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% Multiple hypergeometric functions and 9-$j$ coefficients
% J. Van der Jeugt, Sangita N. Pitre and K. Srinivasa Rao
% J. Phys. A: Math. Gen 27 (1994), 5251-5264.
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\def\ninej#1#2#3#4#5#6#7#8#9{\left\{ \begin{array}{ccc}
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\def\si{\sigma}
\def\ps{\psi}
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\begin{document}
\begin{center}
{\Large \bf Multiple hypergeometric functions and 9-$j$ coefficients}\\[2cm]
{\bf J.\ Van der Jeugt$^{a,}$\footnote{Senior Research
Associate of N.F.W.O. (National Fund for Scientific Research of Belgium).},
Sangita N.~Pitre$^a$ and K.\ Srinivasa Rao$^b$}
\end{center}
\vskip 5mm
\noindent
(a) Department of Applied Mathematics and Computer Science,
University of Gent,\\ Krijgslaan 281-S9, B-9000 Gent, Belgium.\\
(b) The Institute of Mathematical Sciences, Madras 600 113, India.\\
E-mail : Joris.VanderJeugt@rug.ac.be and rao@imsc.ernet.in.
\vskip 1cm
PACS : 02.20.+b, 02.30.+g, 02.90.+p, 03.65.-w\\
printed in J.\ Phys.\ A {\bf 27} (1994) 5251-5264.
\vskip 2cm
\noindent {\bf Abstract:}
It is well known that the 9-$j$  recoupling
coefficient appearing in quantum theory
of angular momentum has 72 symmetries. However, the triple sum series
expression for the 9-$j$ coefficient exhibits none of these symmetries.
Here a {\it stretched\/} 9-$j$ coefficient,
for which a closed form (single
term) expression exists, is considered and it is investigated to what
kind of summation theorems the triple sum series reduces for any of
the 72 symmetries. Apart from well known single summation theorems for
hypergeometric functions, this analysis gives rise to new
summation theorems for double and triple hypergeometric functions.

\newpage

\section{Introduction}

The basic idea of this paper was announced in (Srinivasa Rao and
Van der Jeugt 1994), here referred to as~\one. In that paper,
the doubly stretched 9-$j$ coefficient
\beq
\ninej{a}{b}{c}{d}{e}{f}{a+d}{a+d+i}{i}
\label{1}
\eeq
was considered. This coefficient has a closed form expression
(Sharp 1967). On the other hand, 9-$j$ coefficients can also be
determined by means of the triple sum series of Jucys and Bandzaitis
(1977). This triple sum series can be written down for (\ref{1})
directly, or for any of the 71 remaining
symmetries (Jahn and Hope 1954)
of (\ref{1}). Due to the inherent asymmetry of the
triple sum series, all these 72 expressions would appear to
be different. In~\one, it was observed that for certain
symmetries of (\ref{1}), the triple sum would reduce to a
single sum, and expressing that these single sums are equal
to the closed form expression for (\ref{1}) thus leads to
well known summations theorems for generalized hypergeometric
functions. Moreover, it was pointed out that apart from yielding
manifestations of well known summation theorems, a complete study
of all 72 symmetries opens the scope for finding genuinely new
summation theorems. In the present paper, we present such a
complete analysis, and indeed some summation theorems for double
and triple hypergeometric functions arise which we have not found
in the literature, and hence we believe they are new results.

In Section~2, we recall the notation for generalized hypergeometric
functions, and define multiple hypergeometric functions. In Section~3,
the triple sum series is considered for the 72 symmetries of the
coefficient (\ref{1}). It turns out that 
of the 72 cases, four  yield a
single term (or, a closed form expression); 16 yield a single sum
of three different types; 20 yield a double sum
of six different types; and  32 yield a triple sum of four different 
types. All the single
sums are manifestations of well known summation theorems.
Two of the six double summations and two of the four triple summations
cannot be summed using known results, and they give rise
to new summation theorems for special double and triple hypergeometric
functions, while the rest can be summed using known single 
summation theorems. 
These summation formulae, and some further specializations,
are discussed in Section~4, followed by some conclusions.

\section{Generalized and multiple hypergeometric functions}

The Gauss function or hypergeometric function,
nowadays usually represented by the symbol
$_2F_1[{a,b\atop c};z]$, is defined as
\beq
{ }_2F_1\left[ {a,b\atop c} ; x\right] = \sum_{n=0}^\infty
{(a)_n (b)_n \over (c)_n} {x^n\over n!}.
\eeq
Herein, $a,b$ and $c$ are the (complex) parameters, $x$ is the
(complex) variable of the function, and the common notation
for the Pochammer symbol has been used~:
\beq
(a)_n=a(a+1)(a+2)\cdots(a+n-1)=\frac{\Gamma(a+n)}{\Gamma(a)}, 
\qquad (a)_0=1,
\eeq
where $\Gamma $ is the classical $\Gamma $-function.
For a historical introduction to the Gauss function, and a survey of its
properties, we refer to the first monograph on hypergeometric series by
Bailey (1935) and to Slater (1966). The idea of extending the
number of parameters in the Gauss function occurred for the first
time in the work of Clausen (1828), and these generalized hypergeometric
functions were studied by Saalsch\"utz (1890), Dixon (1903) and Dougall (1907).
Much of the theory was summarized and extended by Bailey (1935; see
also Slater 1963). The standard notation
and definition for a generalized hypergeometric function is~:
\beq
{ }_AF_B\left[ {a_1,\ldots, a_A \atop b_1,\ldots,b_B} ; x \right] =
\sum_{n=0}^\infty { (a_1)_n \cdots (a_A)_n \over (b_1)_n \cdots (b_B)_n }
 {x^n \over n!}.
 \label{AFB}
\eeq
In a more compact notation, devised by Burchnall and Chaundy (1941) in the
case of multiple hypergeometric functions, the whole list of parameters
$(a_1,\ldots,a_A)$ is not written explicitly but simply denoted by $(a)$.
Thus, (\ref{AFB}) is written as $_AF_B\left[ {(a)\atop (b)};x\right]$.

