%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%     LATEX SOURCE FILE                                               %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article} 
\usepackage{amssymb,latexsym}
\headheight=0mm
\headsep=0mm
\oddsidemargin=1mm
\evensidemargin=1mm
\textheight=225mm
\textwidth=155mm
%
% definitions concerning automatic numbering of theorems, lemmas, etc.
%
\newtheorem{theo}{Theorem}
\newtheorem{defi}[theo]{Definition}
\newtheorem{lemm}[theo]{Lemma}
\newtheorem{coro}[theo]{Corollary}
\newtheorem{rema}[theo]{Remark}
\newtheorem{prop}[theo]{Proposition}
%
% SPECIAL DEFINITIONS FOR THIS PAPER
%
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\beas{\begin{eqnarray*}}
\def\eeas{\end{eqnarray*}}
\def\nn{\nonumber}
\def\ds{\displaystyle}
\def\al{\alpha}
\def\be{\beta}
%\def\Z{{\tt Z}}
\def\Z{\mathbb{Z}}
% the end-of-proof box :
\def\mybox{\hfill\llap{$\Box$}}
%
%
\begin{document}
\setlength{\parindent}{0pt}
\begin{center}
{\Large \bf The invariance groups of certain single}\\[4mm]
{\Large \bf and double hypergeometric series}\\[3cm]
{\bf J.\ Van der Jeugt }\\[2mm]
Department of Applied Mathematics and Computer Science,\\
University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium.\\
E-mail~: Joris.VanderJeugt@rug.ac.be
\end{center}

%\addtolength{\baselineskip}{2mm}
%\addtolength{\abovedisplayskip}{1mm}
%\addtolength{\belowdisplayskip}{1mm}
\addtolength{\parskip}{2mm}

\begin{abstract}
Transformation formulas for classical hypergeometric series can 
often be seen as abstract transformations acting upon both the
parameters and the variable of the series (although in many cases
the variable itself is 1).
Together with trivial transformations, deduced from numerator and
denominator permutations, such transformation formulas give rise to 
a group of transformations -- the invariance group. 
For example, it was already established by Hardy that the invariance
group of Thomae's ${}_3F_2$ transformation is the symmetric
group $S_5$. More recently, invariance groups of other known
transformation formulas were determined, and in the same spirit
invariance groups of basic series transformations were 
investigated~\cite{30}.

In this contribution, the notion of invariance group will be introduced
and illustrated by means of some (single) hypergeometric series
transformations.
Furthermore, symmetry applications of the establishment of such invariance
groups are discussed.

Then a transformation formula for a terminating double Clausenian
hypergeometric series of unit argument (one of the proper Kamp\'e de
F\'eriet series, $F^{1:2;2}_{0:2;2}(1,1)$) is established.
This formula is the double series analogue of Whipple's (or
Sheppard's) terminating ${}_3F_2$ transformation.
The invariance group is determined and shown to be a subgroup of
a double copy of the symmetries of the square.
The basic analogue of the transformation formula is also given,
yielding a double series analogue of Sears' ${}_3\Phi_2$ transformation,
with a similar invariance group.
\end{abstract}

\section{Introduction}

The beauty of mathematics, or more specifically, the beauty of 
mathematical formulas,
is a truly subjective matter --- as is beauty in general.
A formula or identity that strikes me as beautiful, may not be so for you, 
and vice versa.
Even in mathematics there are no objective ways of measuring beauty.

Most people, however, would agree that symmetry yields beauty.
If an object posesses many symmetries, it is appealing and attractive.
This statement cannot be demonstrated better than by the beautiful Mughal 
decorations that
are so prominent in the old monuments of the city of Lucknow, capital of 
the nawabs,
and host of this Conference on Special Functions and its Applications.

In mathematics an object is said to be symmetric if it is invariant under 
the action of certain
transformations. These transformations could be reflections or rotations 
if we are dealing
with a geometrical object, or simply linear transformations if we are 
dealing with an algebraic
object or expression.
The set of all transformations that leave an object invariant forms a group, 
often referred to
as the invariance group.
So the invariance group describes the symmetries of the object under 
investigation.

In the present paper we shall illustrate and discuss invariance groups
related to transformation formulas of (basic) hypergeometric
series. The first part is dealing only with well-known transformation
formulas. So here no new transformation formulas are obtained~: only
the recognition of their invariance group --- i.e.\ of their
symmetries and hence their internal beauty --- is established. In the
second part we deduce a new transformation formula for a double series
and its basic analogue, and also here it is shown that this
transformation formula gives rise to an interesting invariance group.

The notation used in this paper is the standard notation of
Gasper and Rahman~\cite{6} for hypergeometric series ${}_pF_q$ and 
basic hypergeometric series ${}_p\Phi_q$; for double hypergeometric
series (or their basic analogues) we shall use a notation which is
close to that of Srivastava and Karlsson~\cite{24}.

\section{An introductory example} 

In this section we shall begin with one of the simplest examples of
a hypergeometric series transformation, namely some known transformations
for the ${}_2F_1$ series.

Consider first Pfaff's transformation, 
sometimes called Saalsch\"utz's theorem 
(see e.g.~\cite{2,19})~:
\beq
{}_2F_1\left( {a,b\atop c};x\right) = (1-x)^{-a}
{}_2F_1\left( {a,c-b\atop c};{x\over x-1}\right).
\label{pfaff}
\eeq
Another classical transformation is Euler's formula~\cite{2,19}~:
\beq
{}_2F_1\left( {a,b\atop c};x\right) = (1-x)^{c-a-b}
{}_2F_1\left( {c-a,c-b\atop c};x\right).
\label{euler}
\eeq
Both transformations have of course a region of convergence.
In this paper, however, we are merely interested in the actual form
of the transformations, and we shall not give the convergence
conditions explicitly (which can be found in the standard literature).

