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%     Invariance groups of transformations of                         %
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\begin{center}
{\Large \bf Invariance groups of transformations}\\[4mm]
{\Large \bf of basic hypergeometric series}\\[3cm]
{\bf J.\ Van der Jeugt\footnote{Research
Associate of the Fund for Scientific Research -- Flanders
(Belgium).} }\\[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\\[2mm]
{\bf and}\\
{\bf K.~Srinivasa Rao}\\[2mm]
Institute of Mathematical Sciences,\\
CIT Campus, Madras 600113, India.\\
E-mail~: rao@imsc.ernet.in. 
\end{center}

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\vspace{1 cm}
\noindent {\bf Abstract}\\
We show that certain two-term transformation formulas between basic
hypergeometric series can easily be described by means of invariance
groups.  
For the transformations of non-terminating ${}_3\phi_2$ series,
and those of terminating balanced ${}_4\phi_3$ series, these invariance
groups are symmetric groups.  For transformations of ${}_2\phi_1$ series
the invariance group is the dihedral group of order~12.
Transformations of terminating ${}_3\phi_2$ series are described by
means of some subgroup of $S_6$, and finally the invariance group of
transformations of very-well-poised non-terminating ${}_8\phi_7$ series
is shown to be isomorphic to the Weyl group of a root system of type
$D_5$.

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

\vspace{1cm}

\newpage
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\section{Introduction}

This article deals with two-term transformation formulas
for basic hypergeometric series. Our aim is to identify
the (finite) invariance group structures underlying known
transformation formulas. Examining these invariance groups
is not only an interesting problem by itself but it enables
a whole list of (known) transformation formulas to be 
summarized as elegant one-line statements!

Anyone working with (basic) hypergeometric series is bound 
to encounter sooner or later transformation formulas which
express a series into another.
A summary of the most commonly used transformation
formulas can be found in Appendix~III of 
Gasper and Rahman~\cite{GR}. 
A required, specific transformation formula, not listed in 
this appendix can be derived by iterative use of one or 
more of the given transformation formulas. 
This can be done by hand, or even simpler by a computer algebra package
such as {\tt HYP} or {\tt HYPQ} that performs these transformations 
consecutively~\cite{Krattenthaler}. 
This process of iteration is not new. Already in 1923
Whipple~\cite{Whipple} showed that by iterating Thomae's $_3F_2$
transformation formula, 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. 
In some way, this group
comes naturally into the picture, once a transformation formula
between two (basic) hypergeometric series is translated into a linear 
transformation between the set of parameters (and/or variables) of the
two series. 
The application of two transformation formulas then
translates into the composition (or product) of two linear
transformations, and this is how a group structure emerges, see for
example~\cite{BLS,Rao}. 
This group is then called the invariance group
of the transformations of (basic) hypergeometric series.

For the transformations of non-terminating ${}_3\phi_2$ series,
and those of terminating balanced ${}_4\phi_3$ series, it is
established in Section~2 that the invariance groups are the symmetric
groups, $S_5$ and \ $S_6$, respectively. 
This result is not surprising, and
essentially the same as in the classical case of 
the transformations of the (ordinary) hypergeometric series 
of unit argument, ${}_3F_2(1)$ and
${}_4F_3(1)$, respectively~\cite{BLS}. 
 Our aim in proving these results is twofold: that the results
 can be established simply, {\it \`a la} Hardy [6] and, more
 importantly, they pave the way for a study of other cases. 
In Section~3, Heine's transformations for
${}_2\phi_1$ series are considered, and it is shown that the invariance
group is the dihedral group of order~12, $D_{12}$. 
Next,
transformations of terminating ${}_3\phi_2$ series are considered. The
invariance group is a non-simple group of order~72, which can be
described in terms of extended symmetries of the hexagon. Finally, in
Section~5 it is established that the invariance group of
transformations of very-well-poised non-terminating ${}_8\phi_7$ series
is $WD_5$, a group of signed permutations isomorphic to the Weyl group
of a root system of type $D_5$.

