% LATEX SOURCE FILE % Group theoretical basis for the terminating 3F2(1) series % K. Srinivasa Rao, J. Van der Jeugt, J. Raynal, R. Jagannathan and % V. Rajeswari % J. Phys. A: Math. Gen. 25 (1992), 861-876. % \documentstyle[12pt]{article} \textheight=23,5cm \textwidth=15,5cm \headsep=0cm \topmargin=0cm \oddsidemargin=0cm \parindent=30pt \parskip=\medskipamount \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\beas{\begin{eqnarray*}} \def\eeas{\end{eqnarray*}} \def\para{\parallel} \def\nn{\nonumber} \def\p{\prime} \hyphenation{Rijks-uni-ver-si-teit} \hyphenation{pa-ra-metri-za-tion} \begin{document} \addtolength{\baselineskip}{0.6\baselineskip} \addtolength{\topsep}{-.3\topsep} % \begin{center} {\LARGE Group theoretical basis for the terminating $_3F_2(1)$ series}\\[2cm] K. Srinivasa Rao$^{\dagger \ast}$\,,\ J. Van der Jeugt$^{\dagger\ast\ast}$\,,\ J. Raynal$^{\ddagger}$\,, \\ R. Jagannathan$^{\ast}$ and V. Rajeswari$^{\ast}$\\[.5cm] $^\dagger$ {\em Laboratorium voor Numerieke Wiskunde en Informatica,\\ Rijksuniversiteit Gent, Krijgslaan 281--S9, B-9000 Gent, Belgium}\\ $^\ddagger$ {\em Service de Physique Th\'eorique, CE--Saclay,}\\[-2mm] {\em F-91190 Gif-sur-Yvette CEDEX, France}\\ $^\ast$ {\em Institute of Mathematical Sciences, Madras-600\,113, India} \end{center} \vspace{1cm} \noindent Short title~: Group theory for ${}_3F_2$ \\ \noindent PACS numbers~: 02.20, 02.90, 03.65 \\ \vspace{1cm} \noindent Abstract~: \begin{quotation} It is shown that a recursive use of the transformation for a terminating $_3F_2(1)$ series used by Weber and Erdelyi (1952), which belongs, as shown by Whipple (1925), to a set of equivalent $_3F_2(1)$ functions obtained by Thomae (1879), results in a 72 element group associated with 18 terminating series. The generators, conjugacy classes, invariant subgroups, characters and dimensions of irreducible representations for this group are presented. \end{quotation} \vfill \noindent-----------------------------------\\ $^{\ast\ast}$ {\footnotesize Research Associate, N.F.W.O. (Belgium)} \newpage \section{Introduction} The terminating generalized hypergeometric functions of unit argument, the $_3F_2(1)$ and the $_4F_3(1)$, have been related to the Wigner $(3-j)$ and the Racah $(6-j)$ coefficients, respectively, in literature (see, for example, Smorodinskii and Shelepin, 1972; Biedenharn and Louck, 1981a, 1981b). Starting with the van der Waerden (1932) form for the $3-j$ coefficient, and resorting to the comprehensive work of Whipple (1925) on the symmetries of the $_3F_2(1)$ functions, Raynal (1978) obtained ten different forms for the $3-j$ coefficient. A set of six $_3F_2(1)$s of the van der Waerden form has been shown (Srinivasa Rao, 1978) to be necessary and sufficient to account for the 72 symmetries of the $3-j$ coefficient. With a transformation for a terminating $_3F_2(1)$ series, used by Weber and Erdelyi (1952), Rajeswari and Srinivasa Rao (1989) derived from the van der Waerden set of six $_3F_2(1)$s, three other sets of $_3F_2(1)$s corresponding to the Wigner (1940), Racah (1942) and Majumdar (1955) forms. They also studied the consequences of relating the Majumdar form of $_3F_2(1)$ for the $3-j$ coefficient to the discrete orthogonal Hahn polynomial to obtain recurrence relations satisfied by the $3-j$ coefficient. Recently, Beyer, Louck and Stein (1987) showed that an identity due to Thomae (1879) between two $_3F_2(1)$ series, together with invariance under separate permutations of numerator and denominator parameters, implies that the symmetric group $S_5$ is an invariance group of the non-terminating series. In the same paper, Bailey's transformation for the terminating Saalsch\"utzian $_4F_3$ series (Bailey 1935, p.~56) is used to study the symmetry group of two-term relations for this series, which is also $S_5$. Using the relation between the $6-j$ coefficient and the terminating Saalsch\"utzian $_4F_3(1)$, and applying this new symmetry group, it is shown (Louck {\sl et al} 1987) that the classical group of 144 symmetries (the Regge symmetries of the $6-j$ coefficient) are extended to a group of 23040 symmetries by extending the domain of these coefficients. Clearly, this extended domain contains also ``unphysical'' arguments for the $6-j$ coefficient. Note that the extended symmetry group of order 23040 had already been encountered by D'Adda {\sl et al} (1972,1974) in their unified treatment of $SU(2)$ and $SU(1,1)$ $6-j$ coefficients. It would be of interest to make a similar analysis for the $3-j$ coefficient and its representing series, the terminating $_3F_2(1)$. Let us point out here that the group $S_5$ of Beyer {\sl et al} (1987) is the symmetry group of two-term relations for the non-terminating $_3F_2(1)$ function. Since it is the terminating $_3F_2(1)$ series which is related to $3-j$ coefficients, we study in this article the corresponding symmetry group of two-term relations for the same. A transformation for a terminating $_3F_2(1)$ series given by Weber and Erdelyi (1952), when used recursively, is shown to generate all the 18 terminating series on which are superposed the trivial $S_2 \times S_2$ symmetry -- a consequence of the invariance of a given terminating $_3F_2(1)$ series to the permutation of its two numerators (note~: the third numerator parameter determines