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\def\eeas{\end{eqnarray*}}
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\def\nj#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\} }
\def\sj#1#2#3#4#5#6{\left\{ \begin{array}{ccc}
  #1 & #2 & #3 \\  #4 & #5 & #6 \end{array} \right\} }
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\begin{document}
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\begin{center}
{\Large \bf Transformations of single and double hypergeometric series
from the triple sum series for the 9-$j$ coefficient}\\[2cm]
{\bf K.\ Srinivasa Rao\footnote{Department of Applied 
Mathematics and Computer Science,
University of Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium
(Permanent address of KSR~: Institute of
Mathematical Sciences, C.I.T.\ Campus, Madras 600 113, India. E-mail~:
rao@imsc.ernet.in).} and J.\ Van der Jeugt\footnote{Department of Applied 
Mathematics and Computer Science,
University of Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium. Research
Associate of the Fund for Scientific Research -- Flanders 
(Belgium). E-mail~: Joris.VanderJeugt@rug.ac.be.} }
\end{center}

%PACS : 02.20.+b, 02.30.+g, 02.90.+p, 03.65.-w
\vskip 1cm
\begin{abstract}
The Bailey transform for a
Saalsch\"utzian ${}_4F_3(1)$ and a transformation of a Kamp\'e de
F\'eriet function into a Saalsch\"utzian ${}_4F_3(1)$ or its Bailey
transform are derived in a novel way in this article: From  the
the simplest known formula for the 9-$j$ coefficient, due to
Ali\v sauskas-Jucys-Bandzaitis, which is the triple sum series. It 
becomes a
single, double or (remains a) triple series when one of the angular
momenta is set to zero, due to its inherent lack of
symmetry. This then is equated to a $_4F_3(1)$ representation of the
6-$j$ coefficient to which the 9-$j$ coefficient reduces when
any one of the nine angular momenta is zero. 
\end{abstract}

\vskip 1cm
\noindent Key words~: Transformations, hypergeometric series in one and
two variables, 9-$j$ recoupling coefficient. 

\noindent Running head~: Hypergeometric series and the 9-$j$
coefficient. 

\noindent Mailing address~: Department of Applied Mathematics and
Computer Science, University of Gent, Krijgslaan 281-S9, B9000 Gent,
Belgium. Tel~: ++32 9 2644812; Fax~: ++32 9 2644995.

\newpage

\noindent {1. INTRODUCTION}

The triple sum series of Ali\v sauskas, Jucys and Bandzaitis 
(Ali\v sauskas and Jucys, 1971; Jucys and Bandzaitis, 1977) is the
simplest known algebraic form for the 9-$j$ angular momentum recoupling
coefficient ubiquitous in atomic, molecular and nuclear physics. This
triple sum series has been shown (Srinivasa Rao and Rajeswari, 1989) to
be a particular case of the extremely general triple hypergeometric
series studied by Lauricella (1893), Saran (1954) and Srivastava
(1964). That this formula does not exhibit any one of the 72 symmetries
of the 9-$j$ coefficient was realized while an algorithm was proposed
(Srinivasa Rao, Rajeswari and Chiu, 1989) for its numerical computation,
where an inherent disadvantage was converted into an advantage. In this
article, another consequence of this lack of symmetry is exploited.

There are two hierarchic formulae for the 9-$j$ coefficient. The
earliest of these and the most straightforward one expresses the 9-$j$
coefficient as a (double) sum over the projection quantum numbers of
the product of six 3-$j$ coefficients. It is this formula which
exhibits the 72 symmetries of the 9-$j$ coefficient (upto an overall
sign factor) and also reveals how the 9-$j$ coefficient reduces to a
6-$j$ coefficient when any one of its nine angular momenta is set equal
to zero. It has been established (Srinivasa Rao and Rajeswari, 1993, for
references) that the 6-$j$ coefficient can be expressed in one of seven
ways as a terminating Saalsch\"utzian ${}_4F_3(1)$.

The triple sum series for the 9-$j$ coefficient, however, does not
always reduce to a 6-$j$ coefficient or a single sum series, when any
one of the nine angular momenta is set equal to zero, as a consequence
of its highly asymmetric nature referred to. Of the nine cases (for
setting an angular momentum parameter in the 9-$j$ coefficient to
zero), two cases manifestly reduce to single sum series and the other
seven cases result in (six) double and (one) triple sum series. An
analysis of these special cases is shown here to yield directly either
the well-known Bailey transform for a Saalsch\"utzian ${}_4F_3(1)$ or a
transformation of a certain Kamp\'e de F\'eriet double series denoted
by $F^{0:3;3}_{1:1;1}(1,1)$ into a Saalsch\"utzian ${}_4F_3(1)$ or its
Bailey transform.