A proof of Euler's transformation can be given by 
using Pfaff's transformation twice~:
\bea
{}_2F_1\left( {a,b\atop c};x\right) &=& 
(1-x)^{-a} {}_2F_1\left( {a,c-b\atop c};{x\over x-1}\right)\label{1}\\
&=&
(1-x)^{-a} {}_2F_1\left( {c-b,a\atop c};{x\over x-1}\right)\label{2}\\
&=&
(1-x)^{-a} (1-{x\over x-1})^{-c+b}{}_2F_1\left( {c-b,c-a\atop c};x\right)
\label{3}\\
&=&
(1-x)^{c-a-b}{}_2F_1\left( {c-a,c-b\atop c};x\right). \label{4}
\eea

So, we have used two ``basic'' or ``elementary'' transformations, 
which we
call $g_1$ and $g_2$. Pfaff's transformation corresponds to $g_1$, and
the trivial transformation permuting the two numerator parameters
is called $g_2$~:
\[
\begin{array}{ll}
(g_1) & \ds {}_2F_1\left( {a,b\atop c};x\right) = (1-x)^{-a}
{}_2F_1\left( {a,c-b\atop c};{x\over x-1}\right), \\[3mm]
(g_2) & \ds
{}_2F_1\left( {a,b\atop c};x\right) = 
{}_2F_1\left( {b,a\atop c};x\right).
\end{array}
\]
Note that in the four steps (\ref{1})-(\ref{4}), we have used
respectively $g_1$, $g_2$, $g_1$ and finally $g_2$ again.
So Euler's transformation corresponds to the product $g_2g_1g_2g_1$
(since these are transformations, the action is from the right
to the left). 

What happens if we use these two elementary transformations
$g_1$ and $g_2$ iteratively in order to generate new transformations?
It is easy to see that this process of iteration stops after a while,
and no futher new transformations are obtained. In this example,
the iterative use gives rise to eight transformations, namely
\beas
{\bf 1} &&{}_2F_1\left( {a,b\atop c};x\right)\\
g_1 && (1-x)^{-a} {}_2F_1\left( {a,c-b\atop c};{x\over x-1}\right)\\
g_2 &&{}_2F_1\left( {b,a\atop c};x\right)\\
g_2g_1 && (1-x)^{-a} {}_2F_1\left( {c-b,a\atop c};{x\over x-1}\right)\\
g_1g_2 && (1-x)^{-b} {}_2F_1\left( {b,c-a\atop c};{x\over x-1}\right)\\
g_1g_2g_1 && (1-x)^{c-a-b}{}_2F_1\left( {c-b,c-a\atop c};x\right) \\
g_2g_1g_2 && (1-x)^{-b} {}_2F_1\left( {c-a,b\atop c};{x\over x-1}\right)\\
g_2g_1g_2g_1 && (1-x)^{c-a-b}{}_2F_1\left( {c-a,c-b\atop c};x\right) 
\eeas

All this is very simple and elementary, and the reader may wonder
what is the purpose of doing so.
The new observation made here is that this iteration process is
actually related to a group structure. This is not only the case
for the present example, but also for many other classes of
series transformations.

Observe that the elementary transformations satisfy~:
\beq
(g_1)^2 =1, \qquad (g_2)^2 =1, \qquad
(g_1g_2)^4 =1 \quad (\Leftrightarrow g_1g_2g_1g_2=g_2g_1g_2g_1).
\label{gr}
\eeq
A set of generators, together with the relations that they satisfy,
determine a group completely~\cite{7}.
The group $G$ generated by two elements $g_1$ and $g_2$,
subject to the relations (\ref{gr}), is known.
It is the dihedral group $D_8$, also known as the group of ``symmetries
of the square.'' It is a subgroup of $S_4$, the permutation group on
four elements.

The group $G$ is acting on the parameters and variable(s) of the 
hypergeometric series. In this case there are four elements $a$, $b$, $c$
and $x$ on which the group transformations act.
The first elementary transformation, apart from introducing a factor,
maps the parameters $(a,b,c,x)$ as follows~:
\[
(a,b,c,x)  \rightarrow  (a,c-b,c, {x\over x-1}).
\]
The second elementary transformation is simply~:
\[
(a,b,c,x) \rightarrow  (b,a,c,x).
\]

Since we now know that $G$ is a subgroup of $S_4$, we can represent
these two generating transformations as certain elements of $S_4$
by rewriting the parameters and variables. 

Writing
\[
a=x_1+x_4, b=x_3+x_4, c=x_1+x_2+x_3+x_4, x=1+{x_2+x_4\over x_1+x_3},
\]
the first elementary transformation becomes
\beas
&&(x_1+x_4, x_3+x_4, x_1+x_2+x_3+x_4, 1+{x_2+x_4\over x_1+x_3}) \rightarrow\\
&&(x_1+x_4, x_1+x_2, x_1+x_2+x_3+x_4, 1+{x_1+x_3\over x_2+x_4})
\eeas
i.e.\ this is the permutation $(1\, 4) (2\, 3)$.
The second elementary transformation becomes
\beas
&&(x_1+x_4, x_3+x_4, x_1+x_2+x_3+x_4, 1+{x_2+x_4\over x_1+x_3}) \rightarrow\\
&&(x_3+x_4, x_1+x_4, x_1+x_2+x_3+x_4, 1+{x_2+x_4\over x_1+x_3})
\eeas
i.e.\ this is the permutation $(1\, 3)$.