the termination of the series) and its two denominator parameters. This 72 element finite group of transformations has nine conjugacy classes and correspondingly nine irreducible representations (irreps)~: four of dimension one, one of dimension two and four of dimension four. The three generators for this 72 element group are given. The smallest invariant or normal subgroup, $H_9$ (say), of this finite group is of order 9, and it is imbedded in an 18 element invariant subgroup, $H_{18}$ and $H_{18}$, in turn, is imbedded in three 36 element invariant subgroups. In terms of Whipple's parameters (1925) for the $_3F_2(1)$, it is shown that $H_9$ is isomorphic to the product of two cyclic groups of order 3. The 72 element group $G_T$ is shown to be the invariance group of $_3\tilde{F}_2$, which is a rescaling of the terminating $_3F_2$. Thus it is the group generating all two-term relations for this series. The phase factor appearing in such a two-term relation is shown to be equal to an irreducible character of $G_T$, motivating the construction of the complete character table for $G_T$. Using the van der Waerden form for the $3-j$ coefficient, the implication of the symmetry group $G_T$ on $3-j$ symbols is investigated. The conclusion is similar to that for the $6-j$ symbols (Louck {\sl et al} 1987)~: the classical group of 72 symmetries (the Regge symmetries of the $3-j$ coefficient) are extended to a group of 1440 symmetries by extension of the domain of these coefficients. Again, this extended domain contains ``unphysical'' arguments. In section 2, the essential notation required is given. In section 3, starting with a matrix representing the Weber-Erdelyi transformation for a terminating $_3F_2(1)$ series, the procedure for generating the 72 element group $G_T$ is described and the Whipple parametrization introduced. In section 4, the structure of the group $G_T$\,, its conjugacy classes, its irreps and their corresponding characters, and the invariant subgroups of $G_T$ are presented. In section 5, comments and conclusions regarding a scaling transformation which makes $G_T$ an invariance group of the terminating $_3F_2(1)$ series\,, the use of the symmetry group in the context of the $3-j$ coefficient, etc. are made. Finally, in an Appendix, the 18 transformations of the $_3F_2(1)$ are stated explicitly, in the Whipple notation and in a scaled, invariant form. \vspace{.5cm} \section{Notation} Whipple (1925) introduced six parameters $r_i$, $i=0,1,2,3,4,5$, such that~: \be \sum^5_{i=0} r_i=0 \ee and let \be \alpha_{lmn} = {1\over 2} + r_l + r_m + r_n\;, \qquad \beta_{mn} = 1+ r_m -r_n\,. \ee With these he defined the function~: \be F_p(l;mn) = {1\over \Gamma(\alpha_{ijk},\beta_{ml},\beta_{nl})}\,{}_3F_2 { \alpha_{imn},\alpha_{jmn},\alpha_{kmn};1 \choose \beta_{ml},\beta_{nl} }\,, \ee where $i,j$ and $k$ are used to represent those three numbers out of the six integers 0,1,2,3,4,5 not already represented by $l,m$ and $n$. The function $_3F_2(1)$ is the generalized hypergeometric function (cf. Slater, 1966) of unit argument having $\alpha_{imn}, \alpha_{jmn},\alpha_{kmn}$ as its three numerator parameters and $\beta_{ml},\beta_{nl}$ as its two denominator parameters. By changing the signs of all the $r_i$ parameters and using the constraint (1), Whipple defined another function~: \be F_n(l;mn) = {1\over \Gamma(\alpha_{lmn},\beta_{lm},\beta_{ln})}\,{}_3F_2 {\alpha_{ljk},\alpha_{lik},\alpha_{lij};1 \choose \beta_{lm},\beta_{ln} }\,. \ee In (3) and (4) use is made of the notation~: \be \Gamma(x,y,z,\ldots) = \Gamma(x)\Gamma(y)\Gamma(z)\ldots \ee By permutation of the suffixes $l,m,n$ over the six integers 0,1,2,3,4,5, sixty $F_p$ functions and sixty $F_n$ functions can be written down. If there is no negative integer in the numerator parameters, these series converge only if the real parts of $\alpha_{ijk}$ in (3) and $\alpha_{lmn}$ in (4) are positive. For the sake of brevity the unit argument of the generalized hypergeometric series will not be displayed and it will be denoted as \( _3F_2 { a,b,c\choose d,e } \) or $_3F_2(a,b,c\,;\,d,e)$, the three numerator and the two denominator parameters being the variables. \noindent(\underline{Note}~: The use of $n$ as a suffix for the $F_n$ function and also as an index for $\alpha$ and $\beta$ is continued here as in the literature.) \vspace{.5cm} \section{Terminating series} Consider the transformation for a terminating $_3F_2$ used by Weber and Erdelyi (1952)~: \be _3F_2 { a,b,-N\choose d,e } = {\Gamma(d,d+N-a) \over \Gamma(d+N,d-a)} {}_3F_2 {a,e-b,-N \choose 1+a-d-N,e} \,. \ee This formula is one of a set (cf. Bailey 1935) obtained by Whipple (1925). If the five parameters of the $_3F_2$ on the l.h.s. of (7) are denoted by the column vector~: \be \vec{x} = (a,b,1-N,d,e)\,, \ee then the parameters of the $_3F_2$ on the r.h.s. of (7) are obtained when the matrix~: \be g_1 = \left[ \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0\\ 0 &-1 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 0\\ 1 & 0 & 1 &-1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{array} \right] \ee operates on $\vec{x}$. Note that $1-N$ is used instead of $-N$, as a component of the column vector $\vec{x}$, since it represents the number of terms in a terminating series. However, $_3F_2(a,b,-N;d,e)$ will be