In section~2, the basic formulae required in our study are given and in
section~3, the nine cases where a 9-$j$ coefficient reduces to a 6-$j$
coefficient are analysed and the resulting transformation formulae
presented. In section~4, the results are discussed.

\newpage
\noindent {2. FORMULAE FOR THE 9-$j$ COEFFICIENT}

The hierarchic formula which can be derived from the recoupling of four
angular momenta in two different ways: ${\vec l}_1+{\vec s}_1={\vec
j}_1$, ${\vec l}_2+{\vec s}_2={\vec j}_2$ and ${\vec j}_1+{\vec j}_2={\vec
J}$ called the $jj$-coupling; ${\vec l}_1+{\vec l}_2={\vec
L}$, ${\vec s}_1+{\vec s}_2={\vec S}$ and ${\vec L}+{\vec S}={\vec J}$
called the $LS$-coupling scheme, expresses the $LS$-$jj$
transformation coefficient or the 9-$j$ coefficient as a sum over two
projection quantum numbers of six 3-$j$ coefficients, each of which
corresponds to a coupling of two angular momenta to give a third
(Edmonds, 1957). However, the hierarchic formula often used in all
numerical work by atomic, molecular and nuclear physicists is~:
\beq
\nj{a}{b}{c}{d}{e}{f}{g}{h}{i} = \sum_x (-1)^{2x}(2x+1)
 \sj{a}{d}{g}{h}{i}{x} \sj{b}{e}{h}{d}{x}{f} \sj{c}{f}{i}{x}{a}{b} 
\label{1}
\eeq
which is a single sum over a product of the three 6-$j$ coefficients.
The 6-$j$ coefficient can be expressed in terms of the ${}_4F_3(1)$'s
in two different ways (Srinivasa Rao and Rajeswari, 1993, and references
therein)~:
\bea
&&\sj{a}{b}{e}{d}{c}{f} = (-1)^{E+1}\De(abe)\De(cde)\De(acf)\De(bdf)
\label{2} \\ 
&&\quad\times\Ga(1-E)[\Ga(1-A,1-B,1-C,1-D,F,G)]^{-1} {\
}_4F_3(A,B,C,D;E,F,G;1) \nn
\eea
where
\beas
&&A=e-a-b,\quad B=e-c-d,\quad C=f-a-c,\quad D=f-b-d,\\
&&E=-a-b-c-d-1,\quad F=e+f-b-c+1,\quad G=e+f-a-d+1
\eeas
and two other members belonging to this set I of three ${}_4F_3$'s
correspond to (\ref{2}) being written out explicitly for
$\sj{e}{b}{a}{f}{c}{d}$ and $\sj{e}{a}{b}{f}{d}{c}$. Equivalently,
\bea
&&\sj{a}{b}{e}{d}{c}{f} = (-1)^{A'-2}\De(abe)\De(cde)\De(acf)\De(bdf)
\label{3}\\ 
&&\quad\times\Ga(A')[\Ga(1-B',1-C',1-D',E',F',G')]^{-1} {\
}_4F_3(A',B',C',D';E',F',G';1) \nn
\eea
where
\beas
&&A'=a+b+e+2,\quad B'=a-c-f, \quad C'=b-d-f,\quad D'=e-c-d,\\
&&E'=a+b-c-d+1,\quad F'=a+e-d-f+1,\quad G'=B+e-c-f+1
\eeas
and three other members belonging to this set II of ${}_4F_3(1)$'s
correspond to (\ref{3}) being written out explicitly for
$\sj{c}{d}{e}{b}{a}{f}$, $\sj{a}{c}{f}{d}{b}{e}$ and
$\sj{d}{b}{f}{a}{c}{e}$. These sets have been shown to be necessary and
sufficient to account for the 144 symmetries of the 6-$j$ coefficient.
Solving the equations for the parameters of the $_4F_3(1)$ on the 
rhs of (2), taking into account the Saalsch\"utzian condition on them:
$A+B+C+D+1 = E+F+G$, for the parameters of the 6-$j$ coefficient and 
using these in (3) and equating the rhs of (2) and (3) leads to 
the ``reversal'' formula for the ${}_4F_3(1)$. In fact, Srinivasa Rao 
and Rajeswari (1993) established that the two sets of $_4F_3(1)$'s
for the 6-$j$ coefficient are related to each other by this reversal 
formula. In 
(\ref{2}) and (\ref{3}), the function $\De(xyz)$ represents~:
\beq
\De(xyz)=[(-x+y+z)!(x-y+z)!(x+y-z)!/(x+y+z+1)!]^{1/2}
\eeq
which reflects the triangle inequality condition satisfied by three
angular momenta and it vanishes if $x,y,z$ do not form a vector triad.

