%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % LATEX SOURCE FILE % % of the paper % % Invariance groups of transformations of % % basic hypergeometric series % % % % by J. Van der Jeugt and K. Srinivasa Rao % % (to be submitted to J. Math. Phys.) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage{amssymb,latexsym} \headheight=0mm \headsep=0mm \oddsidemargin=1mm \evensidemargin=1mm \textheight=225mm \textwidth=155mm % % definitions concerning automatic numbering of theorems, lemmas, etc. % \newtheorem{theo}{Theorem} \newtheorem{defi}[theo]{Definition} \newtheorem{lemm}[theo]{Lemma} \newtheorem{coro}[theo]{Corollary} \newtheorem{rema}[theo]{Remark} \newtheorem{prop}[theo]{Proposition} % % SPECIAL DEFINITIONS FOR THIS PAPER % \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\beas{\begin{eqnarray*}} \def\eeas{\end{eqnarray*}} \def\nn{\nonumber} \def\ds{\displaystyle} \def\al{\alpha} \def\be{\beta} \def\Z{{\tt Z}} %\def\Z{\mathbb{Z}} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{1} % the end-of-proof box : \def\mybox{\hfill\llap{$\Box$}} % % \begin{document} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{1} \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} \addtolength{\baselineskip}{2mm} \addtolength{\abovedisplayskip}{1mm} \addtolength{\belowdisplayskip}{1mm} \addtolength{\parskip}{2mm} \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 % \renewcommand{\thesection}{\Roman{section}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem{GR} G.\ Gasper and M.\ Rahman, {\em Basic Hypergeometric Series} (Cambridge Univ.\ Press, Cambridge, 1990). \bibitem{Krattenthaler} C.\ Krattenthaler, HYP and HYPQ~: Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively $q$-binomial sums and basic hypergeometric series, {J.\ Symbolic Comput.} {\bf 20} (1995) 737--744. Also available on {\tt http://radon.mat.univie.ac.at/People/kratt}. \bibitem{Whipple} F.J.W.\ Whipple, A group of generalized hypergeometric series~: relations between 120 allied series of the type $F[a,b,c; d,e]$, {Proc.\ London Math.\ Soc.} (2) {\bf 23} (1925) 104--114. \bibitem{BLS} W.A.\ Beyer, J.D.\ Louck and P.R.\ Stein, Group theoretical basis of some identities for the generalized hypergeometric series, {J.\ Math.\ Phys.} {\bf 28} (1987) 497--508. \bibitem{Rao} K.\ Srinivasa Rao, J.\ Van der Jeugt, J.\ Raynal, R.\ Jagannathan and V.\ Rajeswari, Group theoretical basis for the terminating $_3F_2(1)$ series, {J.\ Phys.\ A~: Math.\ Gen.} {\bf 25} (1992) 861--876. \bibitem{Louck} J.D.\ Louck, W.A.\ Beyer, L.C.\ Biedenharn and P.R.\ Stein, Symmetries of some hypergeometric series~: implications for 3$j$- and 6$j$-coefficients, in {\em XV International Colloquium on Group Theoretical Methods in Physics}, ed.\ R.\ Gilmore (World Scientific, Singapore, 1987). \bibitem{BieLou} For an overview of the relations between 3-$j$ and 6-$j$ coefficients and hypergeometric series, see~: L.C.\ Biedenharn and J.D.\ Louck, {\em The Racah-Wigner Algebra in Quantum Theory. Encycl.\ of Math.\ and its Appl., vol.\ 9.} (Addison-Wesley, Reading, MA, 1981), Topic~11, p.~429. \bibitem{Raji1} V.\ Rajeswari and K.\ Srinivasa Rao, Four sets of $_3F_2(1)$ functions, Hahn polynomials and recurrence relations for the 3-$j$ coefficients, {J.\ Phys.\ A~: Math.\ Gen.} {\bf 22} (1989) 4113--4123. \bibitem{Raji2} V.\ Rajeswari and K.\ Srinivasa Rao, Generalized basic hypergeometric functions and the $q$-analogues of 3-$j$ and 6-$j$ coefficients, {J.\ Phys.\ A~: Math.\ Gen.} {\bf 24} (1991) 3761--3780. \bibitem{Raji3} V.\ Rajeswari and K.\ Srinivasa Rao, Note on the Explicit Forms for the Clebsch-Gordan Coefficient of the Quantum Group $SU_q(2)$, {J.\ Phys.\ Soc.\ of Japan} {\bf 60} (1991) 3583--3584. \bibitem{Hamermesh} M.\ Hamermesh, {\em Group Theory, and its application to physical problems} (Addison-Wesley, Reading, 1962). \bibitem{Thomae} J.\ Thomae, \"Uber die Funktionen welche durch Reihen von der Form dargestellt werden~: $1+{p p' p'' \over 1 q' q''}+ \cdots$, {J.\ Reine Angew.\ Math.} {\bf 87} (1879) 26--73. \bibitem{Bailey} W.N.\ Bailey, {\em Generalized Hypergeometric Series} (Cambridge Univ.\ Press, Cambridge, 1935). \bibitem{Hardy} G.H.\ Hardy, {\em Ramanujan. Twelve lectures on subjects suggested by his life and work} (Cambridge Univ.\ Press, Cambridge, 1940). \bibitem{Sears} D.B.\ Sears, On the transformation theory of basic hypergeometric functions, {Proc.\ London Math.\ Soc.} (2) {\bf 53} (1951) 158--180. \bibitem{JK} G.\ James and A.\ Kerber, {\em The Representation Theory of the Symmetric Group}, Encyclopedia of Mathematics and its Applications, vol.~16 (Addison-Wesley, Reading, 1981). \bibitem{Heine} E.\ Heine, {\em Handbuch der Kugelfunctionen. Theorie und Anwendungen}, vol.~1 (Reimer, Berlin, 1878). \bibitem{Budden} F.J.\ Budden, {\em The Fascination of Groups} (Cambridge Univ.\ Press, Cambridge, 1972). \bibitem{Bailey2} W.N.\ Bailey, An identity involving Heine's basic hypergeometric series, {J.\ London Math.\ Soc.} {\bf 4} (1929) 254--257. \bibitem{Humphreys} J.E.\ Humphreys, {\em Introduction to Lie algebras and representation theory}. Graduate Texts in Mathematics, vol.~{\bf 9} (Springer, Berlin, 1978). \end{thebibliography} \end{document}