denoted by $_3F_2(\vec{x})$. Using (6) again, with the roles of $d$ and $e$ interchanged, to transform the r.h.s. of (6), Weber and Erdelyi obtained the transformation~: \be _3F_2 {a,b,-N \choose d,e } = { \Gamma(d,e,e+N-a,d+N-a) \over \Gamma(d+N,e+N,d-a,e-a) } {}_3F_2 { a,1-s,-N \choose 1-b+d-s,1-b+e-s }\,, \ee where $s=d+e-a-b+N$. The question arises as to whether this recursive use of the Weber-Erdelyi transformation (6) can be continued. In fact, such a procedure when continued results in a group of 72 transformations, which are the 18 terminating $_3F_2$ series (see Appendix) on which are superposed the $ a \leftrightarrow b$, $d \leftrightarrow e $ and ($a \leftrightarrow b$, $d \leftrightarrow e$) interchanges. Let $g_2$ be the matrix \be g_2 = \left[ \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{array} \right] \ee which interchanges $a$ and $b$ when it operates on $\vec{x}$ and denote by $g_3$ the matrix~: \be g_3 = \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 \end{array} \right] \ee which interchanges $d$ and $e$ when it operates on $\vec{x}$. By forming all possible products of all possible powers of $g_1,\,g_2$ and $g_3$, a group of 72 transformation matrices can be generated which provides a $5\times 5$ representation for the terminating series, with (7) as the basis. Thus, $g_1,\, g_2$ and $g_3$ are the generators of a group $G_T$ for the transformations of a terminating $_3F_2$ series, with $g^2_i={\bf 1}$, for $i=1,2,3$. A similarity transformation, $u^{-1}g_i u$, with~: \be u=\left[ \begin{array}{ccccc} 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 2 & 1 & 0\\ 0 & 0 & 2 & 0 & 1 \end{array} \right] \qquad {\rm and} \qquad u^{-1} = {1\over 3} \left[ \begin{array}{ccccc} 3 & 0 &-1 & 0 & 0\\ 0 & 3 &-1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 &-2 & 3 & 0\\ 0 & 0 &-2 & 0 & 3 \end{array} \right] \,, \ee block diagonalizes the generators, and hence all the $g\in G_T$\,, thereby reducing the generators for the $5\times 5$ representation into the generators for a one-dimensional identity irrep (due to $-N$ being kept fixed in (6)) and the generators for a four-dimensional faithful irrep given by~: \be \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 &-1 & 0 & 1\\ 1 & 0 &-1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right]\,,\quad \left[ \begin{array}{cccc} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right] \quad {\rm and} \qquad \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{array} \right]\,. \ee In terms of Whipple's parameters and the definitions for $F_p$ and $F_n$ series given by (3) and (4), respectively, the transformation (6) can be written as~: \be F_p(0;45) = (-1)^N { \Gamma(\alpha_{015},\alpha_{025}) \over \Gamma(\alpha_{123},\alpha_{124}) } F_n(5;02)\,, \ee where $\alpha_{345}=-N$. (See Appendix and eq.(4.3.3.6) in Slater, 1966.) In the Whipple parameter basis, where \be \vec{x}\,' = (r_0,\,r_1,\,r_2,\,r_3,\,r_4,\,r_5) \ee is represented as a column vector, the transformation (14) is equivalent to the $6\times 6$ transformation matrix~: \be g'_1 = \left[ \begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 &-1 & 0 & 0\\ 0 & 0 & 0 & 0 &-1 & 0\\ 0 &-1 & 0 & 0 & 0 & 0\\ 0 & 0 &-1 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 \end{array} \right]\,. \ee The permutation of the two numerator parameters $a$ and $b$ in the $_3F_2$, in terms of Whipple parameters is equivalent to an interchange of $r_1$ and $r_2$, which is induced by the matrix~: \be g'_2 = \left[ \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right]\,, \ee operating on the basis vector $\vec{x}\,'$. Similarly, the permutation of the two denominator parameters $d$ and $e$ in the $_3F_2$\,, is equivalent to the interchange of $r_4$ and $r_5$\,, induced by~: \be g'_3 = \left[ \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0 \end{array} \right]\,. \ee These three $6 \times 6$ matrices generate a six-dimensional reducible representation for $G_T$. This 6-dimensional representation, in the Whipple parameter basis, $\vec{x}\,'$, can be reduced by the similarity transformation, $u'^{-1}g'_iu'$, with~: \be u'= \left[ \begin{array}{rrrrrr} 1 & 1 & 0 & 0 & 0 & 1\\ 1 & -1 & 1 & 0 & 0 & 1\\ 1 & 0 & -1 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 0 & -1\\ 1 & 0 & 0 & -1 & 1 & -1\\ 1 & 0 & 0 & 0 & -1 & -1 \end{array} \right] \; {\rm and} \; u'^{-1} = {1\over 6} \left[ \begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1\\ 4 & -2 &-2 & 0 & 0 & 0\\ 2 & 2 & -4 & 0 & 0 & 0\\ 0 & 0 & 0 & 4 &-2 & -2\\ 0 & 0 & 0 & 2 & 2 & -4\\ 1 & 1 & 1 &-1 & -1 & -1 \end{array} \right]\,, \ee which block diagonalizes the generators $g'_1,\,g'_2$ and $g'_3$, and hence all the $g'\in G_T$\,. It results in two one-dimensional irreps, one of which is the identity irrep, and a four-dimensional faithful irrep with generators~: \be \left[ \begin{array}{rrrr} 0 & 0 & 0 & 1\\ 0 & 0 &-1 & 1\\ 1 &-1 & 0 &0\\ 1 & 0 & 0& 0 \end{array} \right] \quad , \quad \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0\\ 1 &-1 &0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0& 1 \end{array} \right] \quad {\rm and} \qquad \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0\\ 0 &1 &0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 1& -1 \end{array} \right]\,. \ee From (16)-(18) it follows that $G_T$ is a subgroup of the permutation group $S_6$. Indeed, the generators $g'_i$ of $G_T$ can be represented by $6\times 6$ permutation matrices (including an overall minus-sign for $g'_1$). If we use the cycle notation for an element of $S_6$ represented by a $6\times 6$ permutation matrix, we see from (16)-(18) that \be \begin{array}{l} g'_1 = -(05)(13)(24)\,,\\[.2cm] g'_2 = (12)\,,\\[.2cm] g'_3 = (45)\,, \end{array} \ee where a minus sign for $g'_1$ is included in order to remember that in the Whipple parameter representation this generator is actually a permutation matrix multiplied by $-1$. In the following section it will be very useful to represent elements of $G_T$ by means of the above cycle notation, especially for distinguishing between conjugacy classes with the same order. \vspace{.5cm} \section{Structure of $G_T$ and its irreps} Two elements $h$ and $h'$ of a group $G$ are said to be conjugate if there exists a $g\in G$ such that $h'=ghg^{-1}$\,. This defines an equivalence relation on $G$, the equivalence classes being called the conjugacy classes. Analysis of $G_T$\,, reveals that there are 9 conjugacy classes $K_1,\ldots,K_9$\,. A conjugacy class is represented by one of its elements. In the following table are given the list of all the conjugacy classes $K_i$\,, a representative element (given in terms of the generators, and as a permutation matrix in cycle notation), the order $k_i$ of $K_i$ (i.e. the number of elements of $K_i$), and the order of the elements of $K_i$ (i.e. the smallest integer $s$ such that $g^s ={\bf 1}$, for $g\in K_i$). \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline class & order $k_i$ of $K_i$ & order of $g\in K_i$ & \multicolumn{2}{c|}{representative of $g\in K_i$} \\ \hline $K_1$ & 1 & 0 & $g^2_1 ={\bf 1}$ & {\bf 1}\\ $K_2$ & 4 & 3 & $g_1g_2g_1g_3$ & (345)\\ $K_3$ & 4 & 3 & $(g_2g_1g_3)^2$ & (012)(345)\\ $K_4$ & 6 & 2 & $g_1$ & --(05)(13)(24)\\ $K_5$ & 6 & 2 & $g_2$ & (12)\\ $K_6$ & 9 & 2 & $g_2g_3$ & (12)(45)\\ $K_7$ & 12 & 6 & $g_2g_1g_3$ & --(051324)\\ $K_8$ & 12 & 6 & $g_1g_3g_2g_1g_2$ & (021)(34)\\ $K_9$ & 18 & 4 & $g_1g_2$ & --(05)(1423)\\ \hline \end{tabular} \end{center} \vspace{.4cm} Following the general theory of group representations (cf.\ Wybourne, 1970 or Messiah, 1964), the table of characters for the irreps of $G_T$ has been obtained. As there are 9 conjugacy classes, there are 9 inequivalent irreps, which are denoted by $D^{(1)},\ldots,D^{(9)}$. Four irreps are of dimension 1, one is of dimension 2, and four are of dimension 4. It is only the 4-dimensional irreps which are faithful. The following table lists the characters~: \begin{center} \begin{tabular}{|c|rrrrrrrrr|} \hline & $K_1$ & $K_2$ & $K_3$ & $K_4$ & $K_5$ & $K_6$ & $K_7$ & $K_8$ & $K_9$\\ \hline $D^{(1)}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ $D^{(2)}$ & 1 & 1 & 1 &--1 & 1 & 1 &--1 & 1 &--1 \\ $D^{(3)}$ & 1 & 1 & 1 & 1 &--1 & 1 & 1 &--1 &--1 \\ $D^{(4)}$ & 1 & 1 & 1 &--1 &--1 & 1 &--1 &--1 & 1 \\ $D^{(5)}$ & 2 & 2 & 2 & 0 & 0 &--2 & 0 & 0 & 0 \\ $D^{(6)}$ & 4 & 1 &--2 & 0 & 2 & 0 & 0 &--1 & 0 \\ $D^{(7)}$ & 4 & 1 &--2 & 0 &--2 & 0 & 0 & 1 & 0 \\ $D^{(8)}$ & 4 &--2 & 1 & 2 & 0 & 0 &--1 & 0 & 0 \\ $D^{(9)}$ & 4 &--2 & 1 &--2 & 0 & 0 & 1 & 0 & 0 \\ \hline \end{tabular} \end{center} \vspace{.4cm} Simply by looking at the traces of $g_1$ and $g_2$, and comparing with the columns $K_4$ and $K_5$ (of which $g_1$ and $g_2$ are representatives) in the character table, it is possible to conclude that the representation generated by $g_i$ $(i=1,2,3)$ is equivalent to \[ D^{(1)} \oplus D^{(6)}\,, \] and that the Whipple parameter representation generated by $g'_i\,(i=1,2,3)$ is equivalent to \[ D^{(1)} \oplus D^{(2)} \oplus D^{(6)}\,. \] As a consequence, the irreducible representation matrices (13) and (20) for the generators of $G_T$ are equivalent and both can be labelled by $D^{(6)}$. The next property to analyse is the simplicity of $G_T$. All the invariant subgroups $H$ of $G_T$ have been found. Among these there are proper abelian invariant subgroups, hence $G_T$ is neither simple nor semi-simple. Recall that a subgroup $H$ is an invariant subgroup (self-conjugate subgroup, normal divisor) if $G_THG^{-1}_T=H$. To find invariant subgroups, one can form unions of conjugacy classes and check if they close under the group multiplication law. The following inclusion table gives a complete list of the invariant subgroups of $G_T$ (the subscript denoting the order of $H$)~: \be H_9 \subset H_{18} \left\{ \begin{array}{l} \subset H_{36} \subset G_T \,,\\[.2cm] \subset H'_{36} \subset G_T\,,\\[.2cm] \subset H''_{36} \subset G_T \,, \end{array} \right. \ee where \be \begin{array}{l} H_9 = K_1 \cup K_2 \cup K_3\,,\\[.2cm] H_{18} = H_9 \cup K_6\,,\\[.2cm] H_{36} = H_{18} \cup K_9\,,\\[.2cm] H'_{36} = H_{18} \cup K_4 \cup K_7\,,\\[.2cm] H''_{36} = H_{18} \cup K_5 \cup K_8\,. \end{array} \ee It should be noted that, in terms of the three generators $g_i$ (or $g'_i$) introduced previously, one can write \be K_6=g_2g_3H_9\,, \quad K_9=g_1g_2H_{18} \,, \quad K_4 \cup K_7 = g_1 H_{18} \,, \quad K_5 \cup K_8 = g_2 H_{18}\,, \ee such that the invariant subgroups (23) can be characterized as follows in terms of $H_9$ and the three generators~: \be \begin{array}{rcl} H_{18} &=& H_9 \cup g_2g_3H_9\,,\\[.2cm] H_{36} &=& H_9 \cup g_2g_3H_9 \cup g_1g_2H_9 \cup g_1g_3H_9\,,\\[.2cm] H'_{36}&=& H_9 \cup g_2g_3H_9 \cup g_1H_9 \cup g_1g_2g_3H_9\,,\\[.2cm] H''_{36}&=& H_9\cup g_2g_3H_9 \cup g_2H_9 \cup g_3H_9\,. \end{array} \ee The smallest invariant subgroup, $H_9$\,, is easy to characterize. In fact $H_9 = C_3 \times C_3$\,, the direct product of two cyclic groups on three elements. In terms of the Whipple parametrization, the generators of the two $C_3$'s are (012) and (345). It is now obvious that $H_9$ is an abelian invariant subgroup of $G_T$. It should be noticed that all the invariant subgroups of $G_T$ can be found using the character table and the fact that those elements $h$ of $G_T$ with $\phi(h)=\phi({\bf 1})$, where $\phi$ is a (not necessarily simple) character of $G_T$, form an invariant subgroup (Ledermann 1977, Theorem~2.7). Conversely, having the list of all invariant subgroups of $G_T$, one can reconstruct the character table. Indeed, the first character $\chi^{(1)}$ is trivial. Next, if $N$ is one of $H_{36}$, $H'_{36}$ or $H''_{36}$, $G/N$ is the 2 element group $C_2$, with non-trivial simple character $(1,-1)$. Using the ``lifting process'' (Ledermann 1977, Theorem~2.6), one obtains the simple characters $\chi^{(2)}$, $\chi^{(3)}$ and $\chi^{(4)}$ from $H''_{36}$, $H'_{36}$ and $H_{36}$ respectively. This completes the list of simple characters with $\chi^{(i)}_1=1$. In order to find the remaining simple characters, the theory of induced characters can be used. If $H$ is a subgroup of $G$ for which a character $^H\phi$ is known, then \[ ^G\phi_i = {m\over k_i} \sum_w \; ^H\phi(w)\,, \qquad w\in K_i\cap H \] is a character (simple or compound) of $G$. Herein, $m$ is the index of $H$ and $k_i$ is the order of $K_i$. As the simple characters of an abelian group are well known, $H$ is chosen to be $H_9=C_3\times C_3$, thus $m=72/9=8$. Using the trivial character of $H$, $^H\phi^{(1)}=(1,1,1,1,1,1,1,1,1)$, one finds $^G\phi^{(1)}=(8,8,8,0,0,0,0,0,0)$. By means of the inner product for characters of $G_T$, \[ \langle\phi|\psi\rangle = {1\over 72}\sum_{i=1}^9\; k_i\phi_i\psi_i\,, \] it is found that $\langle^G\phi^{(1)}|\chi^{(1)}\rangle = \langle^G\phi^{(1)}|\chi^{(2)}\rangle = \langle^G\phi^{(1)}|\chi^{(3)}\rangle = \langle^G\phi^{(1)}|\chi^{(4)}\rangle =1$. Thus, subtracting $\chi^{(1)},\ldots,\chi^{(4)}$ from $^G\phi^{(1)}$, one obtains $^G\phi'=(4,4,4,0,0,-4,0,0,0)$. Since all one-dimensional irreps have been found and $\langle^G\phi'|^G\phi'\rangle =4$, it follows that $^G\phi'$ is twice a simple character, i.e.~$^G\phi'=2\chi^{(5)}$. The next simple character, $\chi^{(6)}$, is immediately deduced from our defining representation (8), (10) and (11). Using a non-trivial character of $H$, $^H\phi^{(2)}=(1,1,1,\omega,\omega,\omega, \omega^2,\omega^2,\omega^2)$, where $\omega^2+\omega+1=0$, the inducing process leads to $^G\phi^{(2)}=(8,2,-4,0,0,0,0,0,0)$. One can verify that the inner product of $^G\phi^{(2)}$ with $\chi^{(1)}$, $\chi^{(2)}$, $\chi^{(3)}$, $\chi^{(4)}$ and $\chi^{(5)}$ is zero, and that $\langle^G\phi^{(2)}|\chi^{(6)}\rangle =1$. Subtracting $\chi^{(6)}$ from $^G\phi^{(2)}$, one obtains $^G\phi''=(4,1,-2,0,-2,0,0,1,0)$. Since $\langle^G\phi''|^G\phi''\rangle =1$, it is a simple character, i.e.~$^G\phi''=\chi^{(7)}$. Two more simple characters $\chi^{(8)}$ and $\chi^{(9)}$ need to be found. Using the orthogonality property satisfied by the columns of the character table of $G_T$, namely \[ \sum_{l=1}^9\;\chi^{(l)}_i \chi^{(l)}_j = {72\over k_i} \delta_{ij}\,, \] it is a straightforward exercise to complete the character table. \vspace{.5cm} \section{Comments and conclusions} Although in the preceeding sections $G_T$ was generated by three generators, namely the Weber-Erdelyi transformation $g_1$ and the two interchange transformations $a \leftrightarrow b$ ($g_2$) and $d \leftrightarrow e$ ($g_3$) it should be noted that $G_T$ can actually be generated by only two elements. For instance, using the cycle structure notation for the elements of $G_T$\,, the 72 element group $G_T$ is generated by (12) and $-(0524)(31)$, i.e. by $g_2$ and $(g_1g_3)$. In fact there are many other examples of pairs of generators for $G_T$. Using the notation of Section 1, the Weber-Erdelyi transformation (6) can be written in the following form~: \be _3F_2(\vec{x}) = { \Gamma(d,d+N-a) \over \Gamma(d+N,d-a) } {} _3F_2 (g_1\vec{x})\,, \ee whereas the interchange transformations are~: \be _3F_2(\vec{x}) = {}_3F_2(g_2\vec{x})\,, \qquad _3F_2(\vec{x})={}_3F_2(g_3\vec{x})\,. \ee In general, this analysis implies that \be _3F_2(\vec{x}) = ({\rm factor})\,_3F_2(g\vec{x})\,, \qquad \forall\,g\in G_T\,, \ee where this factor is in terms of $\Gamma$-functions, as in (6) or (9). It would be interesting if this factor could actually be determined in terms of the group element $g$. This can indeed be done. The most elegant way to obtain this is to perform a scaling on the $_3F_2(\vec{x})$~: \be _3\tilde{F}_2(\vec{x})= {\Gamma(d+N,e+N) \over \Gamma(d,e)}\,_3F_2(\vec{x})\,. \ee Then the three generating transformations become~: \be \begin{array}{rcl} _3\tilde{F}_2(\vec{x}) &=& (-1)^N\, _3\tilde{F}_2(g_1\vec{x})\,,\\[.2cm] _3\tilde{F}_2(\vec{x}) &=& _3\tilde{F}_2(g_2 \vec{x}) \, =\,{}_3\tilde{F}_2(g_3\vec{x})\,. \end{array} \ee As $G_T$ is generated by $g_1,\,g_2$ and $g_3$\,, the following result holds~: the scaled terminating $_3\tilde{F}_2$ with unit argument satisfies \bea _3\tilde{F}_2(\vec{x})& =& _3\tilde{F}_2(g\vec{x})\quad , \qquad \forall\, g \in G_T\,,\qquad\hbox{(for $N$ even)},\\ _3\tilde{F}_2(\vec{x})& =& \chi^{(2)}(g)\,_3\tilde{F}_2(g\vec{x})\quad , \qquad \forall\, g \in G_T\,,\qquad\hbox{(for $N$ odd)}, \eea where $\chi^{(2)}(g)$ is the character of $g$ in the irrep $D^{(2)}$ (see Section~4). Hence the 72 element group $G_T$ can be seen as the invariance group of the terminating $_3F_2$. If $N$ is odd, then the coefficient in~(32) is $+1$ or $-1$, and it is equal to $-1$ if one of the following equivalent conditions is satisfied~: \begin{itemize} \item[--] $g_1$ appears an odd number of times in the expression of $g$ in terms of $g_1\,, \, g_2$ and $g_3$\,; \item[--] $g$ is a permutation matrix times $-1$ when represented in the Whipple parametrization\,; \item[--] the left and right hand sides of (32) correspond to a $F_p$ and a $F_n$ in terms of the notation of section 2\,. \end{itemize} The use of the Weber-Erdelyi transformation (6) on the van der Waerden $_3F_2$ form for the $3-j$ coefficient $ { j_1 \quad j_2 \quad j_3 \choose m_1 \quad m_2 \quad m_3 } $ or $ {a \quad b \quad c \choose \alpha \quad \beta \quad \gamma }$ was shown by Rajeswari and Srinivasa Rao (1989) to result in the Majumdar, Racah or Wigner $_3F_2$ forms, with or without the superposition of a column permutation and the $m_i \to -m_i$ substitution on them. If use is made of any one of the other transformations explicitly listed in the Appendix, on the van der Waerden $_3F_2$ form for the $3-j$ coefficient, then it can be shown that the result would be one of the 12 terminating $_3F_2$ forms given in Raynal (1978) --- Viz. Eqs. (6), (15)--(17), (26)--(30) and three others which differ from (15)--(17) by exchange of $a$ and $b$ and change of sign for $\alpha,\;\beta,\;\gamma$ in Raynal (1978), which include the Majumdar, Racah, Wigner forms --- or, one of the 12 forms on which is superposed a `classical' symmetry of the $3-j$ coefficient (Viz. permutations of the columns of the $3-j$ coefficient and the $m_i \to -m_i$ substitution). It is well known that one of the van der Waerden forms for the $3-j$ coefficient can be written as follows~: \bea \lefteqn{ \left(\begin{array}{ccc} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{array} \right) = } \nonumber \\ & &\delta(m_1+m_2+m_3,0) (-1)^{j_1-j_2-m_3} \nonumber \\ & &\times [(-j_1+j_2+j_3)!(j_1-j_2+j_3)!(j_2-m_2)! (j_3-m_3)!(j_1+m_1)!(j_3+m_3)!]^{1/2} \nonumber \\ & &\times [(j_1+j_2-j_3)!(j_1+j_2+j_3+1)!(j_1-m_1)! (j_2+m_2)!]^{-1/2} \nonumber \\ & &\times[(j_3-j_1-m_2)!(j_3-j_2+m_1)!]^{-1} \nonumber \\ & &\times _3F_2\left(\begin{array}{c} -j_1+m_1,\quad -j_2-m_2,\quad -j_1-j_2+j_3 \\ 1+j_3-j_1-m_2,\quad 1+j_3-j_2+m_1 \end{array} ; \quad 1 \right). \eea Using~(33), the three generating elements $g_1$, $g_2$ and $g_3$ of $G_T$ lead, respectively, to the following symmetries of the $3-j$ symbol (apart from a phase factor)~: \be \left(\begin{array}{ccc} j_1 & -j_3-1 & -j_2-1 \\ m_1 & m_3 & m_2 \end{array}\right) \;, \ee \be \left(\begin{array}{ccc} (j_1+j_2-m_3)/2 & (j_1+j_2+m_3)/2 & j_3 \\ (j_1-j_2+m_1-m_2)/2 & (j_1-j_2-m_1+m_2)/2 & -j_1+j_2 \end{array}\right) \;, \ee \be \left(\begin{array}{ccc} (j_1+j_2+m_3)/2 & (j_1+j_2-m_3)/2 & j_3 \\ (-j_1+j_2+m_1-m_2)/2 & (-j_1+j_2-m_1+m_2)/2 & j_1-j_2 \end{array}\right) \;. \ee The second and third of these are well known Regge symmetries of the $3-j$ symbol, while the first has unphysical arguments (the $j$-values being negative; the triangular condition is violated). The classical symmetry group of the $3-j$ coefficient contains 72 symmetries, of which (35) and (36) are two elements. Following Louck {\sl et al} (1987), who extended the classical Regge group of 144 symmetries of the $6-j$ symbol by the $_4F_3$ invariance group $S_5$ in order to obtain a new symmetry group of order 23040, one can perform the same process here and extend the 72 classical symmetries of the $3-j$ symbol by the symmetries induced by the 72 element group $G_T$. Since (35) and (36) are Regge symmetries, already contained in the 72 symmetries, this amounts to enlarging these symmetries by the element (34) and to investigating which group $G$ it generates. In particular, (34) contains unphysical transformations of the type $j\to -j-1$ (preserving the angular momentum eigenvalue $j(j+1)$), known as Yutsis mirror symmetries (Yutsis and Bandzaitis 1965). Let us denote by $j_1\to -j_1-1$ by $r'$. It can be shown by recursively using $r'$ and the column permutations of the $3-j$ coefficient that (34) can be transformed into \be \left(\begin{array}{ccc} -j_1-1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{array}\right) \;. \ee The group $G$ can be generated by the classical symmetries together with $r'$. This new group $G$ is of order 1440; it can be interpreted as the extended symmetry group of the $3-j$ coefficient by extending the domain of this coefficient. This extended domain contains unphysical arguments. It should be noticed that this extended symmetry group of order 1440 has been encountered by D'Adda {\sl et al} (1972), in treating $SU(2)$ and $SU(1,1)$ $3-j$ coefficients, and by Husz\'ar (1972). There are two further observations to make. The first is that the ``trivial'' $_3F_2$ symmetry permuting two of the three numerator parameters corresponds to a non-trivial Regge symmetry for the $3-j$ symbol (in fact, this observation is not new~: see Biedenharn and Louck (1981b), p.