It is easy to see that the two permutations $g_1=(1\, 4) (2\, 3)$ and
$g_2=(1\, 3)$ generate the subgroup $D_8$ of $S_4$.
In the following picture, $g_1=(1\, 4) (2\, 3)$ corresponds to a
reflection about the diagonal of the square with sides numbered from 1
to 4, and $g_2=(1\, 3)$ corresponds to a reflection about the line
indicated. These two reflections generate all symmetries of the square.

\[
\vbox{
\unitlength=1mm
%\special{em:linewidth 0.4pt}
%\linethickness{0.4pt}
\begin{picture}(115.00,150.00)
\thicklines
\put(20.00,140.00){\line(1,0){25.00}}
\put(45.00,140.00){\line(0,-1){25.00}}
\put(45.00,115.00){\line(-1,0){25.00}}
\put(20.00,115.00){\line(0,1){25.00}}
\put(80.00,140.00){\line(1,0){25.00}}
\put(105.00,140.00){\line(0,-1){25.00}}
\put(105.00,115.00){\line(-1,0){25.00}}
\put(80.00,115.00){\line(0,1){25.00}}
\put(33.00,143.00){\makebox(0,0)[cc]{1}}
\put(15.00,129.00){\makebox(0,0)[cc]{2}}
\put(33.00,111.00){\makebox(0,0)[cc]{3}}
\put(48.00,129.00){\makebox(0,0)[cc]{4}}
\put(93.00,143.00){\makebox(0,0)[cc]{1}}
\put(77.00,129.00){\makebox(0,0)[cc]{2}}
\put(93.00,112.00){\makebox(0,0)[cc]{3}}
\put(108.00,129.00){\makebox(0,0)[cc]{4}}
\put(33.00,98.00){\makebox(0,0)[cc]{$(1\, 4)(2\, 3)$}}
\put(93.00,98.00){\makebox(0,0)[cc]{$(1\, 3)$}}
\thinlines
\put(10.00,105.00){\line(1,1){45.00}}
\put(70.00,128.00){\line(1,0){45.00}}
\end{picture}
\vskip -100mm
}
\]

All eight elements of $D_8$ ($\subset S_4$) are given by
\[
\vbox{
\unitlength=1mm
\begin{picture}(118.00,143.00)
\thicklines
\put(10.00,140.00){\line(1,0){15.00}}
\put(25.00,140.00){\line(0,-1){15.00}}
\put(25.00,125.00){\line(-1,0){15.00}}
\put(10.00,125.00){\line(0,1){15.00}}
\put(40.00,140.00){\line(1,0){15.00}}
\put(55.00,140.00){\line(0,-1){15.00}}
\put(55.00,125.00){\line(-1,0){15.00}}
\put(40.00,125.00){\line(0,1){15.00}}
\put(70.00,140.00){\line(1,0){15.00}}
\put(85.00,140.00){\line(0,-1){15.00}}
\put(85.00,125.00){\line(-1,0){15.00}}
\put(70.00,125.00){\line(0,1){15.00}}
\put(100.00,140.00){\line(1,0){15.00}}
\put(115.00,140.00){\line(0,-1){15.00}}
\put(115.00,125.00){\line(-1,0){15.00}}
\put(100.00,125.00){\line(0,1){15.00}}
\put(10.00,110.00){\line(1,0){15.00}}
\put(25.00,110.00){\line(0,-1){15.00}}
\put(25.00,95.00){\line(-1,0){15.00}}
\put(10.00,95.00){\line(0,1){15.00}}
\put(40.00,110.00){\line(1,0){15.00}}
\put(55.00,110.00){\line(0,-1){15.00}}
\put(55.00,95.00){\line(-1,0){15.00}}
\put(40.00,95.00){\line(0,1){15.00}}
\put(70.00,110.00){\line(1,0){15.00}}
\put(85.00,110.00){\line(0,-1){15.00}}
\put(85.00,95.00){\line(-1,0){15.00}}
\put(70.00,95.00){\line(0,1){15.00}}
\put(100.00,110.00){\line(1,0){15.00}}
\put(115.00,110.00){\line(0,-1){15.00}}
\put(115.00,95.00){\line(-1,0){15.00}}
\put(100.00,95.00){\line(0,1){15.00}}
\put(18.00,143.00){\makebox(0,0)[cc]{1}}
\put(7.00,133.00){\makebox(0,0)[cc]{2}}
\put(18.00,122.00){\makebox(0,0)[cc]{3}}
\put(28.00,133.00){\makebox(0,0)[cc]{4}}
\put(48.00,143.00){\makebox(0,0)[cc]{2}}
\put(37.00,133.00){\makebox(0,0)[cc]{3}}
\put(48.00,122.00){\makebox(0,0)[cc]{4}}
\put(58.00,133.00){\makebox(0,0)[cc]{1}}
\put(78.00,143.00){\makebox(0,0)[cc]{3}}
\put(67.00,133.00){\makebox(0,0)[cc]{4}}
\put(78.00,122.00){\makebox(0,0)[cc]{1}}
\put(88.00,133.00){\makebox(0,0)[cc]{2}}
\put(108.00,143.00){\makebox(0,0)[cc]{4}}
\put(97.00,133.00){\makebox(0,0)[cc]{1}}
\put(108.00,122.00){\makebox(0,0)[cc]{2}}
\put(118.00,133.00){\makebox(0,0)[cc]{3}}
\put(18.00,113.00){\makebox(0,0)[cc]{1}}
\put(7.00,103.00){\makebox(0,0)[cc]{4}}
\put(18.00,92.00){\makebox(0,0)[cc]{3}}
\put(28.00,103.00){\makebox(0,0)[cc]{2}}
\put(48.00,113.00){\makebox(0,0)[cc]{2}}
\put(37.00,103.00){\makebox(0,0)[cc]{1}}
\put(48.00,92.00){\makebox(0,0)[cc]{4}}
\put(58.00,103.00){\makebox(0,0)[cc]{3}}
\put(78.00,113.00){\makebox(0,0)[cc]{3}}
\put(67.00,103.00){\makebox(0,0)[cc]{2}}
\put(78.00,92.00){\makebox(0,0)[cc]{1}}
\put(88.00,103.00){\makebox(0,0)[cc]{4}}
\put(108.00,113.00){\makebox(0,0)[cc]{4}}
\put(97.00,103.00){\makebox(0,0)[cc]{3}}
\put(108.00,92.00){\makebox(0,0)[cc]{2}}
\put(118.00,103.00){\makebox(0,0)[cc]{1}}
\end{picture}
\vskip -95mm
}
\]