Apart from the mathematical interest, our motivation for studying
these invariance groups also stems from a number of examples
illustrating the usefulness of the 
transformations of the ordinary and basic hypergeometric series in 
physics and quantum groups.
Louck {\em et al} used the invariance group related to the ${}_4F_3(1)$
series to discuss the symmetries of extended 6-$j$ 
coefficients~\cite{Louck,BieLou}.
Rajeswari and Srinivasa Rao
used Thomae transformations of the $_3F_2(1)$ to derive the well-known
Wigner, Racah and Majumdar forms of the 3-$j$ angular momentum 
coefficient~\cite{Raji1}
of quantum mechanics from the Van der Waerden form. They also derived
the $q$-analogue of the Van der Waerden form of the 3-$j$ 
coefficient and the $q$-analogues of the aforesaid Wigner,
Racah and Majumdar forms of the 3-$j$ coefficient~\cite{Raji2}. 
They showed~\cite{Raji3} that the 
transformation theory for basic hypergeometric functions, $_3\phi_2$,
provides the necessary framework to relate the explict forms for the 
Clebsch-Gordan coefficient of the quantum group $SU_q(2)$.

We end this Introduction by fixing some notation.
For (basic) hypergeometric series, our notation is the standard one
of~\cite{GR}. Throughout the paper, the base $q$ satisfies $|q|<1$. The
$q$-shifted factorial is
\[
(a;q)_n = \left\{ \begin{array}{ll}
 1, & n=0; \\[2mm]
 {\ds \prod_{k=0}^{n-1} (1-a q^k)}, & n=1,2,\ldots,\infty, \end{array} 
 \right.
\]
and $(a_1, \ldots, a_m;q)_n = (a_1;q)_n \cdots (a_m;q)_n$. The basic
hypergeometric series appearing here are
\[
{}_{r+1}\phi_r \left[ {a_1,a_2,\ldots,a_{r+1} \atop b_1,b_2,\ldots,b_r}
; q, z \right] = \sum_{k=0}^\infty {(a_1,a_2,\ldots,a_{r+1};q)_k \over
(q,b_1,b_2, \ldots, b_r;q)_k} z^k,
\]
and for $r=1$ we shall usually write ${}_2\phi_1(a_1,\, a_2; b_1;q,z)$.

It is also useful to establish some notation about permutations here.
A permutation on $n$ elements can be denoted by its cycle
structure~(see, e.g.\ \cite[Chapter~1]{Hamermesh}). For example, the
permutation $p\in S_5$ with $p(1)=2$, $p(2)=3$, $p(3)=1$, $p(4)=5$,
$p(5)=4$ is simply denoted by $(1\, 2\, 3)(4\, 5)$. For the action of
$p$ on an array of $n$ symbols, we use by abuse of notation again $p$;
thus for the above example~:
\[
p \cdot (x_1,x_2,x_3,x_4,x_5) = p \cdot x = (x_2,x_3,x_1,x_5,x_4).
\]
With a permutation, an $n\times n$ monomial matrix (this is a matrix
with exactly one non-zero element in every row and column) is associated
with non-zero entries equal to 1. For the above example, this matrix is
\[
\left( \begin{array}{ccccc} 0&1&0&0&0\\ 0&0&1&0&0\\ 1&0&0&0&0\\
0&0&0&0&1\\ 0&0&0&1&0 \end{array} \right) ;
\]
the action $p\cdot x$ can then be interpreted as a 
multiplication of this permutation matrix with a column vector $x$. We
will also encounter ``signed permutations''; here the non-zero entries of
the monomial matrices are $\pm 1$. 
If, for example, the matrix of a
signed permutation $p$ is given by
\[
\left( \begin{array}{ccccc} 0&1&0&0&0\\ 0&0&-1&0&0\\ 1&0&0&0&0\\
0&0&0&0&-1\\  0&0&0&1&0 \end{array} \right),
\]
then we define its action on an array of $n$ variables by
\[
p \cdot (x_1,x_2,x_3,x_4,x_5) = p \cdot x = (x_2,{1\over x_3},x_1,{1
\over x_5},x_4).
\]