The hierarchic formula where the 9-$j$ coefficient is expressed as a
sum over a product of six 3-$j$ coefficients is highly symmetric.
It is the one which reveals, via the symmetries of the 3-$j$
coefficient, the 72 symmetries (upto an overall phase factor) of the
9-$j$ coefficient: its invariance to column and row
permutations and a reflection about the diagonal of the nine parameters
of the symbol on the lhs of (\ref{1}). It is also the formula which
shows explicitly that when any one of the nine angular momenta is set
equal to zero, two of the six 3-$j$ coefficients become reduce to
simple expressions 
and the remaining four 3-$j$ coefficients can be summed over their
projection quantum numbers to yield a 6-$j$ (or Racah) coefficient,
defined as the recoupling coefficient of four angular momenta (Edmonds,
1957; Biedenharn and Louck, 1981; Varshalovich {\em et al}, 1988). Thus,
we have~:
\bea
&&\nj{0}{e}{e}{f}{d}{b}{f}{c}{a}=
\nj{e}{0}{e}{c}{f}{a}{d}{f}{b} = \nj{f}{f}{0}{d}{c}{a}{b}{a}{e}\nn \\
&=&\nj{f}{b}{d}{0}{e}{e}{f}{a}{c}=
\nj{a}{f}{c}{e}{0}{e}{b}{f}{d} = \nj{b}{a}{e}{f}{f}{0}{d}{c}{e} \nn\\
&=&\nj{e}{d}{c}{e}{b}{a}{0}{f}{f}=
\nj{c}{e}{d}{a}{e}{b}{f}{0}{f} = \nj{a}{b}{e}{c}{d}{e}{f}{f}{0} \nn\\
&&\qquad\qquad={(-1)^{b+c+e+f}\over[(2e+1)(2f+1)]^{1/2}}
\sj{a}{b}{e}{d}{c}{f} .
\label{5}
\eea

The simplest known algebraic form for the 9-$j$ coefficient due to
Ali\v sauskas and Jucys (1971) is the triple sum series~:
\bea
&& \nj{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{6}
\eea
where
\beq
\begin{array}{l}
0\leq x\leq\min(x_4,x_5)=X\\
0\leq y\leq\min(y_4,y_5)=Y\\
0\leq z\leq\min(z_4,z_5)=Z
\end{array}
\label{7}
\eeq
with
\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{8}
\eeq
and
$$
(a,b,c)  =  \left[{(a-b+c)!(a+b-c)!(a+b+c+1)!\over (-a+b+c)!}\right]^{1/2}.
$$
The lack of symmetry of the triple sum series is due to the nature of
the eighteen parameters $x_1,x_2,\ldots,p_3$, given in (\ref{8}),
amongst which nine relations exist.

For the sake of typographical felicity, using the following notation
for Pochhammer symbols~:
\beas
&& (\la)_k={\Ga(\la+k)\over \Ga(\la)} =\la(\la+1)\cdots (\la+k-1),\quad
k\geq 0,\\
&&\hbox{and } (\la)_{-k}=(-1)^k/(1-\la)_k, \quad k<0,
\eeas
eq.~(\ref{6}) can be rewritten as~:
\bea
&& \nj{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\\
&&{\Ga(1+x_1,1+x_2,1+x_3,1+y_1,1+y_2,1+z_1,1+z_2,1+p_1)\over
\Ga(1+x_4,1+x_5,1+y_3,1+1+y_4,1+y_5,1+z_3,1+z_4,1+z_5,1+p_2,1+p_3)}\times\nn\\
&&\sum_{x,y,z} {1\over x!y!z!}
{(1+x_2)_x (1+x_3)_x (-x_4)_x (-x_5)_x \over(-x_1)_x}
{(1+y_1)_y (1+y_2)_y (-y_4)_y (-y_5)_y \over(1+y_3)_y} \nn\\
&& \times {(1+z_2)_z (-z_3)_z (-z_4)_z (-z_5)_z \over(-z_1)_z}
  {1\over(-p_1)_{y+z}(1+p_2)_{x+y}(1+p_3)_{x+z} }
\label{9}
\eea
where
\beq \Ga(x,y,\ldots) = \Ga(x)\Ga(y)\cdots. \eeq
Here (\ref{6}) and (\ref{9}) are equivalent ways of writing the triple
sum series and we give both these here, since it is advantegeous to
choose one or the other as the starting point to obtain our results.

