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\newcommand{\sixx}[6]{\left\{ \begin{array}{lll}
                    #1 & #2 & #3 \\
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				   	 a & b & e \\
					 d & c & f 
				   \end{array} \right\}}
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        			   #4,\ #5  
                                  \end{array}\kern-5pt;#6\right)} 
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                                  \end{array}\kern-5pt ;#8\right)} 
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\begin{document}

\newbox\Adr
\setbox\Adr\vbox{
\centerline{\small\uppercase{ K. Srinivasa Rao and C. Krattenthaler}}
\vskip12pt
\centerline{The Institute of Mathematical Sciences, Chennai-600 113, India}
\centerline{E-mail: {\tt\footnotesize rao@imsc.res.in}}
\centerline{and}
\centerline{Institut Girard Desargues, Universit\'e Claude Bernard Lyon-I,}
\centerline{21, avenue Claude Bernard, F-69622 Villeurbanne, Cedex.}
\centerline{E-mail: {\tt\footnotesize kratt@igd.univ-lyon1.fr}}
\centerline{WWW: \footnotesize\tt http://igd.univ-lyon1.fr/\~{}kratt}}

\title[Group Theoretical aspects of Hypergeometric 
Functions]{\large On Group Theoretical aspects
%, Hypergeometric Transformations 
and Symmetries of Angular Momentum Coefficients}

\author{\box\Adr}

\subjclass[2000]{Primary 33C20;
 Secondary 33C05 33C70} 
\keywords{beta integral,
generalized hypergeometric series, Kamp\'e de F\'eriet
functions}


\begin{abstract} In this article we present a review of the work on 
Group theory of ordinary and basic Hypergeometric transformations. We show 
that an application of the transformations of ordinary hypergeometric series 
relates the 6-$j$ coefficient to sets of 
very-well-poised ${}_7F_6(1)$ series.
\end{abstract}

\maketitle

\section{Introduction}


\ni The fact that the coupling / recoupling coefficients of Quantum Theory 
of Angular Momentum (QTAM) were related to the hypergeometric functions 
\cite{rose}, \cite{smoro-shelep} and 
the use  of a balanced (or Saalsch\"utzian) transformation formula to 
obtain what was claimed to be a {\it new} symmetry \cite{minton} for the 
Racah (or 6-$j$) coefficient was the starting point for 
a study of the symmetries of these angular momentum coefficients in 
terms of generalized hypergeometric functions of unit argument. 
Early studies \cite{ksrbook} in this area of QTAM established 
that if the 3-$j$ coefficient is to be written as a ${}_3F_2(1)$, a set of six 
${}_3F_2(1)$s is necessary and sufficient to account for the 72
symmetries of the 3-$j$ coefficient \cite{ksr1}. In the case of the 
6-$j$ coefficient it was shown that there 
exists a set of three ${}_4F_3(1)$s
\cite{ksr2} and an equivalent set of four ${}_4F_3(1)$s \cite{ksr3}.
These two sets were shown to be related to each other through the {\it 
reversal} of hypergeometric series \cite{slater} and this completed an
understanding of the symmetries of the 6-$j$ coefficient in terms of the 
sets of ${}_4F_3(1)$s demonstrating a confirmation of Askey's remark 
\cite{askey} that the 3-$j$ and 6-$j$ coefficients that arise in QTAM are 
{\it hypergeometric functions and many of their elementary properties are best understood
when considered as such.} \\

\ni Wu observed \cite{wu1} that while the 3-$j$ and the 6-$j$ coefficient 
can be related to a $_3F_2(1)$ and $_4F_3(1)$, respectively, the 9-$j$
coefficient cannot be related to a $_7F_6(1)$ 
but, that it is related to 
a {\it new} hypergeometric function \cite{wu2}, 
was yet another starting 
point. This led to the simplest 
known formula for the 9-$j$ coefficient, a triple sum series
due to Ali\v sauskas, 
Jucys and Bandzaitis \cite{alijucband}  being 
related \cite{ksr4}
to a triple hypergeometric series, with unit arguments. 
From a novel way of looking at the symmetries of the 9-$j$ coefficient,
the Bailey transform for a Saalsch\"utzian 
$_4F_3(1)$ and a transformation of a Kamp\'e de F\'eriet 
function into a Saalsch\"utzian $_4F_3(1)$ or its Bailey 
transform were derived \cite{ksr5}.\\