~433). The second, new, observation is that a ``trivial'' $3-j$ symmetry (namely $j_1\to -j_1-1$) corresponds to a non-trivial transformation for the terminating $_3F_2(1)$ series, namely to (6). It is considered relevant to point out the contemporary work of Beyer, Louck and Stein (1987) in the present context. For this purpose, in the Whipple notation (section 2) let $l,\,m,\,n$ be 0, 4, 5, respectively. Then the numerator and denominator parameters which occur in $F_p(0;45)$, given by (3), after elimination of $r_0$ using (1), are related to the five independent Whipple parameters~: \be \vec{r} = (r_1\,,\,r_2\,,\,r_3\,,\,r_4\,,\,r_5) \ee through the transformation~: \be \vec{\alpha} = A\vec{r}\,, \ee where \[ \vec{\alpha} = (\alpha_{145} -{1\over 2}\,,\,\alpha_{245}-{1\over 2}\,,\, \alpha_{345}-{1\over 2}\,,\, \beta_{40}-1\,,\, \beta_{50}-1)\,, \] and \be A = \left[ \begin{array}{ccccc} 1 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1\\ 0 & 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 2 & 1\\ 1 & 1 & 1 & 1 & 2 \end{array} \right]\,. \ee This $5\times 5$ matrix $A$ plays a crucial role in the study of the group structure of two-term identities by Beyer, Louck and Stein (1987). They analyse the group structure of the non-terminating series and establish that the symmetric group $S_5$ is an invariance group of the two-term relation for the $_3F_2$ series due to Thomae (1879) and the invariance of that series to separate permutations of the numerator and denominator parameters of the $_3F_2$\,. In this article, we generated a 72 element group $G_T$ for the terminating $_3F_2(1)$ series, presented the conjugacy classes, irreps and their characters, and the invariant subgroups of $G_T$ and discussed the role of these terminating series for the $_3F_2(1)$ forms of the $3-j$ coefficient. The group $G_T$, of interest for us has been arrived at by a simple recursive use of a given $_3F_3(1)$ transformation and the results presented for the terminating $_3F_2(1)$ series supplement the work of Beyer {\sl et al} (1985). The structure of the invariance group $G_T$ for the terminating $_3F_2(1)$ series has turned out to be more intricate than that of the symmetric group $S_5$ shown to be the invariance group for the non-terminating $_3F_2(1)$ series investigated by Beyer {\sl et al} (1985). Our study contributes to a complete understanding of an interesting aspect overlooked in the work of Beyer, Louck and Stein (1985). \vspace{.5cm} \noindent {\bf Aknowledgement} One of us (K.S.R.) wishes to thank Professor Guido Vanden Berghe and the N.F.W.O. Belgium for hospitality at the Rijksuniversiteit Gent and for financial support, and another (V.R.) acknowledges with thanks the award of a Research Associateship by the University Grants Commission of India. The authors thank Dr.~Hans De Meyer for a careful reading of the manuscript and Ms. P. Mortier for her diligent typing of the same in \LaTeX. \newpage \renewcommand{\theequation}{\Roman{equation}} \setcounter{equation}{0} \begin{center} {\bf APPENDIX} \end{center} \vspace{.5cm} In this Appendix, the 18 terminating $_3F_2$ transformations are written down explicitly as they arise when the Weber-Erdelyi transformation (6) is recursively used. They are expressed then in terms of Whipple parametrization and finally using the scaling transformation which enabled us to show $G_T$ as the invariance group of the terminating $_3F_2$\,. \bea \lefteqn{ _3F_2 {a,b,-N \choose d,e} }\nn\\ && = { (d-a,N) \over (d,N)}\,{}_3F_2 { a,e-b,-N \choose 1+a-d-N,e }\\[.3cm] && =(-1)^N {(1-s,N)\over (d,N)}\,{}_3F_2 { e-a,e-b,-N \choose s-N,e }\\[.3cm] && ={ (d-a,N)(e-a,N) \over (d,N)(e,N) }\,{}_3F_2 { a,1-s,-N \choose 1+a-d-N,1+a-e-N }\\[.3cm] && ={ (d-a,N)(b,N) \over (d,N)(e,N) }\,{}_3F_2 { e-b,1-d-N,-N \choose 1-b-N,1+a-d-N }\\[.3cm] && ={ (d-b,N) \over (d,N) }\,{}_3F_2 { e-a,b,-N \choose 1+b-d-N,e}\\[.3cm] && =(-1)^N { (1-s,N)(b,N)\over (d,N)(e,N)}\,{}_3F_2 { e-b,d-b,-N \choose 1-b-N,s-N }\\[.3cm] && =(-1)^N {(1-s,N)(a,N) \over (d,N)(e,N) }\,{}_3F_2 { e-a,d-a,-N \choose 1-a-N,s-N }\\[.3cm] && =(-1)^N {(d-a,N)(d-b,N) \over (d,N)(e,N) }\,{}_3F_2 { 1-s,1-d-N,-N \choose 1+a-d-N,1+b-d-N }\nn\\ &&\\ && =(-1)^N {(e-a,N)\over (e,N)}\,{}_3F_2 { a,d-b,-N \choose d,1+a-e-N}\\[.3cm] && =(-1)^N {(e-a,N)(e-b,N) \over (d,N)(e,N)}\,{}_3F_2 { 1-s,1-e-N,-N \choose 1+a-e-N,1+b-e-N }\\[.3cm] && =(-1)^N {(a,N)(b,N) \over (d,N)(e,N) }\,{}_3F_2 { 1-d-N,1-e-N,-N \choose 1-a-N,1-b-N }\\[.3cm] && = {}_3F_2 {a,b,-N \choose d,e } \qquad\qquad\qquad {\rm (identity)}\\[.3cm] && = { (d-b,N)(a,N) \over (d,N)(e,N)}\,{}_3F_2 { e-a,1-d-N,-N \choose 1-a-N,1+b-d-N}\\[.3cm] && = {(d-b,N)(e-b,N) \over (d,N)(e,N)}\,{}_3F_2 { b,1-s,-N \choose 1+b-d-N,1+b-e-N }\\[.3cm] && = {(1-s,N) \over (e,N)}\,{}_3F_2 { d-a,d-b,-N \choose d,s-N }\\[.3cm] && = {(b,N)(e-a,N) \over (d,N)(e,N)}\,{}_3F_2 { d-b,1-e-N,-N \choose 1+a-e-N,1-b-N }\\[.3cm] && = { (a,N)(e-b,N) \over (d,N)(e,N)}\,{}_3F_2 { d-a,1-e-N,-N \choose 1-a-N,1+b-e-N }\\[.3cm] && = {(e-b,N)\over (e,N)}\,{}_3F_2 { b,d-a,-N \choose d,1+b-e-N } \eea where $s=d+e-a-b+N$ and $(\alpha,N)=\Gamma(\alpha+N)/\Gamma(\alpha)$. These transformations reduce to five relations when they are written in terms of Whipple parameters and the notation of Whipple given in Section 2. They are~: \renewcommand{\theequation}{A.