\noindent {3. TRANSFORMATIONS}

It is to be noted, from (\ref{7}), that when we set $a,b,c,\cdots,i=0$,
in succession, for $c=0$ and $e=0$, $X=0=Z$ and $X=0=Y$,
respectively, so that (\ref{6}) or (\ref{9}) reduces to a single sum
over $y$ or $z$. The seven other cases, except $h=0$, reduce (\ref{9})
to a double sum series, since only one of $X, Y, Z$ becomes 0. For
$h=0$, which occurs in both $y_4$ and $y_5$, though $b=e$ and $g=i$,
neither $y_4$ nor $y_5$ becomes 0 and so the triple sum remains a
triple sum except for its reduction to a function of six angular
momenta. Obviously, the number of independent angular momenta reduces
to six in all the nine cases.

First, we consider the cases $c=0$ and $e=0$, which reduce (\ref{9}) to
a single sum series in either $y$ or $z$, respectively. For $c=0$, we
have, 
\bea
&& \nj{a}{a}{0}{d}{e}{f}{g}{h}{f}=
{(d,a,g)(a,e,h)(f,g,h) \over(d,e,f)(a,a,0)(f,0,f)} \times \nn\\
&&{\Ga(1+x_1,1+x_2,1+y_1,1+y_2,1+z_1,1+p_1)\over
\Ga(1+x_4,1+y_3,1+y_4,1+y_5,1+z_3,1+z_4,1+p_2)}\times\nn\\
&& {}_4F_3\left( {1+y_1,1+y_2,-y_4,-y_5 \atop -p_1,1+p_2,1+y_3} ;1\right)
\label{11}
\eea
where
\beq
\begin{array}{lll}
x_1=2f        &     y_1=-a+e+h   & z_1=2a              \\
x_2=d+e-f     &     y_2=g+h-f    & z_3=a+d+g+1          \\
x_4=-d+e+f    &     y_3=2h+1     & z_4=a+d-g          \\
p_1=a+d-h+f\quad &  y_4=a+e-h    &             \\
              &     y_5=g-h+f    &          \\
              &     p_2=-a+d-f+h\quad &  \
\end{array}
\label{12}
\eeq
From (\ref{5}) we know that
\beq
\nj{a}{a}{0}{d}{e}{f}{g}{h}{f} = {(-1)^{a+e+f+g}\over
[(2a+1)(2f+1)]^{1/2}} \sj{h}{g}{f}{d}{e}{a}.
\label{13}
\eeq
Using (\ref{2}), and a symmetry of the 6-$j$ coefficient, we rewrite
(\ref{11}) as~:
\bea
&&\nj{a}{a}{0}{d}{e}{f}{g}{h}{f} = 
\frac{(-1)^{a+e+f+g}(-1)^{E+1}}{[(2a+1)(2f+1)]^{1/2}}
\De(aeh)\De(fgh) \De(agd)\De(fed) \times \nn\\
&&\Ga(1-E)[\Ga(1-A,1-B,1-C,1-D,F,G)]^{-1} {\ }_4F_3 \left( {A,B,C,D
\atop E,F,G};1\right),
\label{14}
\eea
where
\bea
&&A=h-a-e,\quad B=h-f-g,\quad C=d-a-g,\quad D=d-e-f,\nn\\
&&E=-a-e-f-g-1,\quad F=d+h-e-g+1,\quad G=d+h-a-f+1,
\label{15}
\eea
and ${}_4F_3(1)$ is a Saalsch\"utzian. Solving (\ref{15}) for
$a,d,e,f,g$ and $h$, we rewrite (\ref{12}) in terms of these and after
simplifications, we get on equating (\ref{14}) and (\ref{11})~:
\bea
{\ }_4F_3 \left( {A,B,C,D\atop E,F,G};1\right)&=&
{\Ga(E+F-A-B-D,F,E+F-A-B-C,F-C-D)\over \Ga(F-C,F-D,1-G,E+F-A-B)} \nn\\
&\times& {\ }_4F_3 \left( {E-A,E-B,C,D
\atop E,E+F-A-B,E+G-A-B};1\right)
\label{16}
\eea
which is exactly the Bailey transformation for the Saalschu\"utzian
$_4F_3(1)$ (cf.\ Slater, 1966, p.~64, Eq.~(2.4.1.7)). 

The case $e=0$ proceeds along the same lines and it also yields
(\ref{16}), the Bailey transform for a Saalsch\"utzian ${}_4F_3(1)$.