\section{Transformations of non-terminating ${}_3\phi_2$ series,
and Sears' transformations of terminating balanced ${}_4\phi_3$ series}

Thomae~\cite{Thomae} obtained a number of transformations for
hypergeometric series of type ${}_3F_2$ with unit argument, see
also~\cite[\S 3.2]{Bailey}, through the calculus of finite differences.
Whipple~\cite{Whipple} introduced a new notation in order to simplify
the numerous formulas obtained by Thomae, and showed there are 120 such
series related to each other. Still, it was not recognized that an
invariance group (the symmetric group $S_5$) can describe all these
relations. To our knowledge, the existence of such an invariance group
was first stated by Hardy~\cite[footnote on page~111]{Hardy}. Being
unaware of Hardy's result, this property was later rediscovered by Beyer,
Louck and Stein~\cite{BLS}.

Here, the $q$-analogues of Thomae's transformations are considered.
These transformations were derived by Sears~\cite[section~10]{Sears},
and two of them are given 
by (III.9) and (III.10) of~\cite{GR}. It is easy to show that for
these transformations of non-terminating ${_3\phi_2}$ series, the
invariance group is $S_5$.

\begin{prop}
The function
\bea
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 \nn\\[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] 
\label{nonterm32}
\eea
is symmetric in the variables $(x_1,x_2,x_3,x_4,x_5)$.
\end{prop}

\noindent {\em Proof.} Clearly, $f(x)$ is invariant under permutations
of $(x_1,x_2,x_3)$ (resp.\ $(x_4,x_5)$), since these do not change the
factor in front of the ${_3\phi_2}$, and on the ${_3\phi_2}$ they have
the effect of permuting numerator (resp.\ denominator) parameters.
Next, consider the permutation $p=(1\, 4\, 3\, 2\, 5)$ (in cycle
notation~: $x_1\rightarrow x_4\rightarrow x_3\rightarrow x_2\rightarrow 
x_5\rightarrow x_1$), 
which is a permutation of 
order~5. Upon relabelling the parameters of the ${_3\phi_2}$ in $f(x)$ by
\[
{_3\phi_2}\left[{ a,\ b,\ c \atop d,\ e} ; q, {de\over abc} \right],
\]
it is easy to see that equation (III.10) of~\cite{GR} is equivalent to
$f(x)=f(p\cdot x)$. So $f(x)$ is invariant under the permutation $p$,
and under the transposition $x_4\leftrightarrow x_5$. But since $p$ is
a permutation of order~5, the group generated by $p$ and this
transposition is the complete group of permutations on 5 elements (see,
for example~\cite[page~5]{JK}), i.e.\ the symmetric group $S_5$. \mybox

For convergence of all the 120 series in~(\ref{nonterm32}) it is
sufficient that all $x_i$ satisfy $\al^{3/2} < |x_i| < \al$, with $\al<1$ 
a positive real number.

Next, we consider transformations of terminating balanced ${}_4\phi_3$
series. This is the $q$-analogue of Bailey's transformations for
terminating Saalsch\"utzian hypergeometric series of type ${}_4F_3$
with unit argument~\cite[\S 7.2(1)]{Bailey}. All 720 transformations of
this type were already obtained by Bailey~\cite[Chapter~VII]{Bailey},
and the fact that the invariance group for these transformations is the
symmetric group $S_6$ was established by Beyer, Louck and
Stein~\cite{BLS}. For the corresponding basic series, the
essential transformation was first given by Sears~\cite[(8.3)]{Sears},
see also (III.15) and (III.16) of~\cite{GR}. Here again, it is not
difficult to show that also for these transformations of terminating
balanced ${_4\phi_3}$ series, the invariance group is the symmetric
group $S_6$.