\ni The above results were reviewed in an earlier contribution to the
{\it Symmetries in Science} series symposium \cite{ksr6}. Section 2 reviews 
the Group theoretical aspects of hypergeometric transformations and the 
24 Kummer solutions of the Gau{\ss} second order ordinary differential
equation being related to the symmetries of the cube. 
Section 4 relates the ${}_7F_6(1)$ forms for the 6-$j$ coefficient with the 
sets of ${}_4F_3(1)$s using a Whipple transformation and presents an 
understanding of the symmetries of the 6-$j$ coefficient in terms of 
these sets.
It is interesting to note that, recently, the symmetry groups of
hypergeometric series gained considerable attention in irrationality
studies of values of the Riemann zeta function at integers (see
\cite{rhvi1,rhvi2,zud}). To be more precise, what is (independently) 
investigated (and
applied) there (in disguise) is the symmetry group of the $_3F_2$
series and of the very-well-poised $_7F_6$ series, respectively.\\

\def\po#1#2{(#1)_#2}
\def\poq#1#2{(#1;q)_#2}
\ni
The hypergeometric notations that we shall use throughout the paper are
$${}_2F_1(a,b;c;x)=
\sum _{k=0} ^{\infty}\frac {(a)_k\,(b)_k} {(c)_k\,k!}z^k$$
for the Gau{\ss} hypergeometric series $_2F_1$, where 
the Pochhammer symbol $(\alpha)_k$ is defined by
$(\alpha)_k  =  \alpha(\alpha+1)(\alpha+2)\cdots (\alpha+k-1)$,
$k>0$, $(\alpha)_0  = 1$,
and
\begin{equation*}  % \label{eq:hyper}
{}_r F_s\!\left(\begin{matrix} a_1,\dots,a_r\\ b_1,\dots,b_s\end{matrix}; 
z\right)=\sum _{k=0} ^{\infty}\frac {\po{a_1}{k}\cdots\po{a_r}{k}}
{\po{b_1}{k}\cdots\po{b_s}{k}\,k!} z^k\ 
\end{equation*}
for the generalized hypergeometric series $_rF_s$.

\ni
Similarly, the basic hypergeometric notations 
that we shall use are
$${}_2\phi_1(a,b;c;x)=
\sum _{k=0} ^{\infty}\frac {(a;q)_k\,(b;q)_k} {(c;q)_k\,(q;q)_k}z^k$$
for the Heine series $_2\phi_1$, where 
the shifted $q$-factorial $(\alpha;q)_k$ is defined by
$(\alpha;q)_k  =  (1-\alpha)(1-q\alpha)(1-q^2\alpha)\cdots (1-q^{k-1}\alpha)$,
$k>0$, $(\alpha;q)_0  = 1$,
and
\begin{equation*}  % \label{eq:hyper}
{}_r \phi_s\!\left(\begin{matrix} a_1,\dots,a_r\\ b_1,\dots,b_s\end{matrix}; 
q,z\right)=\sum _{k=0} ^{\infty}\frac {\poq{a_1}{k}\cdots\poq{a_r}{k}}
{\poq{b_1}{k}\cdots\poq{b_s}{k}\,(q;q)_k} z^k\ 
\end{equation*}
for the generalized basic hypergeometric series $_r\phi_s$.
 
\section{Group theoretical aspects of hypergeometric transformations}

\ni Pfaff's transformation (\cite{pfaff}, p.68) also referred to
as Saalsch\"utz's theorem \cite{saals}:
\be 
{}_2F_1(a,b;c;x)&=& (1-x)^{-a}\ 
{}_2F_1\left(a,c-b;c;\frac{x}{x-1}\right)
\ee
when iterated generates the well-known  Euler transformation formulae
\cite{slater}:
\be
{}_2F_1(a,b;c;x)&=& (1-x)^{-a}\ 
{}_2F_1\left(a,c-b;c;\frac{x}{x-1}\right)\\
&=& (1-x)^{-b}\ 
{}_2F_1\left(c-a,b;c;\frac{x}{x-1}\right)\\
&=& (1-x)^{c-a-b}\ {}_2F_1(c-a,c-b;c;x),
\ee
on each of which is superposed the trivial numerator parameter
permutation symmetry:
\be
{}_2F_1(a,b;c;x)={}_2F_1(b,a;c;x),
\ee
so that we have a set of eight transformations. If we denote
the Pfaff transformation by $g_1$, and the numerator parameter
permutation of the ${}_2F_1$ by $g_2$, then these two elementary
transformations satisfy the properties:
$$g_1^2=1,\qquad g_2^2=1, \qquad (g_1g_2)^4=1, $$
and the group generated by the two elements subjected to these
relations is the  dihedral group 
$\mathcal D_8$ (sometimes also denoted by $\mathcal D_4$, 
depending on the convention, and denoted
$I_2(4)$ in Coxeter group theory \cite{HumpAC}), also known as the group of 
symmetries of the square. It is a subgroup of the symmetric group $S_4$, 
the permutation group on four elements.\\

