% Latex file
% WIGNER'S 9-$j$ COEFFICIENT AND ITS RELATION TO NEW
% SUMMATION THEOREMS FOR HYPERGEOMETRIC SERIES
% S.N. Pitre and J. Van der Jeugt
% in Proceedings of the 4th International Wigner Symposium,
% Guadalajara 1996,
% eds. N.M. Atakishiyev, K.B. Wolf and T.H. Seligman
% World Scientific Company, Singapore, 1996; pp. 270-273.
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\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\ninej#1#2#3#4#5#6#7#8#9{\left\{ \begin{array}{ccc}
  #1 & #2 & #3 \\  #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right\} }
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\begin{document}

\title{WIGNER'S 9-$j$ COEFFICIENT AND ITS RELATION TO NEW
SUMMATION THEOREMS FOR HYPERGEOMETRIC SERIES}
\author{S.N.~Pitre and J.\ Van der Jeugt\footnote{Senior Research
Associate of N.F.W.O. (National Fund for Scientific Research of 
Belgium).},\\
Department of Applied Mathematics and Computer Science,\\
University of Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium. }

\maketitle
\abstract{
It is known from Wigner's early work that the 9-$j$ coefficient has 72
symmetries. Two decades ago, Ali\v sauskas and Jucys obtained a triple
sum series expression for 9-$j$ coefficients exhibiting none of
these symmetries. This inherent lack of symmetry is now turned to
advantage, as we show that it is the inspiring source for obtaining
new relations between hypergeometric series. Concrete results are
transformation and new summation formulas for the double hypergeometric
Kamp\'e de F\'eriet series of type $F^{0:3}_{1:1}(1,1)$.
}

\section{Introduction}

We shall present a hypergeometric series
transformation discovered when
systematically studying Wigner's 9-$j$ coefficient. This
coefficient, depending upon 9 integer or half-integer
parameters, is denoted by
\beq
\left\{ \begin{array}{ccc} j_1 & j_2 & j_{12}\\ j_3 & j_4 &
j_{34}\\ j_{13}& j_{24} & J \end{array} \right\}
\label{ninej}
\eeq
and is related to the two ways in which 4 quantum mechanical angular
momenta can be coupled~\cite{Wigner,Biedenharn}.

From equations expressing the 9-$j$ coefficient in terms of
3-$j$ or 6-$j$ coefficients (also known as Wigner and Racah
coefficients), it follows that the 9-$j$ coefficient has 72
symmetries~\cite{Wigner,Biedenharn}.
On the other hand, there
exists an alternative expression of the 9-$j$ coefficient~: 
the triple sum series expression obtained as a by-result by
Ali\v sauskas and Jucys~\cite{Alisauskas} and independently
proved in Ref.~4. 
In previous work~\cite{Vanderjeugt} it was shown how a certain
``doubly stretched'' 9-$j$ coefficient for which a single term
expression exists can give rise to
multiple hypergeometric summation results.
Presently, we have investigated those symmetries of doubly
stretched 9-$j$ coefficients which give rise to double sums.
The clue
for finding new transformation or summation formulas lies in
comparing the expressions to which the triple sum series reduces
for two different symmetries. As such, we have systematically compared the
appearing double sums with single sums. 
Here, we present one illustrative example.