\begin{prop}
Let $x_1,\cdots,x_6$ be six parameters satisfying
\beq
x_1x_2x_3x_4x_5x_6=q^{1-n}
\eeq
for some non-negative integer $n$. Then the function
\bea
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 \nn\\
&\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]
\label{term43}
\eea
is symmetric in the variables $(x_1,x_2,x_3,x_4,x_5,x_6)$.
\end{prop}

\noindent {\em Proof.} Again $f(x)$ is obviously invariant under
permutations of $(x_1,x_2,x_3)$ (resp.\ $(x_4,x_5,x_6)$).
Next, consider the cyclic permutation
$p=(1\, 6\, 4\, 2\, 5\, 3)$, which is a
permutation of order~6. Upon relabelling the 
parameters of the ${_4\phi_3}$ in $f(x)$ by
\[
{_4\phi_3}\left[{ q^{-n},\ a,\ b,\ c \atop d,\ e,\ f} ; q, q \right],
\]
it is easy to see that equation (III.16) of~\cite{GR} is equivalent to
$f(x)=f(p\cdot x)$. A similar argument as in the previous proof then
implies the current proposition, i.e.\ the invariance group is $S_6$.
\mybox 

Note that in this context equation (III.15) of~\cite{GR} corresponds to
$f(x)=f(p\cdot x)$ with $p=(1\, 4)\,(2\, 3)\,(5\, 6)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Heine's transformations of ${}_2\phi_1$ series}

Heine~\cite{Heine} showed that
\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
see~\cite[(III.1)--(III.3)]{GR} for this and similar transformations.
In (\ref{heine}), $|z|<1$ and $|a|<1$ is required for convergence. When
iterating this transformation, using the symmetry of the numerator
parameters of the ${}_2\phi_1$, a set of 12 transformation formulas is
obtained. This observation is not new, and can already be found
in~\cite[section~11]{Sears}. However, the identification of the
invariance group behind these 12 transformations has never been made.
Here, we shall show that the invariance group is the dihedral group
$D_{12}$ (sometimes also denoted by $D_6$), 
i.e.\ the group of symmetries of the hexagon. In terms of a
hexagon with sides labelled $x_1,\cdots,x_6$, the group consists of all
rotations and reflections that map the hexagon into itself. The
hexagon and two of its symmetries are shown in the following picture~:
\beq
\vbox{
\unitlength=1mm
\special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(115.00,143.00)
\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
}
\label{pic1}
\eeq
The second hexagon is obtained from the first one by a reflection about
the vertical axis; the third one is obtained from the first one by a
rotation through angle $\pi/3$. 
In all, there are 12 such symmetries. The
group $D_{12}$, a subgroup of $S_6$, is generated by the two elements
described above. 

\begin{prop}
The function
\bea
f(x)&=&f(x_1,x_2,x_3,x_4,x_5)= ( x_1x_4,{x_2x_6\over x_1}; q )_\infty
\nn\\[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} )
\label{nonterm21}
\eea
is invariant under the dihedral group $D_{12}$ acting on the variables
$(x_1,x_2,x_3,x_4,x_5,x_6)$. 
\end{prop}

\noindent {\em Proof.} It is clear that the only effect of the
permutation $(2\, 6) (3\, 5)$ (in cycle notation), i.e.\ $x_2
\leftrightarrow x_6$ and $x_3 \leftrightarrow x_5$, on
(\ref{nonterm21}) is a transposition of the two numerator parameters in
the ${_2\phi_1}$. 
Next, consider the permutation $(1\, 2\, 3\, 4\, 5\, 6)$, or explicitly~:
\beq
p\cdot(x_1,x_2,x_3,x_4,x_5,x_6)=(x_2,x_3,x_4,x_5,x_6,x_1).
\eeq
Upon relabelling the parameters of the ${_2\phi_1}$ in $f(x)$ by
${_2\phi_1} (a, b;c;q,z)$, 
it is easy to see that (\ref{heine}) is equivalent to
$f(x)=f(p\cdot x)$. 
The two permutations considered here correspond to the two symmetries
of the hexagon described in~(\ref{pic1}), and they form a set of
generators for the group $D_{12}$. \mybox 

