x_1,\ldots, x_8$. So it remains to determine the subgroup of
$S_8$ generated by the three elements $(1\;3)$, $(5\;7)$ and
$(1\;2)(3\;4)(5\;6)(7\;8)$. Since the generators never mix
$x_1, x_2, x_3, x_4$ with $x_5, x_6, x_7, x_8$, we are in fact dealing
with a subgroup of $S_4\times S_4$ (the first one permuting indices 
1,2,3 and 4; the second one permuting indices 5,6,7 and 8). 
The invariance group can be further tightened by noting that the
three generators belong to $D_8\times D_8$, where $D_8$ is the 
dihedral group consisting of the 8 ``symmetries of the square''. 
The group $D_8\times D_8$ acts on two squares whose sides are labelled
as follows~:
\vskip 2mm
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\end{picture}
\vskip -135mm
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\]
The elements of $D_8$ (referring to the first dihedral group) consist
of the following 8 permutations (in cycle notation)~:
$()$, $(1\;3)$, $(2\;4)$, $(1\;2)(3\;4)$, $(1\;4)(2\;3)$, $(1\;4\;3\;2)$,
$(1\;3)(2\;4)$ and $(1\;2\;3\;4)$. These correspond to all
rotations and reflections mapping the square into itself.
The double copy $D_8\times D_8$ hence consists of 64 elements. 
However, the invariance group generated by the two trivial
numerator permutations and (\ref{tf}) is not equal
to $D_8\times D_8$, but is only a subgroup of $D_8\times D_8$. 
It is not too difficult to verify that this subgroup 
has 32 elements. Inspection then reveals that this 32-element subgroup
consist of those elements of $D_8\times D_8$ that preserve the
constraint
\[
x_1+x_3+x_5+x_7-x_2-x_4-x_6-x_8=0,
\]
i.e.\ those elements that map $X=x_1+x_3+x_5+x_7-x_2-x_4-x_6-x_8$ into
$\pm X$ by permuting the indices.

We are now in a position to formulate the result about the invariance 
group.

\begin{prop}
Let $x_i$ ($i=0,1,\ldots,8$) be arbitrary parameters such that
$x_1+x_3+x_5+x_7=x_2+x_4+x_6+x_8$ and $1-2x_0+\sum_{i=1}^8 x_i=n$ is a 
nonnegative integer. Then the expression
\beas
&&f(x)=(x_0-x_4-2x_6-x_8)_n\ F^{1:2;2}_{0:2;2}\left[ {-n \atop \ }
{:\atop :} {x_2+x_3,x_1+x_2 \atop x_1+x_2+x_3+x_4,x_0-x_4-2x_6-x_8} 
{;\atop;} \right. \\
&& \left. 
{x_6+x_7,x_5+x_6 \atop x_5+x_6+x_7+x_8,x_0-2x_2-x_4-x_8}
{;\atop ;} 1,1\right]
\eeas
is (upto a sign) invariant under the action of $G$, where $G$ is the
subgroup of $D_8\times D_8$ consisting of those elements mapping
$X=x_1+x_3+x_5+x_7-x_2-x_4-x_6-x_8$ into $\pm X$. The action of an element
$g$ of $G$ is by permuting the indices of $x_1, \ldots x_8$, and we
can write
\[
f(g\cdot x) = \epsilon^n f(x),
\]
where $\epsilon = \pm 1$ is determined by $g(X)=\epsilon X$.
\end{prop}


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\end{document}