\ni  Erd\'elyi and Weber \cite{erdweb} stated that the 
recursive use of the Thomae \cite{thomae} transformation for a 
${}_3F_2(1)$ resulted in a new transformation. This was the starting point 
for a study of the group theory of transformations. Bayer, Louck and 
Stein \cite{bayer} showed that the group of transformations of the 
non-terminating ${}_3F_2(1)$ 
transformations is $S_5$ and that the
group of transformations of the terminating balanced (or Saalsch\"utzian)
${}_4F_3(1)$ series is 
$S_6$. The study of the group theory of the terminating ${}_3F_2(1)$
transformations, obtained by a recursive application of the terminating 
version of the Thomae transformation was done by Srinivasa Rao et al.
\cite{ksr7}. \\

\ni The remark of Hardy \cite{hardy} that the Thomae 
transformation for the non-terminating ${}_3F_2(1)$:
\be
\thrftw{a}{b}{c}{d}{e}{1} = \ds{\frac{\G(d,e,s)}{\G(a,s+b,s+c)}}\ 
\thrftw{d-a}{e-a}{s}{s+b}{s+c}{1} \label{thomae}
\ee
\noindent where $\G(x,y,\dots) =\G(x)\G(y)\cdots$ and $s = d+e-a-b-c$ 
is the parameter excess,

\leftskip 1cm \rightskip 1cm 

\ni {\it is an expression of the theorem that }
$$\frac{1}{\G(\beta_1)\G(\beta_2)\G(\beta_1+\beta_2-\alpha_1-\alpha_2-\alpha_3)}
F\left( \begin{array}{c} \alpha_1,\alpha_2,\alpha_3 \\ \beta_1,\beta_2
\end{array} \right) $$
{\it 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 0cm \rightskip 0cm 

\ni clearly implies a group theoretical interpretation for the
Thomae formula. As in the case of
the terminating $_3F_2$ series, a recursive use of this
transformation (\ref{thomae}) results in our obtaining a
set of {\bf ten} non-terminating Thomae transformations (\cite{ksr8},
Appendix 1). In \cite{ksr9}, the following function was constructed:
\be
\qquad f(x_1,x_2,x_3,x_4,x_5)\ =\ \frac{1}{\Gamma(s,2x_4,2x_5)}   
\ \  \thrftw{2x_1-s}{2x_2-s}{2x_3-s}{\quad 2x_4}{\quad 2x_5}{1}
\ee
where $s=x_1+x_2+x_3-x_4-x_5$. That this function is symmetric in all the
five variables is a consequence of the Thomae transformation for 
non-terminating $_3F_2(1)$ has been proved as follows:
the function $f(\vec{x})\equiv f(x_1,x_2,x_3,x_4,x_5) $
is manifestly invariant for permutations of $(x_1,x_2,x_3)$ and
$(x_4,x_5)$. Consider the permutation $p:\ x_1\rightarrow x_2
\rightarrow x_3 \rightarrow x_4 \rightarrow x_5\rightarrow x_1,$
which is a permutation of order 5. Upon relabeling the parameters
of the $_3F_2(1)$ in $f(\vec{x})$ as $\thrftw{a}{b}{c}{d}{e}{1}$, it is
straight forward to see that corresponding to
\be
f(\vec{x})\ =\ f(p.\vec{x})
\ee
we get:
\be
\thrftw{a}{b}{c}{d}{e}{1}=\ds{\frac{\G(d,s)}{\G(d-a,s-a)}}
\ \thrftw{e-c}{e-b}{a}{e}{s+a}{1},
\ee
\ni belonging to the set of ten Thomae transformations. 
Since $f(\vec{x})$ is invariant 
under this permutation $p$ of order 5, and under the 
transposition, 
$x_4\leftrightarrow x_5$, manifestly, the group 
generated by these two generators (viz. $p$ and the 
transposition) is the complete group of permutations on 5 
elements, i.e. the symmetric group $S_5$. This result is,
obviously, {\it a succinct, quintessential one-line statement}
for the group of Thomae transformations.\\

\ni
From the transformation for the terminating ${}_3F_2$ series:
\be
\thrftw{a}{b}{-n}{d}{e}{1}= \frac{\G(d,d+n-a)}{\G(d+n,d-a}
            \thrftw{a}{e-b}{-n}{a+a-d-n}{e}{1},
\ee
in \cite{ksr7}, the 72-element group $G_T$ of transformations was
generated, and its conjugacy classes, irreducible representations 
and their characters,
as well as the invariant subgroups of $G_T$ were derived, and the
role of these terminating series for the ${}_3F_2(1)$ forms of the
3-$j$ coefficient were discussed.\\