For convergence of all the 12 series in~(\ref{nonterm21}) it is
sufficient that all $x_i$ satisfy $\al^2 < |x_i| < \al$, with $\al<1$ 
a positive real number.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Transformations of terminating ${}_3\phi_2$ series}

Whipple~\cite{Whipple} not only showed that
there are 120 non-terminating ${}_3F_2$ series of unit argument, 
he also established that there are 72 terminating ${}_3F_2(1)$ series
(see also~\cite[\S 3.9]{Bailey}). In~\cite{Rao} the 72-element group
associated with these series transformations was examined. The
$q$-analogues of these transformation formulas were given by
Sears~\cite{Sears}. The essential transformations are summarized
in~\cite[(III.11)--(III.13)]{GR}, and others can be obtained from these
by iteration. For example, from (III.11) and (III.13) one can deduce
that 
\beq
{}_3\phi_2\left[ {q^{-n},\ b,\ c \atop d,\ e};q,q \right] = 
{(c,de/bc;q)_n \over (d,e;q)_n} (b)^n 
{}_3\phi_2\left[ {q^{-n},\ d/c,\ e/c \atop de/bc,\ q^{1-n}/c};q,{q
\over b} \right].
\label{tf32}
\eeq

In the present case the group $G$ associated with the transformations is
somewhat more difficult to describe. By Cayley's and Lagrange's 
theorems~\cite{Budden}, it is obvious that the 72-element group $G$ 
cannot be a subgroup of $S_5$ (since 72 is not a factor of 120), but it
is a subgroup of $S_6$
generated by the two elements $(2\, 4)$ and 
$(1\, 2\, 3\, 4\, 5\, 6)$. 
One way to see that the subgroup generated by these two elements has
order~72 is by 
extending the 12 classical
symmetries of the hexagon described in the previous section. The new
transformations that are allowed on the hexagon are the interchange of
two sides that are next to nearest neighbours, such as $x_1
\leftrightarrow x_3$, $x_2 \leftrightarrow x_4$, $x_3 \leftrightarrow
x_5$, etc. Superposed on the 12 classical symmetries, this enlarges
the total number of allowed transformations to $12\times 6= 72$. 
The two generating elements of $G$, $(2\, 4)$ and $(1\, 2\, 3\, 4\, 5\,
6)$, are described by the second and third hexagons in the following
picture~: 
\beq
\vbox{
\unitlength=1mm
\special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(115.00,143.00)
\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]{$x_4$}}
\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]{$x_3$}}
\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]{$x_2$}}
\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]{$x_5$}}
\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]{$x_6$}}
\put(95.00,135.00){\makebox(0,0)[cc]{$x_1$}}
\end{picture}
\vskip -120mm
}
\label{pic2}
\eeq
Another way to describe the elements of $G$ is by calling $x_2$, $x_4$
and $x_6$ the even labels (or even sides), and $x_1$, $x_3$
and $x_5$ the odd labels (or odd sides) of the hexagon. Then the
elements of $G$ consist of the symmetries of the hexagon on which
permutations of even labels (resp.\ odd labels) are superposed. As a
consequence, $G$ has a subgroup $H$ consisting of permutations of
$(x_1, x_3, x_5)$ and of $(x_2, x_4, x_6)$. Thus $H$ is isomorphic to
$S_3\times S_3$ and its order is 36.