The eight transformations can now be expressed as follows~:\\
{\em The function
\beas
F({\bf x}) &=& F(x_1,x_2,x_3,x_4) =
(-{x_2+x_4\over x_1+x_3})^{(x_1-x_2+x_3+3x_4)/4} \\
&& \times {}_2F_1\left({x_1+x_4, x_3+x_4 \atop x_1+x_2+x_3+x_4};
1+{x_2+x_4\over x_1+x_3}\right)
\eeas
is invariant under the group $G=D_8$, i.e.
\[
F(g\cdot {\bf x}) = F({\bf x}) ,\qquad \forall g\in G.
\]
}

In such a case,
$G$ is called the {\em invariance group} of the hypergeometric
series transformation. Thus for the ${}_2F_1$ case, the
invariance group is the group of symmetries of the square.

Note that in general the group permuting numerator and denominator parameters
will always be a subgroup of $G$.

\section{Other hypergeometric series and basic analogues}

The invariance groups of series transformations are thus obtained
by iteration of known transformation formulas, together with the trivial
numerator and denominator permutations.

This process of iteration is not new. Already in 1923
Whipple~\cite{31} showed that by iterating Thomae's $_3F_2$
transformation formula~\cite{25}, one obtains a set of 120 such series, and he
tabulated the parameters of these 120 series. He did not, however,
recognize any group structure behind these 120 series. 
The first person to recognize the group structure was Hardy~\cite{8}.
He observed~:
\[
{1\over \Gamma(\beta_1)\Gamma(\beta_2)\Gamma(\beta_1+\beta_2-\alpha_1-
\alpha_2-\alpha_3)} 
{}_3F_2\left({\alpha_1,\alpha_2,\alpha_3\atop
 \beta_1,\beta_2};1 \right)
\]
is a symmetric function of the five arguments
\[
\beta_1, \beta_2, \beta_1+\beta_2-\alpha_2-\alpha_3, 
\beta_1+\beta_2-\alpha_3-\alpha_1, \beta_1+\beta_2-\alpha_1-\alpha_2.
\]
This can also be rephrased as
\beas
&&{1\over \Gamma(2x_4)\Gamma(2x_5)\Gamma(x_1+x_2+x3-x_4-x_5)} \times \\
&&{}_3F_2\left({x_1-x_2-x_3+x_4+x_5,-x_1+x_2-x_3+x_4+x_5,
-x_1-x_2+x_3+x_4+x_5\atop
 2x_4,2x_5};1 \right)
\eeas
is a symmetric function in $x_1,\ldots,x_5$.
So the invariance group of the ${}_3F_2$ of unit argument is,
rather surprisingly,  $S_5$, the permutation group on 5 elements (and
not simply $S_3\times S_2$, as one would expect from the numerator
and denominator symmetry).

For basic hypergeometric series, a number of examples were worked
out in~\cite{30}, and we shall just report some of these examples here.

The $q$-analogue of Thomae's transformation is Sears's 
transformation~\cite{17}
for the ${}_3\Phi_2$ series. In this case, the statement is as follows~:\\
{\em
The function
\beas
F(x)&=&F(x_1,x_2,x_3,x_4,x_5)= ( {x_1x_2x_3\over x_4x_5}, x_4^2,
x_5^2 ; q )_\infty \\[1mm]
&\times& {_3\Phi_2}\left({ {x_1\over x_2x_3} x_4x_5,\ {x_2\over x_1x_3}
x_4x_5,\ {x_3\over x_1x_2} x_4x_5 \atop x_4^2,\ x_5^2} ; q, {x_1x_2x_3
\over x_4x_5} \right) 
\eeas
is symmetric in the variables $(x_1,x_2,x_3,x_4,x_5)$.
}

{\em Proof.} Let us sketch the three steps of the proof here very briefly.
\begin{itemize}
\item Clearly, $F(x)$ is invariant under permutations of $(x_1,x_2,x_3)$ 
or $(x_4,x_5)$ (since these permute numerator parameters and denominator
parameters only).
\item Consider the permutation $g=(1\, 4\, 3\, 2\, 5)$ (in cycle
notation), which is a permutation of order~5. 
Relabelling the parameters of the ${_3\Phi_2}$ 
in $F(x)$ by
\[
{_3\Phi_2}\left({ a,\ b,\ c \atop d,\ e} ; q, {de\over abc} \right),
\]
the statement $F(x)=F(g\cdot x)$ is equivalent to
\[
{_3\Phi_2}\left({ a,\ b,\ c \atop d,\ e} ; q, {de\over abc} \right)=
{(b,de/ab,de/bc;q)_\infty \over (d,e,de/abc;q)_\infty}\ 
{_3\Phi_2}\left({ d/b,\ e/b,\ de/abc \atop de/ab,\ de/bc} ; q, b \right),
\]
which coincides with Sears' transformation.
\item
$F(x)$ is invariant under the permutation $g$,
and under the transposition $x_4\leftrightarrow x_5$. 
Since $g$ is a permutation of order~5, the group generated by $g$ and this
transposition is the complete group of permutations on 5 elements
(this is a well-known group theory property, see~\cite{11}),
i.e.\ the symmetric group $S_5$. 
\end{itemize}


Another interesting invariance group is found for Bailey's transformation 
of the terminating Saalsch\"utzian ${}_4F_3(1)$
series, and of Sears' basic analogue. 
The invariance group for Bailey's transformation was in fact first given
by Beyer, Louck and Stein~\cite{3}.
The basic analogue is discussed in~\cite{30}, the result reading~:

{\em
Let $x_1,\cdots,x_6$ be six parameters satisfying
\[
x_1x_2x_3x_4x_5x_6=q^{1-n}
\]
for some non-negative integer $n$. Then the function
\beas
F(x)&=&F(x_1,x_2,x_3,x_4,x_5,x_6) \\
&=&q^{\left(n \atop 2\right)} 
(x_1x_2x_3x_4, x_1x_2x_3x_5, x_1x_2x_3x_6  ; q )_n /(x_1x_2x_3)^n \\
&\times& {_4\Phi_3}\left({ q^{-n},\ x_2x_3,\ x_1x_3,\
x_1x_2 \atop x_1x_2x_3x_4,\ x_1x_2x_3x_5,\ x_1x_2x_3x_6} ; q, q \right)
\eeas
is symmetric in the variables $(x_1,x_2,x_3,x_4,x_5,x_6)$.
}\\
So therefore the invariance group is $G=S_6$, the symmetric group consisting
of permutations on 6 variables.

An even simpler example is provided by Heine's transformation of 
${}_2\phi_1$ series.
This transformation reads~\cite{9}~:
\beq
{}_2\Phi_1 (a,\, b; c; q, z) = {(a,bz;q)_\infty \over
(c,z;q)_\infty} {}_2\Phi_1 (c/a,\, z; bz; q, a),
\label{heine}
\eeq
where $|z|<1$ and $|a|<1$ for convergence.
Iterating this transformation, and using the symmetry of the numerator
parameters of the ${}_2\Phi_1$ yields a set of 12 transformation formulas
(this is known already since a long time, see e.g.\ Sears~\cite{17}).
What is new however, is the fact that there is also an invariance group
behind these 12 transformations. The
identification of the invariance group was performed in~\cite{30}. It is
the dihedral group $D_{12}$ (sometimes also denoted by $D_6$), 
i.e.\ the group of symmetries of the hexagon. 
This group consists of all
rotations and reflections that map the hexagon into itself. 
E.g., labelling the sides of the hexagon by $x_1,\ldots, x_6$, the
second figure below is a reflection of the first hexagon about a vertical
axis, and the third figure is a rotation of the first hexagon (keeping
the shape invariant). In total there are twelve such symmetries of
the hexagon, corresponding to the 12 elements of the group $D_{12}$,
of which only three are shown here.
\[
\vbox{
\unitlength=1mm
%\special{em:linewidth 0.4pt}
%\linethickness{0.4pt}
\begin{picture}(115.00,143.00)
\thicklines
\put(20.00,138.00){\line(1,0){10.00}}
\put(30.00,138.00){\line(3,-5){4.67}}
\put(34.67,130.00){\line(-3,-5){4.67}}
\put(30.00,122.00){\line(-1,0){10.00}}
\put(20.00,122.00){\line(-3,5){4.67}}
\put(15.33,130.00){\line(3,5){4.67}}
\put(60.00,138.00){\line(1,0){10.00}}
\put(70.00,138.00){\line(3,-5){4.67}}
\put(74.67,130.00){\line(-3,-5){4.67}}
\put(70.00,122.00){\line(-1,0){10.00}}
\put(60.00,122.00){\line(-3,5){4.67}}
\put(55.33,130.00){\line(3,5){4.67}}
\put(100.00,138.00){\line(1,0){10.00}}
\put(110.00,138.00){\line(3,-5){4.67}}
\put(114.67,130.00){\line(-3,-5){4.67}}
\put(110.00,122.00){\line(-1,0){10.00}}
\put(100.00,122.00){\line(-3,5){4.67}}
\put(95.33,130.00){\line(3,5){4.67}}
\put(25.00,140.00){\makebox(0,0)[cc]{$x_1$}}
\put(65.00,140.00){\makebox(0,0)[cc]{$x_1$}}
\put(105.00,140.00){\makebox(0,0)[cc]{$x_2$}}
\put(35.00,135.00){\makebox(0,0)[cc]{$x_2$}}
\put(75.00,135.00){\makebox(0,0)[cc]{$x_6$}}
\put(115.00,135.00){\makebox(0,0)[cc]{$x_3$}}
\put(35.00,125.00){\makebox(0,0)[cc]{$x_3$}}
\put(75.00,125.00){\makebox(0,0)[cc]{$x_5$}}
\put(115.00,125.00){\makebox(0,0)[cc]{$x_4$}}
\put(25.00,119.00){\makebox(0,0)[cc]{$x_4$}}
\put(65.00,119.00){\makebox(0,0)[cc]{$x_4$}}
\put(105.00,119.00){\makebox(0,0)[cc]{$x_5$}}
\put(15.00,125.00){\makebox(0,0)[cc]{$x_5$}}
\put(55.00,125.00){\makebox(0,0)[cc]{$x_3$}}
\put(95.00,125.00){\makebox(0,0)[cc]{$x_6$}}
\put(15.00,135.00){\makebox(0,0)[cc]{$x_6$}}
\put(55.00,135.00){\makebox(0,0)[cc]{$x_2$}}
\put(95.00,135.00){\makebox(0,0)[cc]{$x_1$}}
\end{picture}
\vskip -120mm
}
\]

The symmetry statement is now as follows~:\\
{\em
The function
\beas
f(x)&=&f(x_1,x_2,x_3,x_4,x_5)= ( x_1x_4,{x_2x_6\over x_1}; q )_\infty
\\[1mm] 
&\times& {_2\Phi_1}( {x_1x_3\over x_2},\  {x_1x_5\over x_6} ;
x_1x_4 ;q,  {x_2x_6\over x_1} )
\eeas
is invariant under the dihedral group $D_{12}$ acting on the variables
$(x_1,x_2,x_3,x_4,x_5,x_6)$. 
}

Other cases being described in~\cite{30} are~:
\begin{itemize}
\item
The invariance group of Whipple's terminating ${}_3F_2(1)$ 
series (now referred to as Sheppard's transformation) is a 
72-element subgroup of $S_6$ (that
can also be described as hexagon transformations).
\item
For Sears's $q$-analogue of these transformations, the invariance group
is the same.
\item
For transformations of very-well-poised ${}_8\Phi_7$ series,
the invariance group (of order 1920) is a subgroup of the group of 
signed permutations
on 5 elements; it coincides with the Weyl group of a root
system of type $D_5$.
\end{itemize}

\section{A double series transformation and its invariance group}

It is worthwhile observing that also double series transformations can
have interesting invariance groups.
The example discussed here corresponds to a new double series
transformation and its basic analogue; some other double series
transformations worth studying from the point of view of invariance groups
were pointed out to me by R.Y.\ Denys at this Conference (see~\cite{18}).