\section{The triple sum series and an application}

Apart from the well known expressions for 9-$j$ coefficients, a new
expression (the triple sum series) was obtained by Ali\v sauskas, Jucys
and Bandzaitis~\cite{Alisauskas,Jucys}. 
The triple sum, exhibiting no symmetries, is given by the following
complicated formula~: 
\bea
&& \ninej{a}{b}{c}{d}{e}{f}{g}{h}{i}=(-1)^{x_5}
{(d,a,g)(b,e,h)(i,g,h) \over(d,e,f)(b,a,c)(i,c,f)} \times \nn\\
&&\sum_{x,y,z} {(-1)^{x+y+z}\over x!y!z!}
{(x_1-x)!(x_2+x)!(x_3+x)!\over(x_4-x)!(x_5-x)!}
{(y_1+y)!(y_2+y)!\over (y_3+y)!(y_4-y)!(y_5-y)!} \nn\\
&& \times {(z_1-z)!(z_2+z)!\over (z_3-z)!(z_4-z)!(z_5-z)!}
  {(p_1-y-z)!\over (p_2+x+y)!(p_3+x+z)!},
\label{tripsum}
\eea
where
\beq
\begin{array}{lll}
x_1=2f        &     y_1=-b+e+h   & z_1=2a              \\
x_2=d+e-f     &     y_2=g+h-i    & z_2=-a+b+c          \\
x_3=c-f+i     &     y_3=2h+1     & z_3=a+d+g+1          \\
x_4=-d+e+f    &     y_4=b+e-h    & z_4=a+d-g            \\
x_5=c+f-i     &     y_5=g-h+i    & z_5=a-b+c         \\
p_1=a+d-h+i\quad   &     p_2=-b+d-f+h\quad & p_3=-a+b-f+i\
\end{array}
\label{xyz}
\eeq
and
\beq
(a,b,c)  =  \left[{(a-b+c)!(a+b-c)!(a+b+c+1)/(-a+b+c)!}\right]^{1/2}.
\eeq
In (\ref{tripsum}), $a,b,\cdots,i$ are all integers or half-integers,
and the three rows and three columns must form triads (i.e.\ 
satisfy triangular conditions).
The summation indices $x,y,z$ in (\ref{tripsum}) assume all
integer values such that the factorials are nonnegative. 

One of the interesting cases to consider is the doubly stretched 9-$j$
coefficient with $e=d+f$ and $g=a+d$.
From the classical properties, it follows that the following symmetries
are equal~:
\beq
\ninej{i}{h}{a+d}{c}{b}{a}{f}{d+f}{d} =
\ninej{i}{f}{c}{a+d}{d}{a}{h}{d+f}{b}.
\eeq
Using the triple sum series~(\ref{tripsum}) for both sides of the last
equation, one obtains the following identity~:
\bea
&&\sum_{x,z} {(-1)^{(x+z)}\over (-a+d+h-i+x+z)!}
{(-a+b+c+x)!(2d+x)!\over x!(a+b-c-x)!(-a+c+d+f-h+x)!} \nn\\
&&\times {(2i-z)!(a+d+h-i+z)!\over z!(1+c+f+i-z)!(a+d-h+i-z)!} \nn\\
&&= (-1)^{a+b-c} {(-a+b+c)!(a+d+h-i)!(1+a+d+h+i)! \over
(a+b-c)!(c+f-i)!(1+c+f+i)!} \nn\\
&&\times\sum_z (-1)^z(a+b-f+i-z)!(2i-z)!(c+f-i+z)!/[z!(c-f+i-z)!\nn\\
&&(a+d-h+i-z)!(1+a+d+h+i-z)!(-a+b+f-i+z)!]. \label{eq}
\eea
This equation is valid for all integer or half-integer values of the
variables that satisfy the triangular relations for the 9-$j$
coefficient. However, we shall see that it gives rise to a
more general identity.