The description of the invariance itself is also slightly more
complicated in this case, mainly because the argument $z$ of the
${}_3\phi_2$ is equal to $q$ only in 36 cases, and different from $q$
in the other 36 cases. 

\begin{prop}
Let $x_1,\cdots,x_6$ be six parameters satisfying
\beq
x_1x_2x_3x_4x_5x_6=q^{1-n}
\eeq
for some non-negative integer $n$. Consider the functions
\bea
f_0(x)&=&f_0(x_1,x_2,x_3,x_4,x_5,x_6)= 
(x_1x_2x_3x_4, x_1x_2x_4x_5 ; q )_n /(x_1x_2x_4)^n \nn\\
&\times& {_3\phi_2}\left[{q^{-n},\  x_1x_2,\  x_1x_4\atop 
x_1x_2x_3x_4,\ x_1x_2x_4x_5} ; q, q \right] ,
\label{term32-0}
\eea
\bea
f_1(x)&=&f_1(x_1,x_2,x_3,x_4,x_5,x_6)= (-1)^n q^{\left( n \atop 2\right)}
(x_1x_2x_3x_4, x_1x_2x_4x_5 ; q )_n (x_6)^n \nn\\
&\times& {_3\phi_2}\left[{ q^{-n},\  x_1x_2,\  x_1x_4\atop 
x_1x_2x_3x_4,\ x_1x_2x_4x_5} ; q, {q\over x_1x_6} \right] .
\label{term32-1}
\eea
Then $f_i(x)=f_i(p\cdot x)$ if $p\in H$ ($i=0,1$), and
$f_0(x)=f_1(p\cdot x)$ (or $f_1(x)=f_0(p\cdot x)$) if $p\in G-H$. 
\end{prop}

\noindent {\em Proof.} The permutation $(2\, 4)$ (in cycle notation)
only transposes two of the numerator parameters in
the ${_3\phi_2}$ of $f_0(x)$ or $f_1(x)$. 
Next, consider the permutation $(1\, 2\, 3\, 4\, 5\, 6)$, or explicitly~:
\beq
p\cdot(x_1,x_2,x_3,x_4,x_5,x_6)=(x_2,x_3,x_4,x_5,x_6,x_1).
\eeq
This is an element of $G-H$, and it is straightforward to verify that
(\ref{tf32}) is equivalent to
$f_0(x)=f_1(p\cdot x)$. 
The two permutations considered here generate $G$, hence $G$ is the
invariance group of the terminating ${_3\phi_2}$ series transformations.
\mybox 

Consider the permutations (in cycle notation)
$p_1=(1\, 3)(2\, 4)$ and $p_2=(1\, 4\, 3\, 2)(5\, 6)$, 
thus $p_1\in H$ and $p_2\in G-H$. 
As an extra
verification, one can check that (III.11), (III.12) and (III.13)
of~\cite{GR} correspond to $f_0(x)=f_0(p_1 \cdot x)$, $f_0(x)= f_1(p_2
\cdot x)$ and $f_1(x)=f_0(p_2 \cdot x)$ respectively.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Transformations of very-well-poised ${}_8\phi_7$ series}

The last type of transformations considered here are for
very-well-poised ${}_8\phi_7$ series. These transformations can be 
obtained as a limiting case of Bailey's terminating ${}_{10}\phi_9$
series transformations~\cite{Bailey2}, see
also~\cite[(III.23),(III.24)]{GR}. Such transformations are the
$q$-analogue of transformations of ``unrestricted well-poised $_7F_6$
series'' of unit argument, see~\cite[\S 7.5]{Bailey}. 