The series under consideration here is a double Clausenian 
series~\cite{13,14};
these series are the ten proper Kamp\'e de F\'eriet
functions $F^{p:r;r}_{q:s;s}$ for which $q+s=2$ and $p+r=3$.
For a general treatment of hypergeometric series in more variables,
see~\cite{1,12,15,24}.

We shall be dealing with the function
\beq
F^{1:2;2}_{0:2;2}\left( {e \atop \ }: {a,b \atop c,d}\ ; {a',b' \atop
c',d'}\ ;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{doublef}
\eeq
in the terminating case, i.e. $e=-n$.

The (new) transformation formula can be considered as a double analogue
of Whipple's terminating ${}_3F_2$ series, and is given by~\cite{29}~:\\
{\em
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
\bea
&&F^{1:2;2}_{0:2;2}\left( {-n \atop \ }: {a,b \atop c,d}\ ; {a',b' \atop
c',d'}\ ;1,1\right) = \nn\\
&& \qquad  {(d-b+b')_n\over (d)_n}
F^{1:2;2}_{0:2;2}\left( {-n \atop \ }: {c-a,b \atop c,d'-b'+b}\ ; 
{c'-a',b' \atop c',d-b+b'}\ ;1,1\right).
\label{tf}
\eea
}

For a proof of this formula, we refer to the $q$-case considered in
the next section.

Iterative use of this transformation formula yields a number of
different transformations~:
\bea
&& F^{1:2;2}_{0:2;2}\left( {-n \atop \ }: {a,b \atop c,d} ; {a',b' \atop
c',d'};1,1\right)  \nn \\
&=& {(c+d-a-b)_n\over (d)_n}
F^{1:2;2}_{0:2;2}\left( {-n \atop \ }: {c-a,c-b \atop c,c+d-a-b} ; {a',b' \atop c',d'-c+a+b};1,1\right) \\
&=& {(d-c'+a'+b')_n\over (d)_n}
F^{1:2;2}_{0:2;2}\left( {-n \atop \ }: {a,b \atop c,d-c'+a'+b'} ; {c'-a',c'-b' \atop c',c'+d'-a'-b'};1,1\right) \\
&=&{(c+d-a-b+a'+b'-c')_n\over (d)_n} \\
&&\times 
F^{1:2;2}_{0:2;2}\left( {-n \atop \ }: {c-a,c-b \atop c,c+d-a-b+a'+b'-c'} ; 
{c'-a',c'-b' \atop c',d'-c+a+b-a'-b'+c'};1,1\right). \nn
\eea

In total there are 32 such transformations, and thus it would be interesting
to find the nature of the invariance group induced by the basic
transformation (\ref{tf}), together with the trivial transformations
corresponding to the permutation of $a$ and $b$, and of $a'$ and $b'$.
Note that permutations of $c$ and $d$, or $c'$ and $d'$, are not 
permitted in this context. This is because there is a constraint 
involving the parameters $d$ and $d'$ (namely $d+d'=1-n$), and the
convention is that the denominator parameters of which the sum is equal
to $1-n$ should always be in the same position.

In the transformation formula (\ref{tf}), there are in total
9 parameters, with one condition $d+d'=1-n$. So a 
relabelling 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
\[
x_1+x_3+x_5+x_7=x_2+x_4+x_6+x_8.
\]

In terms of the new variables, the three elementary transformations
take the following form~:
\begin{itemize}
\item the interchange of $a$ and $b$ corresponds to the permutation $(1\;3)$;
\item the interchange of $a'$ and $b'$ corresponds to the permutation $(5\;7)$;
\item the new transformation (\ref{tf}) is equivalent
with $x_1 \leftrightarrow x_2$, $x_3 \leftrightarrow x_4$, 
$x_5 \leftrightarrow x_6$, $x_7 \leftrightarrow x_8$, thus corresponds to 
the permutation $(1\;2)(3\;4)(5\;6)(7\;8)$.
\end{itemize}

Which group is generated by these three permutations? 
In order to find out, observe the following~:
\begin{itemize}
\item $x_0$ is left unchanged by the three permutations, so $G$ is
a subgroup of  $S_8$ (acting on $x_1,\ldots, x_8$);
\item
the three generators never mix
$x_1, x_2, x_3, x_4$ with $x_5, x_6, x_7, x_8$~; thus
$G\subset S_4\times S_4$ (the first one permuting indices 
1,2,3 and 4; the second one permuting indices 5,6,7 and 8);
\item it is not difficult to see 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 (as encountered in section~2). 
\end{itemize}
 