When one of the numerator parameters $a_j$ is a negative integer, the
function (\ref{AFB}) becomes a terminating series. Some of the most
well known {\em summation theorems} for generalized hypergeometric
functions are of this type; moreover, they are usually for unit
argument $x=1$. Vandermonde's theorem (which is the
terminating form of the famous Gauss theorem, see Slater 1966) reads~:
\beq
{ }_2F_1\left[ {a,-m \atop c};1 \right] = {(c-a)_m\over(c)_m}.
\label{vdm}
\eeq
In such summation theorems it is always understood that the
termination is determined by $-m$ (hence $-a$ and $-c$ do not
belong to $\{0,1,\ldots,m-1\}$), although it is common not to
mention this assumption explicitly. Another famous summation
theorem is due to Saalsch\"utz~:
\beq
{ }_3F_2\left[ {a,b,-m \atop c,d};1\right] =
{(c-a)_m(c-b)_m\over (c)_m(c-a-b)_m},
\label{ss}
\eeq
for $a+b-m+1=c+d$.
Some different summation theorems, which were discovered relatively
recently, are due to Minton (1970) and Karlsson (1971) (see 
eq.~(1.9.3) of Gasper and Rahman, 1990).
Minton's theorem reads~:
\beq
{ }_{r+2}F_{r+1}\left[ {a,b,b_1+m_1,\ldots,b_r+m_r \atop
 b+1,b_1,\ldots,b_r} ; 1\right] =
 {\Gamma(b+1)\Gamma(1-a)\over \Gamma(1+b-a)}
 {(b_1-b)_{m_1} \cdots (b_r-b)_{m_r} \over (b_1)_{m_1}\cdots (b_r)_{m_r}},
\label{minton}
\eeq
where $m_1,\cdots,m_r$ are nonnegative integers.
Using this result, Karlsson deduced the
following zero-balanced terminating series summation~:
\beq
{ }_{r+1}F_{r}\left[ {-m_1-m_2-\cdots-m_r,b_1+m_1,\ldots,b_r+m_r \atop
 b_1,\ldots,b_r} ; 1\right] = (-1)^{m_1+\cdots+m_r}
 {(m_1+\cdots+m_r)! \over (b_1)_{m_1}\cdots (b_r)_{m_r}}.
\label{karl}
\eeq
There are other summation theorems for generalized hypergeometric
functions (see Slater 1966), but the only ones that appear in connection with
this paper are (\ref{vdm}--\ref{karl}).

For multiple hypergeometric functions, depending on more variables
$x,y,\cdots$, the general theory is less advanced than for generalized
hypergeometric functions of one variable. Also the notation is
not uniform and often confusing. Appell was the first
author to study systematically double hypergeometric functions.
The standard work on Appell series is in Appell and Kamp\'e de
F\'eriet (1926).  A rather general double hypergeometric function
is known as the Kamp\'e de F\'eriet function and is defined
by (Kamp\'e de F\'eriet 1921)~:
\beq
F^{A:B}_{C:D}\left[ {(a)\atop(c)} : {(b)\atop(d)};{(b')\atop(d')};
x,y\right] = \sum_{m,n=0}^\infty
{\prod_{j=1}^A (a_j)_{m+n} \over \prod_{j=1}^C (c_j)_{m+n} }
{\prod_{j=1}^B (b_j)_m (b'_j)_n \over \prod_{j=1}^D (d_j)_m(d'_j)_n }
{x^my^n\over m!n!}.
\label{kdf}
\eeq
Herein, the parameters $a_j$ and $c_j$ appear with index $m+n$ in
the Pochammer symbols. They are the coupling parameters and are
responsible for the fact that (\ref{kdf}) cannot be written
as the product of two single hypergeometric functions in $x$ and
$y$ separately. In (\ref{kdf}), the number of parameters with
index $m$ is the same as that with index $n$. In general, this need
not always be the case and therefore we define the following
double hypergeometric function~:
\beq
F^{A:B;B'}_{C:D;D'}\left[ {(a)\atop(c)} : {(b)\atop(d)};{(b')\atop(d')};
x,y\right] = \sum_{m,n=0}^\infty
{\prod_{j=1}^A (a_j)_{m+n} \over \prod_{j=1}^C (c_j)_{m+n} }
{\prod_{j=1}^B (b_j)_m \over \prod_{j=1}^D (d_j)_m }
{\prod_{j=1}^{B'} (b'_j)_n \over \prod_{j=1}^{D'} (d'_j)_n }
{x^my^n\over m!n!}.
\label{myds}
\eeq
This is actually a special case of a very general function
defined by Srivastava and Daoust (1969). For double hypergeometric
functions there appears only one coupling, namely $m+n$. For
triple hypergeometric functions with summation indices $m,n$ and $p$,
couplings of the form $m+n$, $m+p$, $n+p$ or $m+n+p$ can occur,
and hence the general notation becomes more complicated. Inspired
by the general Srivastava and Daoust (1969) notation, we
define here~:
\bea
&&F^{A:B;B';B''}_{C:D;D';D''}\left[ {(a:\theta)\atop(c:\psi)} :
{(b)\atop(d)};{(b')\atop(d')};{(b'')\atop(d'')} ;
x,y,z\right] = \nn\\[2mm]
&&\qquad\sum_{m,n,p=0}^\infty
{\prod_{j=1}^A (a_j)_{\th_j^1m+\th_j^2n+\th_j^3p} \over
 \prod_{j=1}^C (c_j)_{\ps_j^1m+\ps_j^2n+\ps_j^3p} }
{\prod_{j=1}^B (b_j)_m \over \prod_{j=1}^D (d_j)_m }
{\prod_{j=1}^{B'} (b'_j)_n \over \prod_{j=1}^{D'} (d'_j)_n }
{\prod_{j=1}^{B''} (b''_j)_p \over \prod_{j=1}^{D''} (d''_j)_p }
{x^my^nz^p\over m!n!p!},
\label{tri}
\eea
where $(a:\th)$ stands for $(a_1:\th_1^1\th_1^2\th_1^3,\ldots,
a_A:\th_A^1\th_A^2\th_A^3)$ with $\th_j^i\in\{0,1\}$, and similarly
for $(c:\ps)$. In order to illustrate the above notation, we
give one example~:
\beq
F^{0:1;1;1}_{2:0;0;0}\left[ {\hy \atop c_1:110, c_2:011} :
{b\atop\hy };{b'\atop\hy };{b''\atop\hy };x,y,z\right] =
\sum_{m,n,p=0}^\infty {(b)_m(b')_n(b'')_p\over(c_1)_{m+n}(c_2)_{n+p}}
{x^my^nz^p\over m!n!p!}.
\eeq
For some of the existing literature on multiple hypergeometric
functions, the reader is referred to Exton (1976). To our knowledge,
the only summation theorems for multiple hypergeometric functions
are for some very special generalized Kamp\'e de F\'eriet functions
(see Exton 1976, p.~147). In this paper, we find some
interesting summation theorems for double and triple hypergeometric
functions. One of our results, for example, takes the
following simple form in the notation of (\ref{kdf})~:
\beq
F^{0:3}_{1:1}\left[ {\hy \atop\be} : {\al-\ga,\be+s,-r \atop \al+s};
{\ga-\al,\be+r,-s\atop \ga+r};1,1\right]
={(\al)_s(\ga)_r \over (\ga)_s (\al)_r},
\eeq
where $\al,\be,\ga$ are complex numbers and $r,s\in{\tt N}$ determine
the termination of the series.