For $a=0$, the triple sum in (\ref{9}) reduces to a double sum over $x$
and $y$~:
\bea
&&\nj{0}{b}{b}{d}{e}{f}{d}{h}{i} = (-1)^{x_5}
{(d,0,d)(b,e,h)(i,d,h) \over(d,e,f)(b,0,b)(i,b,f)} \times \nn\\
&&{\Ga(1+x_1,1+x_2,1+y_1,1+y_2,1+z_2)\over
\Ga(1+x_4,1+x_5,1+y_3,1+y_4,1+z_3,1+p_2)}\times\nn\\
&& F^{0:3;3}_{1:1;1}\left({\hy\atop 1+p_2}:{1+x_2,-x_4,-x_5\atop -x_1} ;
{1+y_1,1+y_2,-y_4\atop 1+y_3} ;1,1\right)
\label{17}
\eea
where
\beq
\begin{array}{lll}
x_1=2f        &     y_1=-b+e+h   & z_2=2b              \\
x_2=d+e-f     &     y_2=d+h-i    & z_3=2d+1,          \\
x_4=-d+e+f    &     y_3=2h+1     &           \\
x_5=b+f-i\quad&     y_4=b+e-h    &             \\
              &     p_2=-b+d-f+h\quad &  \
\end{array}
\label{18}
\eeq
and we have used the definition of the double hypergeometric function
(cf.\ Van der Jeugt, Pitre and Srinivasa Rao, 1994)~:
\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{19}
\eeq
which is a special case of a very general function defined by
Srivastava and Daoust (1969). We note that in (\ref{17}) though 
$-x_1$ appears as a denominator parameter, the zero due to this will
occur only after the zero in the numerator occurs due to $-x_4, -x_5$.
Similar situations arise later on in (\ref{31}) and (\ref{36}) also.

From (\ref{5}) and (\ref{3}), we have
\bea
&&\nj{0}{b}{b}{d}{e}{f}{d}{h}{i} =
{(-1)^{b+d+f+h}\over[(2b+1)(2d+1)]^{1/2}} \sj{i}{f}{b}{e}{h}{d} \nn\\
&&= {(-1)^{b+d+f+h}\over[(2b+1)(2d+1)]^{1/2}}(-1)^{A_1-2}
\De(ifb)\De(beh) \De(dhi)\De(efd) \times \nn\\
&&\Ga(A_1)[\Ga(1-B_1,1-C_1,1-D_1,E_1,F_1,G_1)]^{-1} {\ }_4F_3 \left(
{A_1,B_1,C_1,D_1 \atop E_1,F_1,G_1};1\right),
\label{20}
\eea
where
\bea
&&A_1=d+h+i+2,\quad B_1=h-e-b,\quad C_1=i-f-b,\quad D_1=d-e-f,\nn\\
&&E_1=h+i-e-f+1,\quad F_1=d+i-b-e+1,\quad G_1=d+h-b-f+1.
\label{21}
\eea
We now introduce a set of parameters~:
\beq
A=1+x_2,\ B=-x_4,\ C=-x_5,\ D=1+p_2,\ E=-x_1,\ E'=1+y_3
\label{22}
\eeq
in terms of which (\ref{17}) and (\ref{20}) are rewritten and on
equating the two, after simplifications, we get~:
\bea
&&F^{0:3;3}_{1:1;1}\left( {\hy\atop D}:{A,B,C \atop E}; {D-A,D-B,D-C\atop
E'} ; 1,1\right) \nn\\
&&=
{\Ga(A+B+C-2D+E',1+B-E,1+C-E,E')\over\Ga(1-E,A-D+E',B+C-D+E',1+B+C-E)}
\nn\\ 
&&\times {\ }_4F_3\left({A+B+C-D-E+E', D-A, B,C \atop D, B+C-D+E',
1+B+C-E}; 1\right)
\label{23}
\eea
which is a transformation of a Kamp\'e de F\'eriet function into a
Saalsch\"utzian ${}_4F_3(1)$.

When $b$ is set equal to zero, the triple sum (\ref{6}) reduces to a
double sum over $x$ and $z$. However, since $z_2=0$ and $z_5=z_1$, the
$z$-sum is~:
\beq
\sum_z {(-1)^z (p_1-z)!\over (z_3-z)!(z_4-z)!(p_3+x+z)!}
\label{24}
\eeq
Changing the summation index $z\rightarrow z_3-z$, we have for the
$z$-sum~:
\bea
&&(-1)^{z_3}\sum_z{(-1)^z(p_1-z_3+z)!\over
z!(z_4-z_3+z)!(p_3+z_3+x-z)!}\nn \\
&=& {(-1)^{z_3}(p_1-z_3)!\over(z_4-z_3)!(p_3+z_3+x)!} {\ }_2F_1\left(
{1+p_1-z_3, -p_3-z_3-x \atop 1+z_4-z_3};1\right) 
\label{25}
\eea
which can be summed using the Gauss summation theorem to give for the
rhs of (\ref{25})~:
\beq
{(-1)^{z_3}(p_1-z_3)!\Ga(z_3+z_4-p_1+p_3+x) \over (p_3+z_3+x)!
\Ga(z_4-p_1, 1+z_4+p_3+x)}
\label{26}
\eeq
where $z_3=a+d+g+1$, $z_4=a+d-g$, $p_1=a+d-e+i$ and $p_3=-a-f+i$. This
reduces the double sum to a single sum over $x$ and the resulting
${}_4F_3(1)$ when equated to the corresponding ${}_4F_3(1)$ from
(\ref{5}), can be shown to be nothing but the Bailey transform for
terminating ${}_4F_3(1)$'s given by (\ref{16}).