\section{Hypergeometric series}

Due to the double summation in the above expression, we recall the
definition of a certain type of double hypergeometric series.
The Kamp\'e de F\'eriet function $F_{1:1}^{0:3}$, a
generalization of Appell series, is defined as
follows~: 
\beq
F_{1:1}^{0:3}\left[ {\hy \atop D}:{A,B,C\atop
E};{A',B',C'\atop E'};u,v\right] =
\sum_{m,n=0}^\infty {(A,B,C)_m (A',B',C')_n \over
(D)_{m+n}(E)_m(E')_n} {u^m v^n \over m!n!},
\label{kdf}
\eeq
where the notation is classical~: $(A)_m$ is a
Pochammer symbol, and $(A,B,C)_m$ = $(A)_m$ $(B)_m$ $(C)_m$.
Relabelling the 7 independent parameters in~(\ref{eq}),
\beq
\begin{array}{l}
A = -a-d+h-i,\;
B = -1-c-f-i,\;
C = 1+a+d+h-i,\\
D = 1-a+d+h-i,\;
E = -2i,\;
B' = -a-b+c,\;
C' = 1-a+b+c,
\end{array}
\eeq
and introducing the 2 dependent parameters
$A' = D-A$ and $E' = D+E-A-B-C$,
one finds that (\ref{eq}) can be rewritten in the following form~:
\bea
&&F_{1:1}^{0:3}\left[ {\hy \atop D}:{A,B,C\atop
E};{A',B',C'\atop E'};1,1\right]= \nn\\
&&\qquad\Gamma\left[ {D,E',D+E'-A'-B'-C' \atop
A',D+E'-A'-B',D+E'-A'-C'} \right]\nn\\
&&\qquad\times {\ }_4F_3\left[{A,E-B,E-C,D+E'
-A'-B'-C' \atop E,D+E'-A'-B', D+E'-A'-C'};1\right].
\label{kdf1} 
\eea
Herein, $\Gamma$ is the classical Gamma function with the
convention 
\[
\Gamma\left[{A_1,A_2,\ldots \atop B_1,B_2,\ldots}\right] =
{\Gamma(A_1)\Gamma(A_2)\ldots \over
\Gamma(B_1)\Gamma(B_2)\ldots}; 
\]
and ${}_4F_3$ is a generalized hypergeometric
function. From the theory of 9-$j$ coefficients, it thus follows that  
eq.~(\ref{kdf1}) holds under the special conditions when $a,b,c,\ldots$
are integers or half-integers satisfying the triangular conditions. But
it turns out that (\ref{kdf1}), written in this form, is more generally
valid. In fact, we have shown~\cite{pv} that (\ref{kdf1}) is valid for
all complex values of $A,B,\ldots$ (under appropriate convergence
conditions). 

Some limiting cases of (\ref{kdf1}) are worth
considering.  We mention two of the most interesting summation results
following from further specialization of (\ref{kdf1})~:
\bea
&&F_{1:1}^{0:3}\left[ {\hy \atop D}:{A,B,C\atop
E};{D-A,D-B,-C\atop D+E-A-B-C};1,1\right]=\nn\\
&&\qquad\Gamma\left[ E,E+D-A-B-C \atop E-C,E+D-A-B \right],
\label{res1} 
\eea
\bea
&&F_{1:1}^{0:3}\left[ {\hy \atop D}:{A,B,C\atop
E};{D-A,D-B,D-C\atop D+E-A-B-C};1,1\right]=\nn\\
&&\qquad\Gamma\left[ E,E+D-A-B-C,E-D \atop E-A,E-B,E-C \right].
\eea

For other transformation and summation formulas, and their proofs,
we refer the reader to Ref.~6.

\section*{References}
\begin{thebibliography}{99}
\bibitem{Wigner}
{E.\ Wigner}, ``On the matrices which reduce the Kronecker products
of representations of S.R. groups'', in {\em Quantum Theory of
Angular Momentum}, eds.\ L.C.\ Biedenharn and H.\ Van Dam
(Academic Press, New York, 1965).

\bibitem{Biedenharn}
{L.C.\ Biedenharn and J.D.\ Louck}, {\em Angular Momentum in Quantum
Physics} (Addison-Wesley, Reading, 1981).

\bibitem{Alisauskas}
{S.J.\ Ali\v sauskas and A.P.\ Jucys},
{\em J. Math. Phys.} {\bf 12}, 594 (1971).

\bibitem{Jucys}
{A.P.\ Jucys and A.A.\ Bandzaitis}, {\em Angular Momentum in Quantum
Physics} (Mokslas, Vilnius, 1977).

\bibitem{Vanderjeugt}
{J.\ Van der Jeugt, S.N.\ Pitre and K.\ Srinivasa Rao}, 
{\em J. Phys. A~: Math. Gen.} {\bf 27}, 5251 (1994).

\bibitem{pv}
{S.N.\ Pitre and J.\ Van der Jeugt}, ``Transformation and summation
formulas for Kamp\'e de F\'eriet series $F^{0:3}_{1:1}(1,1)$,'' submitted
to {\em J.\ Funct.\ Anal.\ Appl.} (1995).

\end{thebibliography}
\end{document}