\ni The two-term relation for the terminating balanced (or Saalsch\"{u}tzian)
$_4F_3$ of unit argument is:
\be
\fft{A}{B}{C}{E}{F}{G} & =& {\pa{F-C}{G-C}{F}{G}} \nn \\
& &\times  \fft{E-A}{\quad
E-B}{\quad C\quad }{E}{E+F-A-B}{E+G-A-B} .\label{bailey}
\ee
%where the Pochhammer symbol is defined for $n\geq 1$ as:
%\be
%(a)_n = {\frac{\Gamma (a+n)}{\Gamma (a)}} =
%a (a+1) (a+2) \cdots (a+n-1), \quad {\rm with}\quad  (a)_0 = 1.
%\ee

\ni A recursive use of this terminating $_4F_3$ transformation (\ref{bailey})
results in a set of {\bf twenty} transformations (\cite{ksr8}, Appendix 2). In this
case, the function:
\be 
f(x_1,x_2,x_3,x_4,x_5,x_6)\, = \, (x_1+x_2+x_3+x_4, x_1+x_2+x_3+x_5,
x_1+x_2+x_3+x_6)_n \nn \\
\times \fft{x_1+x_2}{x_2+x_3}{x_3+x_1}{x_1+x_2+x_3+x_4}{x_1+x_2+x_3+x_5}
{x_1+x_2+x_3+x_6}
\ee
with $(x,y,\dots)_n = (x)_n(y)_n\cdots $ and 
$x_1+x_2+x_3+x_4+x_5+x_6= 1-n$, for some non-negative integer $n$,
being symmetric in all the six variables
$x_1,x_2,x_3,x_4,x_5,x_6$ is a consequence of the 
${}_4F_3(1)$ transformation.\\

\ni Hardy's clue enabled us \cite{leivens} to look for the 
invariance groups for all the known transformations of basic 
hypergeometric series. A summary of these results is presented 
below:  \\

\ni The Heine transformation, 
\be
{}_2\phi_1(a,b;c;q,x) = \frac{(a,bz;q)_\infty}{(c,z;q)_\infty}
{}_2\phi_1(c/a,z;bz;q,a),
\ee
when iterated yields a set of 12 transformations \cite{sears}.
The invariance group is the dihedral group 
$\mathcal D_{12}$ -- the group of 
symmetries of the hexagon 
(sometimes also denoted as $\mathcal D_6$ or $I_2(6)$). We
construct the function:
\be
\qquad f(x_1,x_2,x_3,x_4,x_5,x_6)=\left(x_1x_4,\frac{x_2x_6}{x_1};q\right)_\infty
{}_2\phi_1\left(\frac{x_1x_3}{x_2},\frac{x_1x_5}{x_6};x_1x_4;q,
\frac{x_2x_6}{x_1}\right),
\ee 
whose invariance under the dihedral group acting on the six 
variables $x_1,\dots, x_6$ generates the set of 12 Heine 
transformations.\\

  
\ni The function:
\be 
f(x)&=&f(x_1,x_2,x_3,x_4,x_5)  \\
   &=& \left( 
{x_1x_2x_3}/{x_4x_5},x_4^2,x_5^2;q\right)_\infty
\nn\\
&&
\kern1cm\times{}_3\phi_2\left( \begin{tabular}{c}
$
{x_1x_4x_5}/{x_2x_3},
{x_2x_4x_5}/{x_1x_3},
{x_3x_4x_5}/{x_1x_2}$ \\
$x_4^2,x_5^2$\\
\end{tabular}
;q,\frac{x_1x_2x_3}{x_4x_5}\right),\nn
\ee
is symmetric in the five variables $x_1,\dots,x_5$. 
Let $p$ be the cyclic permutation $p=(14325)$, that is,
the permutation 
$x_1\rightarrow x_4\rightarrow x_3\rightarrow x_2\rightarrow x_5\rightarrow x_1$:
\be 
p.(x_1,x_2,x_3,x_4,x_5)=(x_4,x_5,x_2,x_3,x_1).
\ee
On relabeling the parameters of the ${}_3\phi_2$s for the 
equation: $f(x)=f(p.x)$, where $x$ is the five component vector,
we get the transformation:
\be 
{}_3\phi_2\left( \begin{tabular}{c}
$a,b,c$ \\ $d,e$ \\
\end{tabular};q,\frac{de}{abc}\right) =
\left(b,
{de}/{ab},{de}/{ac}
\right)_\infty 
\ {}_3\phi_2\left( \begin{tabular}{c}
$
{d}/{b},{e}/{b}
,s$ \\ 
$
{de}/{ab},{de}/{ac}
$ \\
\end{tabular};q,b\right),
\ee
which is the $q$- analogue of the non-terminating Thomae
transformation. The defined function is manifestly invariant
under the permutations $(x_1,x_2,x_3)$ and $(x_4,x_5)$. $f(x)$
is also invariant under the tansposition $x_4\leftrightarrow x_5$,
as well as the 
above cyclic permutation $p$ of order 5. Hence, the group generated 
by $p$ and the transposition is $S_5$ which is thus the invariance
group of the $q$-analogue of the Thomae non-terminating transformation.\\