\section{The 9-$j$ coefficient}

The 9-$j$ coefficient, or $ls$-$jj$ transformation coefficient,
plays an important role in quantum theory of angular momentum
(Wigner 1940, Biedenharn and Louck 1981). It is either given
as a double sum over a product of six 3-$j$ coefficients, or as a single
sum over a product of three 6-$j$ coefficients. From the expression
in terms of 3-$j$ coefficients and the symmetries of the 
3-$j$ coefficient, it can be established  that the 9-$j$
coefficient has 72 symmetries. Another expression for the
9-$j$ coefficient is the triple sum series of Jucys and
Bandzaitis (1977). This expression has proved to be useful in
numerical computations (Srinivasa Rao {\em et al} 1989, 1993), however,
it does not exhibit any of the 72 symmetries. The Jucys-Bandzaitis
triple sum is given by~:
\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)!\over (-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.\ 
$(a,b,c)$ is
a triad if $-a+b+c$, $a-b+c$ and $a+b-c$ are nonnegative integers).
Moreover, the summation indices $x,y,z$ in (\ref{tripsum}) assume all
integer values such that the factorials are nonnegative. Explicitly,
this means~:
$$
\begin{array}{l}
0\leq x\leq\min(x_4,x_5),\\
\max(0,-p_2-x)\leq y\leq\min(y_4,y_5),\\
\max(0,-p_3-x)\leq z\leq\min(z_4,z_5,p_1-y).
\end{array}
$$

Sharp (1967) classified stretched 9-$j$ coefficients (a
triad $(a,b,c)$ is stretched if one of the numbers $-a+b+c$, $a-b+c$,
$a+b-c$ is zero). In particular, he derived a single term expression
for the following doubly stretched 9-$j$ coefficient (see also
Varshalovich {\em et al} 1975)~:
\beq
\ninej{a}{b}{c}{d}{e}{f}{a+d}{a+d+i}{i}.
\label{9j}
\eeq
This particular 9-$j$ coefficient was also the subject of~\one.
If one uses the triple
sum expression (\ref{tripsum}) on the following particular symmetry of (\ref{9j}),
the values of the numbers (\ref{xyz}) are such that the triple
sum reduces to a single term, and one finds~:
\bea
&&\ninej{a}{b}{c}{d}{e}{f}{a+d}{a+d+i}{i}
= \ninej{a+d}{a+d+i}{i}{a}{b}{c}{d}{e}{f} = \nn\\[2mm]
&&(-1)^{d-e+f}
{(a+d+i,b,e)\over (a,b,c)(d,e,f)(i,c,f)}
\left[ {(2a)!(2d)!(2i)!\over
(2a+2d+1)(2a+2d+2i+1)!}\right]^{1/2} .
\label{sharp}
\eea
Thus, for a particular symmetry of (\ref{9j}), the triple sum
expression reduces to a single term. Due to the asymmetry of (\ref{tripsum}),
the triple sum does not always reduce to a single term for
other symmetries
of (\ref{9j}). This was the basic observation in~\one~:
consider any symmetry of (\ref{9j}) and use (\ref{tripsum});
this (usually) gives rise to a single, double or triple sum
series which has to be
equal to the single term (\ref{sharp}), hence a summation theorem
follows. A few of the symmetries were already considered in~\one.
It was shown that in some
cases the triple sum reduces to a single sum, and that this single
sum was a manifestation of the Vandermonde, Saalsch\"utz or
Karlsson theorem. An example where the triple sum reduces to a
double sum and another where the triple sum remained a triple sum
were given in~\one.

In this paper, we give a complete classification of the summations
that appear for the 72 symmetries and study the consequent 
summation theorems. The results are easy to obtain,
but require a careful analysis of the parameters (\ref{xyz}) for
each of the 72 symmetries of (\ref{9j}).