The group $D_8\times D_8$ acts on two squares whose sides are labelled
as follows~:
\[
\vbox{
\unitlength=1mm
%\special{em:linewidth 0.4pt}
%\linethickness{0.4pt}
\begin{picture}(57.00,152.00)
\thicklines
\put(10.00,150.00){\line(1,0){15.00}}
\put(25.00,150.00){\line(0,-1){15.00}}
\put(25.00,135.00){\line(-1,0){15.00}}
\put(10.00,135.00){\line(0,1){15.00}}
\put(10.00,150.00){\line(0,0){0.00}}
\put(40.00,150.00){\line(1,0){15.00}}
\put(55.00,150.00){\line(0,-1){15.00}}
\put(55.00,135.00){\line(-1,0){15.00}}
\put(40.00,135.00){\line(0,1){15.00}}
\put(18.00,152.00){\makebox(0,0)[cc]{1}}
\put(48.00,152.00){\makebox(0,0)[cc]{5}}
\put(8.00,143.00){\makebox(0,0)[cc]{2}}
\put(38.00,143.00){\makebox(0,0)[cc]{6}}
\put(18.00,133.00){\makebox(0,0)[cc]{3}}
\put(48.00,133.00){\makebox(0,0)[cc]{7}}
\put(27.00,143.00){\makebox(0,0)[cc]{4}}
\put(57.00,144.00){\makebox(0,0)[cc]{8}}
\end{picture}
\vskip -135mm
}
\]

It is immediately clear that the three generating elements leave the
two squares invariant. The group $D_8\times D_8$ has 64 elements.
The group $G$, generated by the three permutations, has in fact only
32 elements. It can be determined that 
$G$ 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.

So we finally come to the following result~:\\
{\em
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\\ 
&\times& F^{1:2;2}_{0:2;2}\left( {-n \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} ; \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};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 $g(X)=\epsilon X$.
}

\section{The basic transformation formula}

In this last section we shall give the basic analogue of the
transformation formula (\ref{tf}) and its proof.

For the double $q$-series appearing here, our notation is inspired
by that of Srivastava and Karlsson~\cite{24}~: 
\bea
&&\Phi^{1:2;2}_{0:2;2}\left( {e \atop \ }: {a,b \atop c,d}\ ; 
{a',b' \atop c',d'}\ ;{q;x,y \atop \nu} \right) = \nn \\
&&\sum_{j,k=0}^\infty 
{(e;q)_{j+k}(a;q)_j(b;q)_j(a';q)_k(b';q)_k \over (q;q)_j (c;q)_j
(d;q)_j(q;q)_k(c';q)_k(d';q)_k} 
{x^j y^k (q^\nu)^{jk}}.
\eea

The transformation formula now reads~:\\
{\em
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 \ }: {a,b \atop c,d}\ ; 
{a',b' \atop c',d'}\ ; {q;{cdq^n\over ab},{c'd'q^n\over a'b'}
\atop -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 \ }: {c/a,b \atop c,d'b/b'} ; 
{c'/a',b' \atop c',db'/b};{q;q,q \atop 0}\right).
\label{qtf}
\eea
}

{\bf Proof.}
The lhs of (\ref{qtf}) can be written as
\[
\sum_{k=0}^n {(q^{-n};q)_k(a';q)_k(b';q)_k \over (q;q)_k(c';q)_k(d';q)_k}
\left({c'd'q^n\over a'b'}\right)^k \;{}_3\Phi_2 \left( 
{q^{-n+k}, a, b \atop c, d }; q,{cdq^{n-k}\over ab} \right).
\]
Applying Sears' formula on the ${}_3\Phi_2$ yields
\beas
&& \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 }; 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
\[
{(d/b;q)_{n-k} \over (d';q)_k(d;q)_{n-k}(bq^{1-n+k}/d;q)_j} =
{(d'b;q)_n \over (d';q)_n} {b^{k-n} \over (d'b;q)_j(d'bq^j;q)_k}.
\]
Substituting this in the last double sum gives
\beas
&&{(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 }; q, {b c'd'q^n\over a'b'}\right)
\eeas
Applying once more Sears' formula on this ${}_3\Phi_2$ 
leads to
\beas
&&{(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 }; 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.


Observe that by further specialization of the parameters, some reduction
and summation formulas can be obtained.
Indeed, with $b'=abq^{1-n}/cd$, the rhs of (\ref{qtf}) simplifies
and reduces to~:
\beas
&&\Phi^{1:2;2}_{0:2;2}\left( {q^{-n} \atop \ }: {a,b \atop c,d} ; 
{a',abq^{1-n}/cd \atop c',q^{1-n}/d};
{q;{cdq^n/ab},{cc'q^n/aba'} \atop -1}\right) = \\
&&{(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 }; q, q\right).
\eeas

With $c'=q^{1-n}/c$, this further reduces to
\beas
&&\Phi^{1:2;2}_{0:2;2}\left( {q^{-n} \atop \ }: {a,b \atop c,d} ; 
{a',abq^{1-n}/cd \atop q^{1-n}/c,q^{1-n}/d};
{q;{cdq^n/ab},{q/aba'} \atop -1}\right) = \\
&&{(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 }; q, q\right).
\eeas
Finally, choosing $a'=1/a$, the last ${}_3\Phi_2$ can be summed and yields~:
\[
\Phi^{1:2;2}_{0:2;2}\left( {q^{-n} \atop \ }: {a,b \atop c,d} ; 
{1/a,abq^{1-n}/cd \atop q^{1-n}/c,q^{1-n}/d};
{q;{cdq^n/ab},{q/b} \atop -1}\right) = 
a^n {(c/a;q)_n(d/a;q)_n \over (c;q)_n(d;q)_n}.
\]

\section{Conclusion}

In this contribution, we have considered and determined some
invariance groups for single hypergeometric series
and their $q$-analogues. It should be emphasized that, as far as
the single series are concerned, no new transformation formulas
are obtained~: we only discover ``new group
structures behind old formulas''.


The expression of the symmetry enables a whole list of (known) 
transformation formulas 
to be summarized as elegant one-line statements.
In my opinion, the symmetry aspect puts these transformation formulas
on a higher level as far as beauty is concerned.
Observe that the knowledge of the symmetry group gives sometimes 
a good understanding of why certain transformations can be deduced
from others (such as in the first example of this paper).

The knowledge of the invariance group is also useful to 
discover symmetry properties of quantities that are
being expressed in terms of such a hypergeometric series, such as
the  $3j$- or $6j$-coefficients~\cite{21}.