\ni Let $x_1,\dots,x_6$ satisfy the condition:
$ x_1x_2x_3x_4x_5x_6 = q^{1-n}$ for some non-negative integer $n$.
Then the function:
\be
f(x)&=&f(x_1,x_2,x_3,x_4,x_5,x_6) \nn \\
&=& q^{n(n-1)/2} (x_1x_2x_3x_4,x_1x_2x_3x_5,x_1x_2x_3x_6;q)_n\frac{1}{(x_1x_2x_3)^n} \\
& & \times {}_4\phi_3\left( \begin{tabular}{c}
$q^{-n},x_2x_3,x_1x_3,x_1x_2$\\
$x_1x_2x_3x_4,x_1x_2x_3x_5,x_1x_2x_3x_6$\nn \\
\end{tabular};q,q\right),
\ee
being symmetric in the variables $x_1,\dots,x_6$ is a manifestation
of the $q$-analogue of the terminating balanced (Saalsch\"utzian) 
${}_4F_3(1)$ transformation. The invariance group is $S_6$ and 
the proof is along the same lines as that given above.\\

\ni The above are the $q$-analogues of the results \cite{ksr9} of Bayer, Louck 
and Stein \cite{bayer} for ordinary hypergeometric series transformations.\\

\ni For terminating ${}_3\phi_2$ series, the $q$-analogue of the 
Whipple transformation obtained by Sears \cite{sears} is:
\be 
{}_3\phi_2\left( \begin{tabular}{c}
$q^{-n},b,c$ \\ $d,e$ \\
\end{tabular};q,q\right) &=&  
b^n\, 
\frac{\left(c,
{de}/{bc};q\right)_n}
{(d,e;q)_n}\ 
{}_3\phi_2\left( \begin{tabular}{c}
$q^{-n}, 
{d}/{c},{b}/{c}
$ \\ 
$ 
{de}/{bc},{q^{1-n}}/{c}
$ \\
\end{tabular};q,\frac{q}{b}\right),
\ee 
which when recursively applied yields 72 transformations, as in 
the case of the terminating ${}_3F_2(1)$ investigated in \cite{ksr8}.
This 72-element group cannot be a subgroup of $S_5$ by Cayley's
and Lagrange's theorems (since 72 is not a factor of 120). Thus,
it is shown in \cite{ksr9} that the invariance group of the  
terminating ${}_3\phi_2$ series transformations is a 72-element 
subgroup of $S_6$ generated by the permutations: $(24)$ and $(123456)$.\\

\ni It was also shown in \cite{ksr9} that the transformations of the 
very-well-poised ${}_8\phi_7$ series belong to the invariance group (of
order 1920), which is a subgroup of signed permutations 
on 5 elements,   
coincides with the Weyl group of 
the root system of type $D_5$ (see \cite{HumpAC}).\\