Before summarizing these results, it is convenient to introduce
a short-hand notation for a symmetry of (\ref{9j}). For any
symmetry, either a permutation $(\si_a\si_b\si_c)$ of $(abc)$
appears in a row and then a permutation $(\si_c\si_f\si_i)$ of
$(cfi)$ appears in a column, or vice versa. In the first case,
the 9-$j$ coefficient will be denoted by $(\si_a\si_b\si_c|\si_c\si_f\si_i)$,
and in the second case by $(\si_c\si_f\si_i|\si_a\si_b\si_c)$.
For example~:
\beq
(bca|cif)=\ninej{b}{c}{a}{a+d+i}{i}{a+d}{e}{f}{d},\qquad
(ifc|acb)=\ninej{a+d}{d}{a}{i}{f}{c}{a+d+i}{e}{b}.
\eeq
This notation determines the 9-$j$ coefficient uniquely and
has the advantage that it is much shorter to tabulate than
the 9-$j$ symbol itself.

Among the 72 expressions, there are four single terms, namely
\beq
(abc|icf),\; (abc|ifc),\; (bac|cfi),\;\hbox{ and }\; (bac|fci).
\label{single}
\eeq
The first of these corresponds in fact to the second 9-$j$ coefficient
in (\ref{sharp}).

Then, there are 16 symmetries for which the triple sum
expression reduces to a single sum and these are given
in Table~1.

\begin{table}[htb]
\caption{Symmetries giving rise to single sum expressions.}
\[
\begin{tabular}{|l|l|}
\hline
Vandermonde $_2F_1$ & $\begin{array}{cccccc}
  (cba|cfi) & (cba|fci) & (fic|acb) &
  (cif|acb) & (ifc|cab) & (icf|cab) \end{array}$ \\ \hline
Saalsch\"utz $_3F_2$ & $\begin{array}{cccccc}
  (bca|cif) & (bca|fic) & (fic|bca) &
  (cif|bca) & (icf|acb) & (ifc|acb) \\
  (cfi|cab) & (fci|cab) & &&& \end{array}$ \\ \hline
Karlsson $_4F_3$ & $\begin{array}{cccccc} (abc|cfi) & (abc|fci) &&&& \end{array}$\\
\hline
\end{tabular}
\]
\end{table}

Of these 16 cases, 6 single sums are identified as a Vandermonde $_2F_1$
summation, 8 are identified as a Saalsch\"utz $_3F_2$ summation, and
2 as a Karlsson $_4F_3$ summation. Thus, all the single sums can be
explicitly performed using the theorems (\ref{vdm}), (\ref{ss}) and
(\ref{karl}). On the other hand, it follows also from the symmetries that
all these sums can be written as a single term. In other words,
suppose we did not know theorem (\ref{karl}), then we could have
deduced it (in a restricted form only for integer parameters) from
the equality $(abc|cfi)=(abc|icf)$. This inspired us to consider also
the other symmetries and to investigate whether the corresponding double
or triple sums can be performed using known summation theorems, or whether
they give rise to new results.

Among the remaining symmetries, there are 20 double sums and 32
triple sums. The double sums fall into six different types,
which we have denoted by D1,$\ldots$,D6. Among the 32 triple sums,
there appear only four different types, denoted by T1, T2, T3 and T4.
Tables~2 and 3 give the double and triple sums according to their
type.

\begin{table}[htb]
\caption{Symmetries giving rise to double sum expressions.}
\[
\begin{tabular}{|l|l|l|}
\hline
D1 & $\begin{array}{cccccc}
  (fic|cba) & (cif|cba) & &&&  \end{array}$& new \\ \hline
D2 & $\begin{array}{cccccc}
  (bac|fic) & (bac|cif) &&&&   \end{array}$ & new\\ \hline
D3 & $\begin{array}{cccccc}
  (bca|cfi) & (bca|fci) & (icf|bac) &
  (ifc|bac) &  & \end{array}$ & (\ref{minton})+(\ref{vdm})\\ \hline
D4 & $\begin{array}{cccccc}
  (acb|cfi) & (acb|fci) & (cba|icf) &
  (cba|ifc) & (fic|cab) & (cif|cab) \end{array}$&
     (\ref{vdm})+(\ref{vdm})\\ \hline
D5 & $\begin{array}{cccccc}
  (cab|cfi) & (cab|fci) & (fic|abc) &
  (cif|abc) &  & \end{array}$ & (\ref{minton})+(\ref{ss})\\ \hline
D6 & $\begin{array}{cccccc}
  (cfi|acb) & (fci|acb) & &&&  \end{array}$ &(\ref{vdm})+(\ref{ss}) \\ \hline
\end{tabular}
\]
\end{table}

\begin{table}[htb]
\caption{Symmetries giving rise to triple sum expressions.}
\[
\begin{tabular}{|l|l|l|}
\hline
T1 & $\begin{array}{cccccc}
  (abc|fic) & (abc|cif) & (bca|icf)&
  (bca|ifc) & (acb|fic) & (acb|cif)\\
  (cab|icf) & (cab|ifc) & (bac|icf)&
  (bac|ifc) & (cfi|abc) & (fci|abc)\\
  (cfi|bca) & (fci|bca) & (cfi|cba)&
  (fci|cba) & (fic|bac) & (cif|bac)  \end{array}$ & new\\ \hline
T2 & $\begin{array}{cccccc}
  (acb|icf) & (acb|ifc) & &&&  \end{array}$ & new\\ \hline
T3 & $\begin{array}{cccccc}
  (cab|fic) & (cab|cif) & (icf|cba) &
  (ifc|cba) & (icf|bca) & (ifc|bca) \end{array}$&(\ref{minton})+D1\\ \hline
T4 & $\begin{array}{cccccc}
  (cba|fic) & (cba|cif) & (icf|abc) &
  (ifc|abc) & (cfi|bac) & (fci|bac) \end{array}$&
   (\ref{vdm})+(\ref{ss})+(\ref{ss})\\ \hline
\end{tabular}
\]
\end{table}