The cases $d=0$ and $i=0$ are similar to the case $b=0$ in that they
result in double sums (over $x$ and $y$ or $x$ and $z$, respectively)
and in both cases, the ${}_2F_1$ Gauss summation can be used to perform the
$x$-sum. The resulting single sum series are ${}_4F_3(1)$'s which are
Saalsch\"utzian and on being equated to the corresponding appropriate
${}_4F_3(1)$'s from (\ref{5}), yield the Bailey transformation
(\ref{16}). 

The case $f=0$ results from (\ref{9}) in a genuine double sum over $y$
and $z$ and when it is equated to the appropriate ${}_4F_3(1)$ form for
the 6-$j$ coefficient in (\ref{5}), we obtain the following Kamp\'e de
F\'eriet transformation~:
\bea
&&F^{0:3;3}_{1:1;1}\left( {\hy\atop D}:{A,B,C \atop E}; {D-A,D-B,D-C\atop
E'} ; 1,1\right) \nn\\
&&=(-1)^{B+C-D}{\Ga(A+B+C-2D+E',1+D-B-E',1+D-C-E') \over
\Ga(A-D+E',1+D-E')} \nn\\
&&\times {\Ga(E-B-C,E,1+B-D,1+C-D)\over\Ga(1-D,1-E',E-B,E-C,1+B+C-D)}\nn\\
&&\times {\ }_4F_3\left({1-A+D-E',1+A+B+C-D-E, B,C \atop
1+B+C-E,1+B+C-D, 1+D-E'}; 1\right).
\label{27}
\eea
Examination of the numerator and denominator parameters of the
${}_4F_3(1)$'s in (\ref{23}) and (\ref{27}) shows that two of the
numerator parameters $B$ and $C$, and a denominator parameter $1+B+C-E$
are occuring in both. It can be shown by using the property of the
Pochhammer symbols (or Gamma functions)~:
\beq (1-z-n)_n = (-1)^n (z)_n 
\label{28}
\eeq
that the two ${}_4F_3(1)$'s are related to each other by the Bailey
transformation (\ref{16}). Thus, the transformations (\ref{23}) and
(\ref{27}) are related to each other through the Bailey transformation
for the ${}_4F_3(1)$'s.

For $g=0$, we get a double sum series over $x$ and $z$. However, in the
parameters of the $z$-sum, 
\beq z_1=2a, z_2=-a+b+c, z_3=2a+1=z_1+1, z_5=a-b+c, p_3=-a+b-f+h,
\label{29} \eeq
a numerator and denominator parameter differ
by 1, so that from (\ref{9})~:
\beq
\sum_z {(1+z_2)_z (-z_3)_z (-z_5)_z \over z!(-z_1)_z (1+p_3)_{x+z}} =   
{1\over (1+p_3)_x }{\ }_3F_2\left({a',b',b_1+m_1\atop b'+1,b_1};1\right)
\label{30}
\eeq
where $a'=-z_5$, $b'=-z_1-1$, $b_1=1+p_3+x$ and $m_1=z_2-p_3-x$. The
${}_3F_2$ is summable using Minton's theorem (1970) and results in~:
\[
{1\over (1+p_3)_x}{\Ga(1-a',1+b')\over\Ga(1-a'+b')}{(b_1-b')_{m_1} \over
(b_1)_{m_1}} 
\]
which can be simplified into~:
\[
{\Ga(-z_1,1+z_5,1+p_3)\over\Ga(1+z_2,-z_1+z_5,2+p_3+z_1)}{1\over
(2+p_3+z_1)_x} .
\]
The Pochhammer $x$ factor from this Minton sum reduces the double sum
and results in the Saalsch\"utzian~:
\beq
{}_4F_3\left( {1+x_2,1+x_3,-x_4,-x_5\atop -x_1,1+p_2,
2+p_3+z_1};1\right) 
\label{31}
\eeq
which on being equated to the corresponding ${}_4F_3(1)$ from (\ref{5})
(and (\ref{2})) gives us again the Bailey transformation (\ref{16}).