For our purposes, we define
\bea
&&w(a;b,c,d,e,f)={(aq/b,aq/c,aq/d,aq/e,aq/f,a^2q^2/bcdef;q)_\infty
\over (aq;q)_\infty } \label{defw}\\[1mm]
&& \times{}_8\phi_7\left[ \begin{tabular}{llllllll}
$a$,& $q\sqrt{a}$,& $-q\sqrt{a}$,& $b$, & $c$, & $d$, & $e$, & $f$ \\
  &$\sqrt{a}$,& $-\sqrt{a}$,&$aq/b$,&$aq/c$,&$aq/d$,&$aq/e$,&$aq/f$\\
\end{tabular} ;q,{a^2q^2 \over{bcdef}} \right]. \nn
\eea
Clearly, $w$ is invariant under permutations of $(b,c,d,e,f)$, and
hence $S_5$ is going to be a subgroup of the invariance group. In order
to find the complete invariance group, we make again a relabelling as
follows~: 
\bea
&&f(x_0,x_1,x_2,x_3,x_4,x_5)=w(x_0^3x_1x_2x_3x_4x_5/q;\,
{x_0x_2x_3x_4x_5 \over x_1}, {x_0x_1x_3x_4x_5 \over x_2}, \nn\\ 
&&\qquad\qquad {x_0x_1x_2x_4x_5 \over x_3}, {x_0x_1x_2x_3x_5 \over x_4}, 
{x_0x_1x_2x_3x_4 \over x_5}).
\eea
Thus permutations of $(b,c,d,e,f)$ in $w$ correspond to permutations
of $(x_1,x_2,x_3,x_4,x_5)$ in $f$. On examining the transformations
given in~\cite{GR}, one finds that (III.23) is equivalent to
\beq
f(x_0,x_1,x_2,x_3,x_4,x_5)=f(x_0,x_1,x_2,x_3,1/x_5,1/x_4),
\label{p23}
\eeq
and that (III.24) is equivalent to
\beq
f(x_0,x_1,x_2,x_3,x_4,x_5)=f(x_0,1/x_2,1/x_3,1/x_4,1/x_5,x_1).
\label{p24}
\eeq
This implies that the invariance group will be a subgroup of the group
$WB_5$ of signed permutations on 5 letters. In general, the group $WB_n$
of signed permutations on $n$ letters is the semidirect product of the
symmetric group $S_n$ operating as permutations on a standard basis
$e_i$ and $({\Z}_2)^n$ operating as $e_i \rightarrow \pm e_i$; so its
order is $n!2^n$. $WB_n$ can also be seen as the group of $n\times n$
signed permutation matrices (cfr.\ Section~1, also for the action of
such a signed permutation on the array $(x_1,x_2,x_3,x_4,x_5)$).
Since the invariance group is
generated by the elements of $S_5$ and the elements corresponding to
(\ref{p23}) and (\ref{p24}), we are dealing with a subgroup $WD_5$ of
$WB_5$ consisting of those elements with an even number of $(-1)$'s in
their matrix representation. Therefore the order of $G$ is $5!2^4 =
1920$. We have used the notation $WB_n$ and $WD_n$ because $WB_n$ is
the Weyl group of a root system of type $B_n$ (or $C_n$), and $WD_n$ is
the Weyl group of a root system of type $D_n$~(see, e.g.\
\cite{Humphreys}). 

Thus we have the following result~:
\begin{prop}
The function $f(x_0,x)=f(x_0,x_1,x_2,x_3,x_4,x_5)$ satisfies $f(x_0,x)=
f(x_0,p\cdot x)$ for every element $p$ of $WB_5$ that has an even number
of minus signs in its matrix representation. Hence the invariance group
of the very-well-poised $_8\phi_7$ transformations is the group $WD_5$.
\end{prop}
For convergence of all the 1920 series it is sufficient that all $x_i$
($i=1, \ldots 5$) satisfy $1/\al < |x_i| < \al$ and $|x_0|<1/\al^5$ for
some real number $\al>1$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Remarks}

The seminal remark of Hardy in his {\it Notes on Lecture VII}~\cite{Hardy}:

\leftskip 1cm \rightskip 1 cm

\noindent ``Formula (7.3.3) is equivalent to (1), \S 3.2, of Bailey's tract.
It is an expression of the theorem that 
$${1\over \Gamma(\beta_1)\Gamma(\beta_2)\Gamma(\beta_1+\beta_2-\alpha_1-\alpha_2
-\alpha_3)} F\left( \begin{array}{c}
                   \alpha_1,\alpha_2,\alpha_3\\
                   \beta_1,\beta_2 \end{array} \right)$$
is a symmetric function of the five arguments\\
$$\beta_1, \beta_2, \beta_1+\beta_2-\alpha_2-\alpha_3, 
\beta_1+\beta_2-\alpha_3-\alpha_1, \beta_1+\beta_2-\alpha_1-\alpha_2."$$

\leftskip 0 cm \rightskip 0 cm

\noindent clearly implies a group theoretical interpretation for formula 
(7.3.3), which is a $_3F_2(1)$ transformation of Thomae~\cite{Thomae}:

$${\Gamma(x+y+s+1)\over {\Gamma(x+s+1)\Gamma(y+s+1)}}
  F\left( \begin{array}{c}
           -a,-b,x+y+s+1\\
           x+s+1,y+s+1 \end{array} \right) = $$
$$ = {\Gamma(a+b+s+1)\over {\Gamma(a+s+1)\Gamma(b+s+1)}}
  F\left( \begin{array}{c}
           -x,-y,a+b+s+1\\
           a+s+1,b+s+1 \end{array} \right).$$

Forty seven years later, Beyer, Louck and Stein~\cite{BLS} rediscovered 
that Thomae's identity between two $_3F_2(1)$ hypergeometric series 
of unit argument, together with the trivial invariance under separate 
permutations of the numerator and denominator parameters implies the
symmetry group $S_5$ is an invariance group of this series. They also showed 
that Bailey's identity for Saalsch\"utzian $_4F_3(1)$ series has
$S_6$ as its invariance group. Srinivasa Rao {\it et al}~\cite{Rao} studied
the group theory of terminating $_3F_2(1)$ series --- a case not 
considered by Beyer {\it et al}~\cite{BLS} --- and found all the invariant 
subgroups of the 72-element group.

In this article, taking the cue from Hardy's remark, the group theoretical 
basis of well-known basic hypergeometric transformations has been studied,
for the first time. 
We have constructed explicit functions $f(x)$, where $x$ is a 
multi-dimensional parameter, expressed in 
terms of a basic hypergeometric series for the given transformation, 
and established the appropriate symmetry group. 
The transformations are then given by~:
\beq
f(x) = f(p\cdot x)
\label{eq18}
\eeq
where $p$ is any element of the symmetry group. 
The constructed functions (\ref{nonterm32}), (\ref{term43}) and (\ref{nonterm21}) together with 
(\ref{eq18}) are succint, quintessential one-line statements for  
Sears' non-terminating $_3\phi_2$, Sears' terminating balanced
$_4\phi_3$ and
Heine's $_2\phi_1$ transformations of basic hypergeometric series, 
respectively, which have as their corresponding invariance groups~:
the symmetric groups $S_5$, $S_6$ and the dihedral group $D_{12}$.

In the case of the Sears' terminating $_3\phi_2$ transformations,
the 72-element invariance group $G$ (being a subgroup of $S_6$), a
36-element subgroup $H$ of $G$, and 
two functions (instead of one), $f_0(x)$ and $f_1(x)$ were necessary
to give a complete description.
In terms of these all the transformations were generated by~:
$$ f_i(x) = f_j(p\cdot x), \qquad (i,j=0,1)  $$
where $i=j$ if $p\in H$ and $i\neq j$ if $p\in G-H$.

Finally, in the case of transformations of very-well-poised $_8\phi_7$
series, though the constructed function $f(x)$ is a function of six 
parameters, the invariance group is a subgroup $WD_5$ of $WB_5$, which is 
the Weyl group of a root system of type $B_n$ (or, equivalently, the 
group of $n\times n$ signed permutation matrices).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}

One of us (KSR) wishes to thank Professor G.\ Vanden Berghe and the Department of Applied Mathematics and Computer Science for excellent
hospitality during the author's stay when most of this work was done.

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