It remains to describe these ten types D1,$\ldots$,D6,T1,$\ldots$,T4.
A description is summarized in the last columns of the tables~2 and~3.
The double sums of type D1 and D2 seem to be new, and they will be
given at the end of this section. Type D3 is a double sum with one
coupling, but nevertheless one of the summations can be performed
separately by means of the Minton $_4F_3$ theorem (\ref{minton}).
What remains can then be simplified and reduces to Vandermonde's
theorem in the second summation variable. Type D4 is a double
sum with one coupling, where one of the summations can be performed
first by means of Vandermonde's $_2F_1$ theorem (\ref{vdm}), and
the remaining sum also reduces to a Vandermonde sum. Type D5 is
a double sum, again with one coupling. Here, one of the summations
can be done using Minton's $_4F_3$ theorem, and what remains is
a summation of the Saalsch\"utz $_3F_2$ type. Finally, for the
type D6 sums, one summation can be performed by means of Vandermonde's
theorem, and the remaining summation then reduces to a Saalsch\"utz
$_3F_2$ series.

For the triple sums, T1 and T2 seem to be new and will be given 
explicitly below.
In the triple sum of type T3, one summation can be performed by means
of Minton's $_4F_3$ theorem, and the remaining double sum is of type D1.
And in the triple sum of type T4, one summation can be performed
using Vandermonde's theorem, then a second summation can be performed
using the Saalsch\"utz $_3F_2$ theorem, and finally the third summation
can also be done using the Saalsch\"utz theorem.

To conclude this section, we give the explicit forms of the
summations D1, D2, T1 and T2, which cannot be performed using any
known summation theorems.

The double sum D1 is of the following type~:
\bea
&& \sum_{x,z} {(-1)^{x+z}\over x!z!}
 {(2b-x)!(a-b+d+e+i+x)!(a-b+c+x)!\over(a+b+d-e+i-x)!(-a+b+c-x)!(a-b+d+e-i+x)!}  \nn\\
&&\times {(2f-z)!(c-f+i+z)!\over
   (f+d+e+1-z)!(c+f-i-z)!(a-b-f+i+x+z)!} \nn\\[2mm]
&& = (-1)^{2c}{(a-b+d+e+i)!(a+b+d+e+i+1)!(2i)!\over
    (e+d-f)!(2a+2d+2i+1)!(e+d+f+1)!}.
\label{D1}
\eea
In this sum, the coupling comes from the factor $(a-b-f+i+x+z)$. But
$a-b-f+i$ is nonpositive; indeed, if it was positive then also
$(a-b-f+i) + (d-e+f) > 0$, or $(a+d+i)-b-e > 0$, which is in
contradiction with the fact that $(b,e,a+d+i)$ forms a triad in (\ref{9j}).
Because of this, the summation (\ref{D1}) does not have the term with
$x=z=0$ as starting value, and it cannot directly be rewritten in terms of
one of the double hypergeometric functions of the previous section. However,
we shall see in the next Section that by a simple substitution of the
summation variables, (\ref{D1}) can be cast in the form of a double
hypergeometric function.

The double sum D2 is of the type~:
\bea
&& \sum_{y,z} {(-1)^{y+z}\over y!z!}
 {(e+d-f+y)!\over(2d+1+y)!(2a-y)!(e-d+f-y)!} \times \nn\\
&& {(2b-z)!(a-b+c+z)!(a+b+i+f-y-z)!\over
   (a+b+d+e+i+1-z)!(a+b+d-e+i-z)!(-a+b+c-z)!(a-b+f-i+z)!} \nn\\[2mm]
&& = (-1)^{c+d-e+i}{(2d)!\over(d+e+f+1)!(2a+2d+1)!(d-e+f)!}.
\label{D2}
\eea
Here, there is again a coupling, but now we can assume that $2a$
determines the termination of the $y$-summation and that $-a+b+c$
determines the termination of the $z$-summation. Then, the coupling
itself never becomes zero and the expression (\ref{D2}) can indeed
be written in terms of a double hypergeometric function provided
that $a-b+f-i\geq 0$. This will
be given explicitly in the next section.

The triple sum T1 can be written in the following form~:
\bea
&& \sum_{x,y,z} {(-1)^{x+y+z}(2a+2d+x)!(c+f-i+x)!
 (a+b+d-e+i+y)!(a+b-c+y)!\over
 x!y!z!(-c+f+i-x)!(2b+1+y)!(a-b+d+e+i-y)!(a-b+c-y)!}  \nn\\
&&\times {(-d+e+f+z)!\over(2a+2d+1-z)!(d-e+f-z)!}
  {(p_1-y-z)!\over (p_2+x+y)!(p_3+x+z)!} \nn\\[2mm]
&&= {(-1)^{c+d-e-i}(2a+2d)!(a+b+d-e+i)!(-d+e+f)!\over
 (-c+f+i)!(a-b+c)!(a+b+c+1)!(b+e-a-d-i)!(d-e+f)!}, \nn\\
 &&
\label{T1}
\eea
where $p_1=a-b+c+2d$, $p_2=a+d+b-e-i$ and $p_3=-d+e-i+c$.
The upper bounds for the summation variables $x,y,z$ are given by
$-c+f+i, a-b+c, d-e+f$ respectively. Thus $p_1-y-z$ is always
positive, and provided that $p_2$ and $p_3$ are also positive
(which is possible in this case), the summation (\ref{T1})
can be rewritten as a genuine triple hypergeometric function;
this is presented in the next section.