Finally, the case $h=0$ is the one in which the triple sum remains as a
triple sum, since $X,Y$ and $Z$ are non-zero. However, the set of
parameters (\ref{8}) in this case is~:
\beq
\begin{array}{lll}
x_1=2f        &     y_1=0   & z_1=2a              \\
x_2=b+d-f     &     y_2=0    & z_2=-a+b+c          \\
x_3=c-f+g     &     y_3=1     & z_3=a+d+g+1          \\
x_4=b-d+f    &     y_4=2b    & z_4=a+d-g            \\
x_5=c+f-g     &     y_5=2g    & z_5=a-b+c         \\
p_1=a+d+g\quad   &     p_2=-b+d-f\quad & p_3=-a+b-f+g. \
\end{array}
\label{32}
\eeq
Since $y_1=y_2=0$, $y_3=1$ and $z_3=p_1+1$, the sum over $y$ in
(\ref{6}) becomes~:
\beq
\sum_y {(-1)^y y!(p_1-z-y)!\over(y+1)!(y_4-y)!(y_5-y)!(p_2+x+y)!}.
\label{33}
\eeq
Changing the summation index from $y\rightarrow y_4-y$ and rearranging
this sum results in a ${}_3F_2(1)$ of the form given in (\ref{30}) but
with 
\[
a'=-p_2-y_4-x,\ b'=-y_4-1,\ b_1=1-y_4+y_5,\ m_1=p_1-y_5-z.
\]
Using the Minton theorem to sum this ${}_3F_2(1)$, after
simplifications, the $y$-sum (\ref{33}) becomes~:
\beq
{(-1)^{x_2+y_4+2x+z} \Ga(2+p_1,1-p_2) \over \Ga(2+y_4, 2+y_5)
(-1-p_1)_z (p_2)_x}.
\label{34}
\eeq
The parameters (\ref{32}) reveal~: $p_2=-x_4$ and $1+p_1=z_3$. Due to
these relations, when we use (\ref{34}) for (\ref{33}) in (\ref{6}), we
get for the $h=0$ case~:
\beas
&&\nj{a}{b}{c}{d}{b}{f}{g}{0}{g} = {(-1)^{x_2+y_4}\over\Ga(2+y_4,2+y_5)}
{(d,a,g)(b,b,0)(g,g,0) \over(d,b,f)(b,a,c)(g,c,f)} \times \\
&&\sum_{x,z}{(-1)^z\over x!z!}{(x_1-x)!(x_2+x)!(x_3+x)!(z_1-z)!(z_2+z)!
\over (x_5-x)!(p_3+x+z)!(z_4-z)!(z_5-z)!}
\eeas
which can be rearranged into a Kamp\'e de F\'eriet function~:
\bea
&&\nj{a}{b}{c}{d}{b}{f}{g}{0}{g} = {(-1)^{x_2+x_4+x_5}\over\Ga(2+y_4,2+y_5)}
{(d,a,g)(b,b,0)(g,g,0) \over(d,b,f)(b,a,c)(g,c,f)}
{x_1!x_2!x_3!z_1!z_2!\over x_5!z_4!z_5!}\times \nn\\
&&
F^{0:3;3}_{1:1;1}\left({\hy\atop 1+p_3}:{1+x_2,1+x_3,-x_5\atop -x_1};
{1+z_2,-z_4,-z_5 \atop -z_1};1,1\right).
\label{36}
\eea
Thus, this case $h=0$, also reduces to a double sum series and when
this is equated to the ${}_4F_3(1)$ which corresponds to the 6-$j$
coefficient in (\ref{5}), we get the transformation (\ref{23}) after we
identify~:
\beas
&&A=1-a+b+c,\ B=-a+b-c,\ C=-a-d+g,\\
&&D=1-a+b-f+g,\ E=2a,\ E'=-2f.
\eeas

\noindent {4. DISCUSSION OF RESULTS}

The realization that setting any one of the nine angular momenta in the
9-$j$ coefficient reduces always to a 6-$j$ coefficient, through the
hierarchic formula for the 9-$j$, but that the highly asymmetric triple
sum series form becomes a single series in two cases, a double series
in six cases and remains a triple series in one case, led us to the
study of these nine cases in this article. We have shown that in two
cases, we obtain the Bailey transformation of Saalsch\"utzian
${}_4F_3(1)$'s and in all the other seven cases, the transformation of
a certain Kamp\'e de F\'eriet double series into a Saalsch\"utzian
${}_4F_3(1)$ or its Bailey transform. Since the ${}_4F_3(1)$'s involved
are Saalsch\"utzian, the transformations involved are for functions of
six independent parameters.