{\arabic{equation}}} \setcounter{equation}{0} \bea \Gamma(\alpha_{123},\alpha_{124},\alpha_{125})F_p(0) &=& \Gamma(\alpha_{023},\alpha_{024},\alpha_{025})F_p(1)\\ &=& \Gamma(\alpha_{013},\alpha_{014},\alpha_{015})F_p(2)\\ &=& (-1)^N \Gamma(\alpha_{123},\alpha_{013},\alpha_{023})F_n(3)\\ &=& (-1)^N \Gamma(\alpha_{124},\alpha_{014},\alpha_{024})F_n(4)\\ &=& (-1)^N \Gamma(\alpha_{125},\alpha_{015},\alpha_{025})F_n(5) \eea where (A.1) represents (XIII), (XIV) and (XVII)\,,\\ $\phantom {\rm where\,}$ (A.2) represents (III), (IV) and (XVI)\,, \\ $\phantom {\rm where\,}$ (A.3) represents (VI), (VII) and (XI)\,,\\ $\phantom {\rm where\,}$ (A.4) represents (IX), (X) and (XVIII)\,,\\ $\phantom {\rm where\,}$ (A.5) represents (I), (V) and (VIII),\\ while (XII) is the identity\,; (II) and (XV) corresponds to $F_p(0;45)=F_p(0;35)$ and $F_p(0;45)=F_p(0;34)$, respectively. These relations~: $F_p(0;45)=F_p(0;35)=F_p(0;34)$ represent the fact that for a given $l$, all the ten expressions $F_p(l;mn)$ (as well as, all the ten $F_n(l;mn)$) are equal. It is for this reason that they are denoted simply as $F_p(l)$ or $F_n(l)$ above. The relations (A.1) to (A.5) are the same as (4.3.3.2) to (4.3.3.6) in Slater (1966), who has also tabulated the expressions for $\alpha$ (and $\beta$) in terms of $a,\,b,\,c(=-N),\,d,\,e$ (cf. Table 4.1, Slater, 1966). The transformation (XI) represents the reversal of series. If the scaling transformation (29) is used in the definitions (3) and (4) for the $F_p(l;mn)$ and $F_n(l;mn)$ functions, then for $\alpha_{kmn}=-N$~: \be F_p(l;mn) = {1\over \Gamma(\alpha_{ijk},\alpha_{ijm},\alpha_{ijn}) }\, {}_3\tilde{F}_2 {\alpha_{imn},\alpha_{jmn},-N \choose \beta_{ml},\beta_{nl} } \ee and \be F_n(l;mn) = {1\over \Gamma(\alpha_{lmn},\alpha_{kln},\alpha_{klm}) }\, {}_3\tilde{F}_2 {\alpha_{ljk},\alpha_{lik},-N \choose \beta_{lm},\beta_{ln} }\,. \ee Redefining~: \be \tilde{F}_p(l;mn) = \Gamma(\alpha_{ijk},\alpha_{ijm},\alpha_{ijn})F_p(l;mn)\,, \ee and \be \tilde{F}_n(l;mn) = \Gamma(\alpha_{lmn},\alpha_{kln},\alpha_{klm})F_n(l;mn)\,, \ee for $\alpha_{345}=-N$, the relations (A.1) to (A.5) will now become simply~: \bea \tilde{F}_p(0) &=& \tilde{F}_p(1) = \tilde{F}_p(2)\nn\\ &=& (-1)^N\tilde{F}_n(3)=(-1)^N\tilde{F}_n(4)= (-1)^N\tilde{F}_n(5)\,, \eea since \be \begin{array}{l} \tilde{F}_p(0) = \Gamma(\alpha_{123},\alpha_{124},\alpha_{125})F_p(0)\,, \\[.2cm] \tilde{F}_p(1) = \Gamma(\alpha_{023},\alpha_{024},\alpha_{025})F_p(1)\,, \\[.2cm] \tilde{F}_p(2) = \Gamma(\alpha_{013},\alpha_{014},\alpha_{015})F_p(2)\,, \\[.2cm] \tilde{F}_p(3) = \Gamma(\alpha_{123},\alpha_{013},\alpha_{023})F_n(3)\,, \\[.2cm] \tilde{F}_p(4) = \Gamma(\alpha_{124},\alpha_{014},\alpha_{024})F_n(4)\,, \end{array}\ee and \[ \tilde{F}_p(5) = \Gamma(\alpha_{125},\alpha_{015},\alpha_{025})F_n(5)\,. \] In general, for any $\alpha_{lmn}=-N$\,, the relations among the 18 terminating series would be~: \bea \tilde{F}_p(i) &=& \tilde{F}_p(j) = \tilde{F}_p(k)\nn\\ &=& (-1)^N\,\tilde{F}_n(l) = (-1)^N\,\tilde{F}_n(m)=(-1)^N\,\tilde{F}_n(n)\,. \eea One of us has obtained a relation similar to (A.12) (cf.\ Eq.~(25) in Raynal, 1978). But that relation is different since it is valid for the $3-j$ coefficient when expressed in terms of a scaled $_3F_2$. Of the three generators $g_1\,,\,g_2\,,\,g_3$ for $G_T$\,, in the text, for the generator $g_1$\,, the $5\times 5$ matrix representating the Weber-Erdelyi transformation (6), denoted by (I) above, was chosen. The 72 elements of the $5\times 5$ representation for $G_T$ can also be generated if $g_1$ is anyone of the matrices representing the transformation (V) - (X) or (XVIII). However, if for $g_1$\,, the $5\times 5$ unit matrix representing (XII) is chosen, then it would result in a 4-element subgroup of $G_T$\,. Similarly, choosing (XI) for $g_1$ results in an 8-element subgroup of $G_T$\,; choosing (II), (III), (XIV) or (XV) for $g_1$ results in 12-element subgroups of $G_T$\,; and choosing (IV), (XIII), (XVI) or (XVII) results in $36$ element subgroups of the group $G_T$\,. When $c=\alpha_{345}=-N$ determines the termination of the $_3F_2$ series, from the definition (3) for $F_p$, it follows that $(m,n)$ can take only the three values (3,4), (3,5) or (4,5). Since any one of the numerator parameters of $F_p(l)$ --- Viz. $\alpha_{imn}\,,\,\alpha_{jmn}\,,\,\alpha_{kmn}$ --- can be $\alpha_{345}$, the indices $i,\,j,\,k$ are restricted to 5, 4 or 3, which in turn implies that $l$ can be only 0, 1 or 2. Therefore, $(m,n)$ being any two of 3, 4, 5 ($^3C_2$) and $l$ being any one of 0, 1, 2 ($^3C_1$), it is obvious that $\alpha_{345}$ can occur as a numerator parameter in only $(^3C_1 \times ^3C_2 =)$ 9 series. When $r_i$ is replaced by $-r_i$, instead of the $F_p(l)$ series, the $F_n(l)$ series arise. From the definition (4) for the $F_n(l)$ series, $(j,k),\;(i,k)$ or $(i,j)$ can take the values (3,4), (3,5) or (4,5) so that $l$ can be 5, 4 or 3 ($^3C_1$) and $(m,n)$ can be only (0,1), (0,2) or (1,2). Once again there are only 9 $F_n$ series. 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