Finally, the triple sum T2 is of the type~:
\bea
&& \sum_{x,y,z} {(-1)^{x+y+z}\over x!y!z!}
 {(2b-x)!(a-b+c+x)!(a-b+d+e+i+x)!\over(-a+b+c-x)!(a+b+d-e+i-x)!}\nn\\
&&\times {(c+f-i+y)!(d-e+f+y)!(2i+z)!
 \over(2f+1+y)!(c-f+i-y)!(d+e-f-y)!(2a+2d+1-z)!(2a-z)!}\nn\\
&&\qquad\qquad\times{(p_1-y-z)\over (p_2+x+y)!(p_3+x+z)!} \nn\\[2mm]
&& = {(-1)^{2a+c+d-e+i}(a-b+d+e+i)!(c+f-i)!(2i)!(2d)!(a+b+d+e+i+1)!\over
 (2a+2d+1)!(c-f+i)!(d+e-f)!(c+f+i+1)!(d+e+f+1)!(-c+f+i)!},\nn\\
&&
\label{T2}
\eea
where $p_1=2a+d+e-f$, $p_2=a-b+f-i$ and $p_3=-a-b-d+e+i$.
Again, under certain extra assumptions, this summation can be
rewritten as a triple hypergeometric function, given in the
following section.

Once more, it should be emphasized that the summations (\ref{D1}--\ref{T2})
cannot be simplified using known summation theorems. Also, when writing,
for example, the summation explicitly for some of the other symmetries
of type T1 (as given in Table~3), the actual expression might at first
sight look different from (\ref{T1}). Only when rewritten in the
multiple hypergeometric function notation such apparent different summations
become clearly the same.

\section{New summation formulae}

Since (\ref{D1}) is somewhat special, we shall first treat the hypergeometric
function summation that is related to (\ref{D2}).

The expression (\ref{D2}) is valid for all integer or
half-integer $a,b,c,d,e,f,i$ that satisfy the triangular
relations implied by (\ref{9j}). To write it in a more
general form, it is useful to relabel these seven independent
parameters by the following seven integer parameters~:
\beq
\begin{array}{lcl}
m&=&a+b-c\\
p&=&d-e+f\\
q&=&c-f+i\\
r&=&a-b+f-i
\end{array} \qquad
\begin{array}{lcl}
\al&=&-a-b-d-e-i-1\\
\be&=&-2b\\
\ga&=&d+e-f+1
\end{array}
\label{chD2}
\eeq
Then, (\ref{D2}) can be rewritten in terms of the double hypergeometric
function (\ref{myds})~:
\bea
&&F^{0:4;3}_{1:2;1}\left[ {\hy\atop\al+\ga} :
 {\al,\be+m,q+r+1,-m-p-q\atop \be,r+1} ;\right.\nn\\[2mm]
&&\qquad\qquad\qquad\qquad\qquad
 \left. {\al+\ga+m+p+q,\ga,-m-q-r \atop \ga+p+1} ; 1,1\right] \nn\\[3mm]
&&= (-1)^{m+q}
 {(\ga)_{p}(\al)_{m+q} \over (\be)_{m}(\al+\ga)_{m+p+q}
  (\ga+p+1)_{m+q+r} }
 {(m+q+r)!(m+p+q)! r!\over p!(q+r)!} \nn\\
&&
\label{f1}
\eea
In this form, it turns out that the above expression is valid for
all complex numbers $\al,\be,\ga$ and for all nonnegative integers
$m,p,q,r$, as long as the termination of the series is
determined by $(-m-p-q)$ for the first summation index and by
$(-m-q-r)$ for the second summation index. Thus, (\ref{f1})
is a summation theorem for a special double hypergeometric function.
The validity of (\ref{f1}) has been checked for a large number of
data by means of MACSYMA (1985). In principle, our technique provides
a proof of (\ref{f1}) only for integer values (due to integer or
half-integer angular momenta). However, the more general result (\ref{f1})
can be understood by extending some angular momentum values to the
complex plane and by considering analytic continuations of angular
momentum coefficients (thus, certain couplings correspond to
$su(1,1)$ tensor products rather than to $su(2)$ tensor products).
The property observed here is similar to the extension of some
properties of 6-$j$ coefficients to certain complex
arguments (Raynal 1979). It should be mentioned that a detailed study
of the definition and properties of generalised 9-$j$ coefficients
has not appeared in the literature.

Some summations for special Kamp\'e de F\'eriet functions (\ref{kdf}) can be
obtained from (\ref{f1}) by putting $m=0$ or $q=0$. For
example, when $q=0$, (\ref{f1}) becomes
\bea
F^{0:3}_{1:1}\left[{\hy\atop\al+\ga} :
 {\al,\be+m,-m-p\atop \be} ;
 {\al+\ga+m+p,\ga,-m-r \atop \ga+p+1} ; 1,1\right] &&\nn\\[2mm]
= (-1)^{m}
 {(\ga)_{p}(\al)_{m} \over (\be)_{m}(\al+\ga)_{m+p}
  (\ga+p+1)_{m+r} }
 {(m+r)!(m+p)! \over p!} &&
\label{f2}
\eea