\ni In 1836, Kummer published a set of six distinct solutions for the 
second oder ordinary 
Gau{\ss} differential equation characterized by three
regular singular points. Each of these six solutions has four forms 
which are related to each other by Euler's transformations (2)--(4).
Thus, there are in all 24 solutions of the 
Gau{\ss} hypergeometric equation
which can be found in most classical text books \cite{whitwat}. Recently,
Posser \cite{posser} related these solutions of Kummer to one another 
through a finite group (of order 24, or 48 if one includes the 
mirror symmetries) of transformations. It is remarkable that these
24 solutions have been related \cite{leivens} to the (rotational) group of 
symmetries of the cube. This is achieved through the function:
\be
\qquad f(x)= 
{}_2F_1\left(\frac{1}{2}+x_1+x_2+x_3,\frac{1}{2}+x_1+x_2+x_4;
\ \frac{1}{2}+x_1-x_6;\ -\frac{x_1+x_6}{x_3+x_4}\right),
\ee
where the six variables $x_1,\dots, x_6$ (identified with the six faces
of a cube labeled as $1,\dots,6$  such that labels of the opposite faces
of the cube add to 7) satisfy the constraint:
$$\sum_{i=1}^6 x_i = 0. $$
The symmetries of the cube can then be associated with the 24 permutations 
of a group $ G$ (a subgroup of the 720 permutations 
belonging to $S_6$)
of the indices labeling its faces. Identifying the four arguments of the 
above ${}_2F_1$ with $a,b,c,z$, if we denote by $g\ \epsilon \ G$ one of these 
symmetries,
then $f(g.x)$ will correspond to one of the Kummer solutions.
In \cite{leivens} it is shown that the 24 complete Kummer solutions
(up to a constant) can be related to the permutations which 
generate the symmetries of the cube and that the mirror symmetries, or the 
reflection symmetries of the cube, are the ones which correspond to the 
interchange of the numerator parameters of the ${}_2F_1$ function.
The intimate connection between the symmetries of
the cube and the 24 Kummer solutions of the 
Gau{\ss} differential 
equation, is an aesthetically beautiful result discovered 90 years after
the discovery of the 
Gau{\ss} equation and 66 years after Kummer established 
its complete set of solutions.
      
\section {6-$j$ coefficient in terms of sets of very-well-poised ${}_7F_6$
series}

\ni In \cite{ksr10}, it was shown that the highly symmetric form of the 
Racah or 6-$j$ angular momentum recoupling coefficient, due to 
Regge \cite{regge}, which exhibits the 144 symmetries of the 
coefficient, can be formally cast into a $_5F_4(1)$ form. Though the
${}_5F_4(1)$ exhibits the 144 symmetries, it has the property that 
for real integral and half integral values of the angular momenta,
the numerator and denominator parameters are integers. The 
termination of the series, which occurs due to the numerator 
parameter being a negative integer, unfortunately, occurs 
after the zero due to the denominator parameter! Hence, this formal
expansion represents a divergent series and is not useful.\\

\ni The claim that a new symmetry was found by Minton \cite{bmm} for the 
6-$j$ coefficient which did not satisfy even the triangle inequalities
\cite{vinay} led us to the set I \cite{ksr2} of $_4F_3(1)$s for the 
6-$j$ coefficient. Racah's \cite{racah1} achievment 
was to show that the recoupling
coefficient for three angular momenta, $W(abcd;ef)$,  can be written 
as a single sum series (independent of the 3-$j$ coefficients and hence of
the projection quantum numbers of angular momenta), viz: 
\be
\sixj & = & (-1)^{a+b+c+d}\ W(abcd;ef)\  \label{1} \\
 &=& N \sum_P (-1)^P\ (P + 1)!\ \{\prod_{i=1}^4(P - \alpha_i)!
\prod_{j=1}^3(P - \beta_j)!\}^{-1}, \label{2}
\ee
with
\be
& N = (-1)^{a+b+c+d} \bp(a,b,e)\ \bp(c,d,e)\ \bp(a,c,f)\ \bp(b,d,f), &
 \label{3} \\
&\bp(x,y,z)={\ds \left[ \frac{(-x+y+z)!(x-y+z)!(x+y-z)!}{(x+y+z+1)!}\right]^{1/2}},&\\
& \ax_1 = a+b+e,\ \ax_2 = c+d+e,\ \ax_3 = a+c+f,\ \ax_4 = b+d+f, &
 \nonumber \\
& \ba_1 = a+b+c+d,\ \ba_2 = a+d+e+f,\ \ba_3 = b+c+e+f &
 \label{4}
\ee
and
\be
& P_{min}\ \leq \ P\ \leq \ P_{max}, &
\label{5} \\
& P_{min}\ =\ max(\ax_1,\ax_2,\ax_3,\ax_4),\ \ \  P_{max}\ =
\ min(\ba_1,\ba_2,\ba_3).& 
 \label{6}
\ee
By setting in (\ref{2}), $s=\ba_k-P, k=1,2,3$, in succession, a set I of 
three series expansions, and in turn, the set I of the following three
$_4F_3(1)$s has been obtained in \cite{ksr2}:
\be
\sixj & = & (-1)^{E+1} N\ \G(1-E) \nn \\
& & \times  \left[\G(1-A, 1-B, 1-C, 1-D, F, G)\right]^{-1} \\
& & \times _4F_3(ABCD\ ;\ EFG\ ;\ 1), \nn  \label{7}
\ee
 where
 \be
& A = e-a-b,\ B = e-c-d,\ C = f-a-c,\ D = f-b-d,& \\
&E = -a-b-c-d-1, \ F = e+f-b-c+1, \ G = e+f-a-d+1, & 
 \label{8} \nn
\ee
for $k=1$ and for  $k=2$ and 3, the numerator and denominator parameter sets are:
\be
& A = a-b-e,\ B = d-c-e, \ C = a-c-f, \ D = d-b-f, & \\
& E = -b-c-e-f-1,\ F = a+d-b-c+1, \ G = a+d-e-f+1, & \label{9}
\nn \ee
and 
\be
& A = b-a-e,\ B = c-d-e,\ C = c-a-f,\ D = b-d-f, &  \\
& E = -a-d-e-f-a,\ F = b+c-a-d+1, \ G = b+c-e-f+1. & \label{10}
\nn \ee
The Minton procedure was to apply a Saalsch\"utzian ${}_4F_3(1)$
transformation to the ${}_4F_3(1)$ in (\ref{7}) (with the parameters given by 
(\ref{9})) and identify the transformed ${}_4F_3(1)$ with a 6-$j$ coefficient
using again (\ref{7}). We have shown in \cite{ksr1} 
that this procedure will, {\it at best}, result in a $j\rightarrow 
-j-1$ substitution for one or more of the six angular momenta in the 
6-$j$ coefficient, which violates triangle 
inequalities \cite{vinay}, though it is a mathematically valid symmetry 
in quantum theory of angular momentum.  \\