The Bailey transformation of a terminating ${}_4F_3(1)$ is a well-known
result and it is
derived here from the 9-$j$ recoupling angular momentum coefficient (or
from the Racah-Wigner algebra for quantum theory of angular momentum).
Using the Jucys mirror symmetry, $j\rightarrow -j-1$, Varshalovich
{\em et.\ al.} (1988; see also Vilenkin and Klimyk, 1991) have shown that the 
Bailey transformation for the Saalsch\"utzian $_4F_3(1)$ can be derived
from the $_4F_3(1)$ representations for the 6-$j$ coefficient. On the 
other hand, we have shown (Srinivasa Rao {\em et al.}, 1975) that the use of
the Bailey transformation on a $_4F_3(1)$ representation of the 6-$j$ 
coefficient belonging to set I (2) or set II (3) of $_4F_3(1)$'s results
{\em at best} in the mathematical Jucys symmetry
and otherwise leads to a symmetry like that obtained by Minton (1970) for the
6-$j$ coefficient. 

The Kamp\'e de F\'eriet double series transformation into a
Saalsch\"utzian ${}_4F_3(1)$, or its Bailey transform, can be shown to
be not new. It is, in fact, the terminating form of a
formula given by Karlsson (1994) in his proof of one of our double
hypergeometric summation formulae (cf.\ Van der Jeugt, Pitre and
Srinivasa Rao, 1994, eq.~(33)). The formula in question is (Karlsson,
1994, eq.~(5))~:
\beas
&&F_{1:1;1}^{0:3;3}\left( {\hy \atop d}:{a,b,c\atop
e};{a',b',c'\atop e'};1,1\right)\\
&=&{\Gamma(d,e',a'+b'-d,d+e'-a'-b'-c') \over\Gamma(
a',b',e'-c',d+e'-a'-b')} \\[1mm]
&&\times {\ }_4F_3\left({a,b,e-c,d+e'
-a'-b'-c' \atop e,1+d-a'-b', d+e'-a'-b'};1\right) \\[1mm]
&+&{\Gamma(d,e,a+b-d,d+e-a-b-c) \over
\Gamma(a,b,e-c,d+e-a-b)}\\[1mm]
&&\times {\ }_4F_3\left({a',b',e'-c',d+e
-a-b-c \atop e',1+d-a-b, d+e-a-b};1\right)
\eeas
which holds for $a'=d-a$ and $b'-d=b$, under the condition
$\Re(1-d+c+c')>0$, in addition to the conditions $\Re(d+e-a-b-c)>0$ and
$\Re(d+e'-a'-b'-c')>0$. It is also one of a set of transformation
formulae of Kamp\'e de F\'eriet series given by Pitre and Van der Jeugt
(1996).
Consider now in the above formula the extra relation $c'=d-c$, and
interchange $a$ and $c$ ($a\leftrightarrow c$) in the resulting
formula. In it, if $b$ or $c$ is a negative integer, then the second
term on the rhs vanishes so that we have~:
\beas
&&F_{1:1;1}^{0:3;3}\left( {\hy \atop d}:{a,b,c\atop
e};{d-a,d-b,d-c\atop e'};1,1\right)\\
&=&{\Gamma(d,e',d-b-c,a+b+c-2d+e') \over\Gamma(
d-b,d-c,e'+a-d,b+c-d+e')} \\[1mm]
&&\times {\ }_4F_3\left({b,c,e-a,a+b+c-2d+e' 
\atop b+c-d+e',e,1+b+c-d};1\right) .
\eeas
Bailey transform the ${}_4F_3(1)$ on the rhs and use the property of
the Pochhammer symbols~: $(\al)_{-n}=(-1)^n/(1-\al)_n$ to obtain
(\ref{23}) (with $a,b,c,d,e,e'$ replacing $A,B,C,D,E,E'$).

It has to be noted that the results presented here made use of that
${}_4F_3(1)$ form for the 6-$j$ coefficient, belonging to set I or set
II, which has the larger number of parameters expressible directly in
terms of the parameters (\ref{8}) of the triple sum series. This is
achieved by inspection and with the help of Mathematica. In fine, the
results obtained though not new, the methodology to obtain these
transformations is novel and it is a direct consequence of the triple
sum series of Ali\v sauskas, Jucys and Bandzaitis and the fundamental
property of the 9-$j$ coefficient that it reduces to a 6-$j$
coefficient when any one of its angular momenta is set to zero.



\noindent{ACKNOWLEDGEMENTS}

One of us (KSR) wishes to thank Prof.\ G.\ Vanden Berghe for the
excellent hospitality at the University of Ghent and for his interest
in the work. This work was partly supported by the E.C. (contract No.
CI1*-CT92-0101).
\vskip 1cm

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\end{document}