Now consider the double sum (\ref{D1}). As was explained in the previous
section, there is no term with $x=z=0$, which prevents us from rewriting
(\ref{D1}) directly as a hypergeometric function. However, if we perform
the following transformation of variables~:
\[
x=-a+b+c-u,\qquad z=c+f-i-v,
\]
then (\ref{D1}) becomes
\bea
&& \sum_{u,v} {(-1)^{u+v}\over u!v!}
 {(a+b-c+u)!(c+d+e+i-u)!(2c-u)!\over(2a-c+d-e+i+u)!(-a+b+c-u)!(c+d+e-i-u)!}  \nn\\
&&\times {(-c+f+i+v)!(2c-v)!\over
   (-c+d+e+i+1+v)!(c+f-i-v)!(2c-u-v)!} \nn\\[2mm]
&& = (-1)^{a-b-f+i}{(a-b+d+e+i)!(a+b+d+e+i+1)!(2i)!\over
    (e+d-f)!(2a+2d+2i+1)!(e+d+f+1)!};
\label{D1p}
\eea
now there is a term with $u=v=0$ and one can again rewrite the equation.
It is convenient to define the following parameters~:
\beq
\begin{array}{lcl}
n_0&=&a-b+c\\
n_1&=&-a+b-d+e-i\\
n_2&=&d-e+f\\
n_3&=&c-f+i
\end{array} \qquad
\begin{array}{lcl}
\al&=&-c+d+e+i+1\\
\be&=&-c+f+i+1\\
\ga&=&2a-c+d-e+i+1,
\end{array}
\label{chD1p}
\eeq
and to set $n=n_0+n_1+n_2+n_3$.
Then, (\ref{D1p}) can be rewritten in terms of the double hypergeometric
function (\ref{myds})~:
\bea
&&F^{1:3;2}_{0:3;2}\left[ {-n\atop\hy} :
 {\ga+n_1,\be-\al-n+n_3,-n+n_0\atop \ga,1-\al-n,-n} ;
 %\right.\nn\\[2mm]
 % &&\qquad\qquad\qquad\qquad\qquad \left.
 {\be,-n+n_3 \atop \al+1,-n} ; 1,1\right] \nn\\[3mm]
&&= (-1)^{n_1+n_2}
 {(n-n_0)!(n-n_3)! (\al-\be+1)_{n-n_3} (\be)_{n_3} (\al+\ga+n)_{n_1} \over
  n! (\al+n_0)_{n-n_0} (\al+1)_{n-n_3} (\ga)_{n_1}}\nn\\
&&
\label{f1p}
\eea
In this form, the summation is valid for all nonnegative integers
$n_0, n_1, n_2$ and $n_3$, and for all complex numbers $\al,\be,\ga$
(as long as no denominators become zero). Some summation theorems for
Kamp\'e de F\'eriet functions of type $F^{1:2}_{0:2}$ could be derived
from (\ref{f1p}), for example by putting $\be=1-n_3$, or by considering
the special cases $n_0=0$ or $n_1=0$.

Next, we consider (\ref{T1}), and use the following relabelling~:
\beq
\begin{array}{lcl}
\al&=& -a+b-2d-f-i\\
\be&=& -a+b-d+e-i+1\\
\ga&=&a+b-i-f+1\\
\de&=&2b+2
\end{array}
\qquad
\begin{array}{lcl}
p&=& -c+f+i\\
q&=& a-b+c\\
r&=& d-e+f
\end{array}
\label{chT1}
\eeq
With these new parameters, the triple sum (\ref{T1}) can be rewritten
as a triple hypergeometric function (\ref{tri}) as follows~:
\bea
&&F^{0:3;4;3}_{3:0;1;0}\left[
 {\hy\atop\al+p:011,\; \be+q:101,\; \ga+r:110}:
 {\ga-\al,\be+q+r,-p\atop\hy} ; \right.\nn\\[2mm]
&&\qquad\qquad\qquad\qquad\qquad\left. {\de-\be,\ga+p,\al+r,-q\atop\de} ;
 {\be+p+q,\al-\ga,-r\atop\hy} ; 1,1,1 \right] \nn \\[3mm]
&&\qquad = { (\al)_p(\be)_q (\ga)_r \over (\ga)_p (\de)_q (\al)_r }.
\label{f3}
\eea
In this form, the above summation result is valid for all
complex parameters $\al,\be,\ga,\de$ and for all nonnegative
integers $p,q,r$ (which determine the termination of the three
summation indices). As for the previous case, it has been
verified carefully using MACSYMA.

An interesting symmetric double summation follows from (\ref{f3})
by putting $q=0$. Then, (\ref{f3}) reduces to a Kamp\'e de F\'eriet
function and we obtain~:
\beq
F^{0:3}_{1:1}\left[ {\hy \atop\be} :
{\ga-\al,\be+r,-p\atop \ga+r};{\al-\ga,\be+p,-r \atop \al+p};1,1\right]
={(\al)_p(\ga)_r \over (\ga)_p (\al)_r}.
\label{kdfsum}
\eeq
This terminating series summation can in fact be
further generalized to~:
\beq
F^{0:3}_{1:1}\left[ {\hy \atop b} :
{c-a,b+e,-d\atop c+e};{a-c,b+d,-e \atop a+d};1,1\right]
={\Ga(a+d)\Ga(c+e)\over\Ga(c+d)\Ga(a+e)},
\label{kdfinf}
\eeq
which is now valid for all complex parameters for which the
double series is convergent (that is, for which $\Re(a+d)>0$ and
$\Re(c+e)>0$). We have performed a number of numerical tests
to verify (\ref{kdfinf}).

Finally, we give the triple hypergeometric function result related
to (\ref{T2}). The new parameters are now~:
\beq
\begin{array}{lcl}
\al&=&-a+b+c\\
\be&=&2f+1\\
\ga&=&-f+e+d
\end{array}
\qquad
\begin{array}{lcl}
m&=&c-f+i\\
p&=&a-b+f-i\\
q&=&a+b-c\\
r&=&d-e+f
\end{array}
\eeq
Then, (\ref{T2}) can be rewritten as follows~:
\bea
&&F^{0:4;4;3}_{3:1;1;0}\left[
 {\hy\atop1-\ga-m-p-q:011,\; \be-\al-p-q-r:101,\; p+1:110}:\right.\nn\\[2mm]
&&\qquad\qquad\left. {m+p+1,m+\ga+\be-\al-1,1-\al,-m-q-r\atop 1-\al-q} ;
 {\al+p,r+1,1-\ga,-m\atop\be} ; \right. \nn\\[2mm]
&&\qquad\qquad\left. {\be-\al+m-p,-\ga-m-p-q-r,-m-p-q\atop\hy} ;
 1,1,1 \right] \nn \\[3mm]
&&= {(-1)^{p+q+r}
 (m+q+r)!(m+p+q)!p!(\be+\ga-1)_{m+q}(\ga)_r
 \over (m+p)!r!(\al)_q(\be)_m(\ga)_{m+p+q}
  (\be-\al-p-q-r)_{q+r}}.
\label{f4}
\eea
Again, this summation result is valid for all complex $\al,\be,\ga$
and all nonnegative integers $m,p,q,r$ such that the termination of
the three summation indices is determined by $-m-q-r$, $-m$ and
$-m-p-q$ respectively.

