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hg21 is a package to get contiguity relations
of the hypergeometric function 2F1.

% $OpenXM: OpenXM/src/asir-contrib/packages/doc/hg21-ja.tex,v 1.1 2000/12/16 13:29:46 takayama Exp $
\documentclass{jarticle}

\begin{document}

{\tt u1(a,b,c,z)} $B$NLa$99TNs$r(B $U_1$ $B$H$9$k$H$-(B
(u $B$O(B ``up'' $B$N(B u $B$G$"$k(B),
$$\pmatrix{
     F'(a+1,b,c;z) \cr
     F(a+1,b,c;z) \cr} = U_1 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$,$J$j$?$D(B.

{\tt u2(a,b,c,z)} $B$NLa$99TNs$r(B $U_2$ $B$H$9$k$H$-(B
(u $B$O(B ``up'' $B$N(B u $B$G$"$k(B),
$$\pmatrix{
     F'(a,b+1,c;z) \cr
     F(a,b+1,c;z) \cr} = U_2 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$,$J$j$?$D(B.

{\tt u3(a,b,c,z)} $B$NLa$99TNs$r(B $U_3$ $B$H$9$k$H$-(B
(u $B$O(B ``up'' $B$N(B u $B$G$"$k(B),
$$\pmatrix{
     F'(a,b,c+1;z) \cr
     F(a,b,c+1;z) \cr} = U_3 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$,$J$j$?$D(B.

{\tt d1(a,b,c,z)} $B$NLa$99TNs$r(B $D_1$ $B$H$9$k$H$-(B
(d $B$O(B ``down'' $B$N(B d $B$G$"$k(B),
$$\pmatrix{
     F'(a-1,b,c;z) \cr
     F(a-1,b,c;z) \cr} = D_1 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$,$J$j$?$D(B.

{\tt d2(a,b,c,z)} $B$NLa$99TNs$r(B $D_2$ $B$H$9$k$H$-(B
(d $B$O(B ``down'' $B$N(B d $B$G$"$k(B),
$$\pmatrix{
     F'(a,b-1,c;z) \cr
     F(a,b-1,c;z) \cr} = D_2 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$,$J$j$?$D(B.

{\tt d3(a,b,c,z)} $B$NLa$99TNs$r(B $D_3$ $B$H$9$k$H$-(B
(d $B$O(B ``down'' $B$N(B d $B$G$"$k(B),
$$\pmatrix{
     F'(a,b,c-1;z) \cr
     F(a,b,c-1;z) \cr} = D_3 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$,$J$j$?$D(B.

$B$3$l$i$O(B, $B%,%&%9$ND64v2?4X?t$N$h$/$7$i$l$?8x<0$G$"$k(B.\\
$BNc(B:
\begin{verbatim}
[377] load("hg21")$
[378] u1(a,b,c,z);
[ (b*z-c+a+1)/(-a*z+a) (b)/(-z+1) ]
[ (z)/(a) 1 ]
\end{verbatim}


{\tt hg21\_check()} $B$G$O$3$l$i$NJQ498x<0$,$?$@$7$$$+(B
$B$I$&$+$r(B,
$B$?$H$($P(B,
\verb@  R = d1(a+1,b,c,z)*u1(a,b,c,z); @
$B$,C10L9TNs$+$I$&$+$r$_$k$3$H$K$h$j(B, $B%A%'%C%/$7$F$$$k(B.

$p, q, r$ $B$r@0?t$H$9$k$H$-(B,
$B$3$l$i$N9TNs$r$+$1;;$9$k$3$H$K$h$j(B,
$B<!$N<0$r$_$?$9(B $T$ $B$rF@$k$3$H$,2DG=$G$"$k(B.
$$\pmatrix{
     F'(a+p,b+q,c+r;z) \cr
     F(a+p,b+q,c+r;z) \cr} = T 
  \pmatrix{
     F'(a,b,c;z) \cr
     F(a,b,c;z) \cr} 
$$
$B$?$@$7(B, $B9TNs(B $U_i, D_i$ $B$NJ,Jl$,(B $0$ $B$K$J$i$J$$(B
$B$3$H$,I,MW$G$"$k(B.

\noindent
$BNc(B:
\begin{verbatim}
[379] tam(1/2,1/2,1,3/4);
[Aplus,Bminus,Cplus]=[3,-3,6]
[[ 9402863/1505280 170306533/752640 ]
[ -8131157/430080 -147868387/215040 ],[7/2,-5/2,7]]
[380] 
\end{verbatim}
$B$3$N=PNO$O(B,
$$\pmatrix{
     F'(1/2,1/2,1;3/4) \cr
     F(1/2,1/2,1;3/4) \cr} = T 
  \pmatrix{
     F'(7/2,-5/2,7;3/4) \cr
     F(7/2,-5/2,7;3/4) \cr} 
$$
$B$G$"$k$3$H$r0UL#$9$k(B.
$B$3$3$G(B,
$$ T = \pmatrix{ 9402863/1505280 & 170306533/752640 \cr
                 -8131157/430080 & -147868387/215040 \cr}
$$
$B$H$*$/(B.
$B4X?t(B {\tt tam} $B$NLa$jCM$NBh0l@.J,$,(B $B9TNs(B $T$ $B$G$"$k(B.
(cf. $BEDB<;a$N(B, HG function $B$N@:EYJ]>Z7W;;$N%W%m%0%i%`(B).


$B$J$*(B,  $F'(a,b,c;z)$ $B$O(B $F(a,b,c;z)$ $B$H(B $F(a+1,b,c;z)$
$B$GI=$9$3$H$,2DG=$G$"$k(B.
$B$=$l$K$O<!$N$h$/$7$i$l$?D64v2?4X?t$N8x<0$r;H$($P$h$$(B:
$$ \frac{1}{a} z F'(a,b,c;z) = - F(a,b,c;z) + F(a+1,b,c;z). $$

\end{document}