\ni A hypergeometric series is called 
very-well-poised if the numerator
($a_i$) and denominator ($b_i$) parameters of the hypergeometric series
are such that:
$$ a_i+b_i = 1+a_0, \quad {\rm for}\quad 
i=1,2,\dots, n,$$
and amongst the parameters $a_i$ occurs $1+a_0/2$. 
The abbreviated notation for a 
very-well-poised hypergeometric series 
with argument 1, which we shall use 
here,\footnote{This notation for the 
very-well-poised ${}_7F_6$
as ${}_7V_6$ has been introduced by one of us 
CK in \cite{kratt}, akin to the 
notation ${}_{p+1}W_p$ used for 
very-well-poised basic hypergeometric series in
\cite{gasp-rah}.}
uses its numerator parameters only and it is:
\be 
{}_{r+1}V_r(a_0;\ a_2,a_3,\dots,a_r)\kern8cm   \\
\nn
= {}_{r+1}F_r
\left( \begin{array}{l}
a_0,1+\frac{a_0}{2},\qquad a_2, \qquad \ \quad \ a_3,\qquad \dots, \qquad a_r \\
\qquad \frac{a_0}{2},\quad 1+a_0-a_2,1+a_0-a_3,\dots,1+a_0-a_r
\end{array};1\right). 
\ee
The transformation of a balanced or Saalsch\"utzian ${}_4F_3(1)$ series into a 
very-well-poised ${}_7F_6(1)$ series, with the special form of the second parameter,
was given by Whipple \cite{whipple}:
\be
\ffta{x}{y}{z}{-n}{u}{v}{w}  =  \G \left(
\begin{array}{l}
v+w-x,1+y-u,1+z-u,1-n-u\\
1+y-n-u,1+z-n-u,1+y+z-u,1-u
\end{array} \right) \kern-1cm   \\
\nn
\times{}_7F_6 \left( \begin{array}{l}
a,1+\frac{a}{2},w-x,v-x,\qquad y,\qquad \qquad \quad z,\qquad \qquad \quad -n\\
\qquad \frac{a}{2},\qquad v,\qquad w, 1+z-u-n,1+y-u-n,1+y+z-u \end{array};1 \right),
\ee
in which $a=y+z-u-n=w+v-x-1$ (and $x+y+z-n+1=u+v+w$) and the ${}_7F_6(1)$ in
the expression above can be written in the shortened notation for the 
very-well-poised
hypergeometric series as:
\be
{}_7F_6(-;-;1) \equiv {}_7V_6(a;\ w-x,v-x,y,z,-n). 
\ee 

\ni Recently, 
Ali\v sauskas \cite{alisausk} has claimed that he has found
{\it a new expression for the 6-$j$ coefficient of SU(2)} and it is
(with minor changes in the notation):
\be
\sixj &=& \nabla(a,c,f) \nabla(b,d,f)\nabla(e,d,c)\bp(a,b,e) \nn \\
& & \kern-1cm\times \sum_j \frac{(-1)^{d+c+f-(a+b+e)/2+j}\ (2j+1)}
{\nabla^2(\frac{1}{2}(b+e-a),d,j) \nabla^2(\frac{1}{2}(a-b+e),c,j)    
\nabla^2(\frac{1}{2}(a+b-e),f,j)},
\ee
where
$$ \nabla(x,y,z)= \triangle(x,y,z)\frac{(x+y+z+1)!}{(-x+y+z)!},$$
and he states that the sum is 
a very-well-poised ${}_7F_6(1)$. He
also obtains another expression for the 6-$j$ coefficient by
substituting $a\rightarrow -a-1, b\rightarrow -b-1$, which is also
a very-well-poised ${}_7F_6(1)$. \\