To conclude, by extending certain arguments, originally corresponding
to angular momenta, to the complex plane, we have been able to formulate
four summation theorems, (\ref{f1}), (\ref{f1p}), (\ref{f3}) and (\ref{f4}).
These summations have 7 independent parameters, some of which are
nonnegative integers. By specializing certain parameters in these
expressions, simple summation theorems for some Kamp\'e de F\'eriet
functions can be obtained, such as (\ref{f2}), (\ref{kdfsum})
and (\ref{kdfinf}).

\section*{Appendix}

The summation results were derived from the central formulae
(\ref{D1})--(\ref{T2}). Each of these can still be manipulated to
give interesting side results. We give one example in this Appendix.

Consider (\ref{D2}) as a first expression, and consider (\ref{D2}) with
$c$ replaced by $c+1$ as a second expression.  Adding the right hand sides of
these two expressions obviously gives zero, and since $c$ appears in
only two factors in the left hand sides the sum of these also simplifies
a lot. Thus, we obtain~:
\beas
&& \sum_{y,z} {(-1)^{y+z}\over y!z!}
 {(e+d-f+y)!\over(2d+1+y)!(2a-y)!(e-d+f-y)!} \times \\
&& {(2b-z)!(a+b+i+f-y-z)!\over
   (a+b+d+e+i+1-z)!(a+b+d-e+i-z)!(a-b+f-i+z)!} \times\\[2mm]
&&{(a-b+c+z)!\over(-a+b+c+1-z)!} = 0.
\eeas
Using the same relabelling as in (\ref{chD2}), this can be rewritten
as~:
\beas
&&F^{0:4;3}_{1:2;1}\left[ {\hy\atop\al+\ga} :
 {\al,\be+m-1,q+r+1,-m-p-q\atop \be,r+1} ;\right.\\[2mm]
&&\qquad\qquad\qquad\qquad\qquad
 \left. {\al+\ga+m+p+q,\ga,-m-q-r \atop \ga+p+1} ; 1,1\right] =0.
\eeas
Again, this is valid for all nonnegative integers $m,n,p,q$ and
all complex $\al,\be,\ga$ (as long as the denominators are nonzero).

Another remark which is worth pointing out is that some of the summation
theorems given here have a $q$-analogue. This leads to new results for
double {\em basic} hypergeometric function sums (Gasper and Rahman 1990).
For this purpose, we define the following
double basic hypergeometric function, which is a
$q$-generalization of a Kamp\'e de F\'eriet function~:
\bea
&&\Phi^{0:3}_{1:1}\left[ {-\atop c} : {a_1,a_2,a_3\atop a_4};
 {b_1,b_2,b_3\atop b_4};
x,y;q\right] = \nn\\
&&\sum_{m,n=0}^\infty
{(a_1,a_2,a_3;q)_m \over (a_4;q)_m }{(b_1,b_2,b_3;q)_n \over (b_4;q)_n }
{1\over (c;q)_{m+n}} {x^m\over(q;q)_m}{y^n\over(q;q)_n},
\eea
where the classical notation of Gasper and Rahman (1990) is used~:
\beq
(a;q)_n=\left\{
 \begin{array}{ll}
 1, & n=0, \\
 (1-a)(1-aq)\cdots (1-aq^{n-1}), & n=1,2,\ldots,\\
 \prod_{k=0}^\infty (1-aq^k),\qquad (|q|<1), & n=\infty,
 \end{array} \right.
\eeq
and $(a_1,a_2,\ldots,a_m;q)=(a_1;q)_n(a_2;q)_n\cdots (a_m;q)_n$.
Then, the following $q$-analogues of (\ref{kdfsum}) and (\ref{kdfinf}) 
are valid~:
\beq
\Phi^{0:3}_{1:1}\left[ {-\atop \be} : {\ga/\al,\be q^r,q^{-p}\atop \ga q^r};
 {\al/\ga,\be q^p,q^{-r}\atop \al q^p}; q,q;q\right] =
(\ga/\al)^{p-r} { (\al;q)_p (\ga;q)_r \over (\ga;q)_p (\al;q)_r },
\eeq
where $p$ and $r$ are positive integers; and~:
\beq
\Phi^{0:3}_{1:1}\left[ {-\atop \be} : {\ga/\al,\be\ep,1/\de\atop \ga\ep};
 {\al/\ga,\be\de,1/\ep\atop \al\de}; q,q;q\right] =
{ ( {1\over\ga\de},{1\over\al\ep};q^{-1})_\infty \over
  ( {1\over\al\de},{1\over\ga\ep};q^{-1})_\infty },
\eeq
where $|q|, |\al\de|, |\ga\ep| > 1$.



\section*{Acknowledgements}

It is a pleasure to thank Professors R.\ Askey, R.P.\ Agarwal, G.\ Gasper
and Per W.\ Karlsson for
fruitful exchanges regarding the summation theorems and Professors
G.\ Vanden Berghe and H.\ de Meyer for stimulating discussions.
We would also like to thank the referee for his remarks, which have
extended the results of the paper.
J.\ Van der Jeugt wishes to acknowledge the hospitality of the members
and Director of the Institute of Mathematical Sciences (Madras),
where part of this work was performed.
This research was partly supported by the E.E.C. (contract No.
CI1*-CT92-0101).
\vskip 1cm

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\end{itemize}
\end{document}