We have also determined the invariance group for a double 
hypergeometric series; here the formula itself is believed to
be new, and so is the recognition of the group structure.
Finally, it should be noticed that the transformation formula
(\ref{tf}) given here can be related to
certain symmetries of a ``singly stretched''
$9j$-coefficient~\cite{20,26,28}.

\begin{thebibliography}{99}

\bibitem{1}
P.\ Appell and J.\ Kamp\'e de F\'eriet,
{\em Fonctions Hyperg\'eom\'etriques et Hy\-per\-sph\'e\-ri\-ques~:
Polyn\^omes d'Hermite} 
(Gauthier-Villars, Paris, 1926).

\bibitem{2}
W.N.\ Bailey, 
{\em Generalized Hypergeometric Series}
(Cambridge Univ.\ Press, Cambridge, 1935).

\bibitem{3}
W.A.\ Beyer, J.D.\ Louck and P.R.\ Stein,
Group theoretical basis of some identities for the generalized
hypergeometric series,
{\em J.\ Math.\ Phys.} {\bf 28} (1987) 497--508.

\bibitem{6}
G.\ Gasper and M.\ Rahman,
{\em Basic Hypergeometric Series}
(Cambridge Univ.\ Press, Cambridge, 1990).

\bibitem{7}
M.\ Hamermesh,
{\em Group Theory, and its application to physical problems.}
(Addison Wesley, Reading, 1962).

\bibitem{8}
G.H.\ Hardy,
{\em Ramanujan. Twelve lectures on subjects suggested by his life and
work} 
(Cambridge Univ.\ Press, Cambridge, 1940).

\bibitem{9}
E.\ Heine,
{\em Handbuch der Kugelfunctionen. Theorie und Anwendungen,} vol.~1.
(Reimer, Berlin, 1878).

\bibitem{11}
G.\ James and A.\ Kerber,
{\em The representation theory of the symmetric group.}
(Addison Wesley, Reading, 1981).

\bibitem{12}
J.\ Kamp\'e de F\'eriet, 
Les fonctions hyperg\'eom\'etriques d'ordre sup\'erieur \`a deux
variables, 
{\em C.\ R.\ Acad.\ Sci.\ Paris} {\bf 173} (1921) 401--404.

\bibitem{13}
Per W.\ Karlsson, 
Reduction of double Clausenian hypergeometric functions, 
{\em Int.\ J.\ Math.\ Stat.\ Sci.} {\bf 5} (1996), 33--49.

\bibitem{14}
Per W.\ Karlsson,
Some formulae for double Clausenian functions,
{\em J.\ Comp.\ Appl.\ Math.} {\bf 118} (2000), 203--213.

\bibitem{15}
G.\ Lauricella,
Sulle funzioni ipergeometriche a pi\`u variabili,
{\em Rend.\ Circ.\ Mat.\ Palermo} {\bf 7} (1893) 111--158.

\bibitem{17}
D.B.\ Sears,
On the transformation theory of basic hypergeometric functions,
{\em Proc.\ London Math.\ Soc.\ (2)} {\bf 53} (1951), 158--180.

\bibitem{18}
S.P.\ Singh,
Certain transformation formulae involving basic hypergeometric functions,
{\em J.\ Math.\ Phys.\ Sciences} {\bf 28} (1994), 189--195.

\bibitem{19}
L.J.\ Slater, 
{\em Generalized Hypergeometric Functions}
(Cambridge Univ.\ Press, Cambridge, 1966).

\bibitem{20}
K.\ Srinivasa Rao and J.\ Van der Jeugt, 
Stretched 9-$j$ coefficients and summation theorems, 
{\em J.\ Phys.\ A~: Math.\ Gen.} {\bf 27} (1994) 3083--3090. 

\bibitem{21}
K.\ Srinivasa Rao, J.\ Van der Jeugt, J.\ Raynal, R.\ Jagannathan
and V.\ Rajeswari, 
Group theoretical basis for the terminating $_3F_2(1)$ series, 
{\em J.\ Phys.\ A~: Math.\ Gen.} {\bf 25} (1992) 861--876.

\bibitem{24}
H.M.\ Srivastava and P.W.\ Karlsson, 
{\em Multiple Gaussian Hypergeometric Series}
(Halsted, New York, 1985).

\bibitem{25}
J.\ Thomae,
\"Uber die Funktionen welche durch Reihen von der Form dargestellt
werden~: $1+{pp'p'' \over 1q'q''}+\cdots$,
{\em J.\ Reine Angew.\ Math.} {\bf 87} (1879), 26--73.

\bibitem{26}
J.\ Van der Jeugt, S.N.\ Pitre and K.\ Srinivasa Rao, 
Multiple hypergeometric functions and 9-$j$ coefficients, 
{\em J.\ Phys.\ A~: Math.\ Gen.} {\bf 27} (1994) 5251--5264.

\bibitem{28}
J.\ Van der Jeugt,
Hypergeometric series related to the 9-$j$ coefficient of $su(1,1)$,
{\em J.\ Comp.\ Appl.\ Math.} {\bf 118} (2000), 337--351.

\bibitem{29}
J.\ Van der Jeugt,
Transformation formula for a double Clausenian hypergeometric series,
its $q$-analogue, and its invariance group,
{\em in press} (2001).

\bibitem{30}
J.\ Van der Jeugt and K.\ Srinivasa Rao,
Invariance groups of transformations of basic hypergeometric series,
{\em J.\ Math.\ Phys.} {\bf 40} (1999), 6692--6700.

\bibitem{31}
F.J.W.\ Whipple,
A group of generalized hypergeometric series~: relations between 120
allied series of the type $F[a,b,c;d,e]$,
{\em Proc.\ London Math.\ Soc. (2)} {\bf 23} (1925), 104--114.

\end{thebibliography}

\end{document}