\ni We use the Whipple transformation for the ${}_4F_3(1)$ forms of
the 6-$j$ coefficient, and since in this set of ${}_4F_3(1)$s all the 
four numerator parameters are negative integers, there are four different
ways in which the given transformation can be used to obtain 
very-well-poised
${}_7V_6(1)$ forms for the 6-$j$ coefficient. In the table given below
are the four different permutations and the corresponding ${}_7V_6(1)$ 
forms we obtain for the 6-$j$ coefficient.\\

\begin{center}
\begin{tabular}{cc}
\hline
Parameter permutation & ${}_7V_6$ (parameters) \\
\hline
(4231);(231) & ${}_7V_6 (A+B+C-E;\ G-D,F-D,A,B,C)$ \\   
(3241);(231) & ${}_7V_6 (A+B+D-E;\ G-C,F-C,A,B,D)$ \\   
(2341);(231) & ${}_7V_6 (A+C+D-E;\ G-B,F-B,A,C,D)$ \\   
(1234);(231) & ${}_7V_6 (B+C+D-E;\ G-A,F-A,B,C,D)$ \\   
\hline
\end{tabular}
\end{center}

\medskip

\ni These are the only independent ${}_7V_6$ forms for the 6-$j$ 
coefficient. Of these, the one corresponding to the second of 
the three ${}_4F_3(1)$ forms (\ref{9}) for the 6-$j$ coefficient corresponds to
the `new' expression reported by 
Ali\v sauskas. \\

\ni In our approach we have shown \cite{ksr1} that there are two sets
of ${}_4F_3(1)$s which are related to each other by `reversal' of 
series, which completely maps the set I of three ${}_4F_3(1)$s -- explicitly 
given here in this section -- onto the set II of four ${}_4F_3(1)$s. 
From the table given above, it is clear that we get for each of the 
${}_4F_3(1)$s belonging to set I of hypergeometric functions, a set of 
four ${}_7V_6(1)$s. It is straight forward to then observe that each of these
${}_7V_6(1)$s will account for only 12 of the 144 symmetries of the
6-$j$ coefficient. For instance, the first entry in the table above
clearly shows that the symmetries exhibited are due to the permutations of 
the  parameters $A,B,C$ ($S_3$) and the parameters $F,G$ ($S_2$). It is also
clear from an examination of the table that the 12 symmetries corresponding 
to each of the four ${}_7V_6(1)$s are distinctly different. Thus, we can conclude, that
if we want to express the 6-$j$ coefficient as a ${}_7V_6(1)$, then {\bf a set of 
12 \ ${}_7V_6(1)$s is necessary and sufficient to account for the 144 symmetries}
of the 6-$j$-coefficient. \\

\ni Equivalently, corresponding to the set II of four ${}_4F_3(1)$s, one can write down 
a set of 16 ${}_7V_6(1)$s, each of which will account for only 9 of 
the 144 symmetries of the 6-$j$ coefficient, enabling us to draw 
the conclusion that if the 6-$j$ coefficient is represented as a 
${}_7V_6(1)$ derived by the use of the Whipple transformation on the 
${}_4F_3(1)$ belonging to set II of ${}_4F_3(1)$s for the 6-$j$ coefficient,
then a set of 16 ${}_7V_6(1)$s is necessary and sufficient to account for
the 144 symmetries of the 6-$j$ coefficient.\\

\ni This completes our understanding of the 144 symmetries of the 6-$j$ 
coefficient in terms of the equivalent sets of 12 or 16 ${}_7V_6(1)$s. The 
`new' expression obtained by 
Ali\v sauskas is just one member of the set of 12 
${}_7V_6(1)$s.\\
        

\ni {\bf Acknowledgements:} One of us (KSR) wishes to thank Professor Bruno 
Gruber for the kind invitation to the XIII Symmetries in Science
Symposium and for the excellent hospitality. It was the second pleasurable 
experience he had and it gave him an opportunity to meet again `friends
of Gruber' and also spend a week in Vienna en route, during which period
a part of this work was done.
 
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\end{document}













	
