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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "read `hsum6.mpl`;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%LPackage~\"Hypergeometric~Summation
\",~Maple~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TCopyright~2001,~Wolf
ram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "read `rules.mpl`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 6 "(15 b)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "DR2:=diffru
leup(hyperterm([],[gamma+n],-x,k),k,y(n,x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$DR2G/--&%\"DG6#\"\"#6#%\"yG6$,&%\"nG\"\"\"F1!\"\"%\"
xG,$*&,(%&gammaGF1F0F1F1F2F2-F-6$F0F3F1F2" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "subs(gamma=c-n,DR2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/--&%\"DG6#\"\"#6#%\"yG6$,&%\"nG\"\"\"F/!\"\"%\"xG,$*&,&%\"cGF/F
/F0F0-F+6$F.F1F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "RE:=s
ubs(gamma=c-n,recursiondown(hyperterm([],[gamma+n],-x,k),k,y(n,x)));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/-%\"yG6$,&%\"nG\"\"\"F+F+%\"x
G,&**,&%\"cGF+F+!\"\"F+F0F+-F'6$F*F,F+F,F1F+**F/F+F0F+-F'6$,&F*F+F+F1F
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0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "DR1:=subs(gamma=c-
n,map(f->collect(f,y),subs(RE,subs(n=n+1,DR2))));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$DR1G/--&%\"DG6#\"\"#6#%\"yG6$%\"nG%\"xG,&*(,&%\"cG\"
\"\"F5!\"\"F5F0F6-F-F.F5F6*(F3F5-F-6$,&F/F5F5F6F0F5F0F6F5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "(18 a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DR1:=diffruledown(hyperterm([],[gam
ma+2*n],-x,k),k,y(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DR1G/-%
%diffG6$-%\"yG6$%\"nG%\"xGF-,&**,0*&\"\"%\"\"\")F,\"\"#F3!\"\"*&\"\"'F
3F,F3F3*(F2F3%&gammaGF3F,F3F6*&\"\"$F3F:F3F3*$)F:F5F3F6F5F6F-F3F3F-F6,
(F:F3*&F5F3F,F3F3F5F6F6F)F3F3*(,(F:F3*&F5F3F,F3F3F3F6F3F-F6-F*6$,&F,F3
F3F6F-F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(y(n,x),x
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{XPPMATH 20 "6#/-%%diffG6$-%\"yG6$%\"nG%\"xGF+,&**,**&\"\"$\"\"\"%\"cG
F1F1*$)F2\"\"#F1!\"\"F5F6F+F1F1F'F1F+F6,&F2F1F5F6F6F1*(,&F2F1F1F6F1-F(
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}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "DR2:=s
ubs(gamma=c-2*n,diffruleup(hyperterm([],[gamma+2*n],-x,k),k,y(n,x)));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DR2G/--&%\"DG6#\"\"#6#%\"yG6$,&
%\"nG\"\"\"F1!\"\"%\"xG,&*&,&%\"cGF1F+F2F2-F-F.F1F2**-F-6$F0F3F1F3F1,&
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "DR1:=diffruledown(hyperterm(
[alpha+n],[c],x,k),k,y(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DR
1G/-%%diffG6$-%\"yG6$%\"nG%\"xGF-,&*(F-!\"\",*%&alphaG\"\"\"%\"cGF0F-F
3F,F3F3F)F3F3*(F-F0,(F4F0F2F3F,F3F3-F*6$,&F,F3F3F0F-F3F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(alpha=a-n,DR1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6$%\"nG%\"xGF+,&*(F+!\"\",(%\"a
G\"\"\"%\"cGF.F+F1F1F'F1F1*(F+F.,&F2F.F0F1F1-F(6$,&F*F1F1F.F+F1F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "RE:=recursionup(hyperterm([a
lpha+n],[c],x,k),k,y(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/
-%\"yG6$,&%\"nG\"\"\"F+!\"\"%\"xG,&*(,**&\"\"#F+%&alphaGF+F+F-F+%\"cGF
,*&F2F+F*F+F+F+,(F4F,F3F+F*F+F,-F'6$F*F-F+F+*(,&F3F+F*F+F+F6F,-F'6$,&F
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "DR2:=di
ff(y(n-1,x),x)=map(factor,subs(alpha=a-n,subs(n=n-1,collect(subs(RE,op
(2,DR1)),y))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DR2G/--&%\"DG6#
\"\"#6#%\"yG6$,&%\"nG\"\"\"F1!\"\"%\"xG,&*(,&%\"aGF1F1F2F1-F-F.F1F3F2F
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "DR1:=diffruledown(hyper
term([a],[gamma-n],x,k),k,y(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$DR1G/-%%diffG6$-%\"yG6$%\"nG%\"xGF-,&F)\"\"\"*(,(%\"aGF/F,F/%&gamma
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" {MPLTEXT 1 0 20 "subs(gamma=c+n,DR1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%diffG6$-%\"yG6$%\"nG%\"xGF+,&F'\"\"\"*(,&%\"cG!\"\"%\"aGF-F-
F0F1-F(6$,&F*F-F-F1F+F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "RE:=recursionup(hyperterm([a],[gamma-n],x,k),k,y(n,x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#REG/-%\"yG6$,&%\"nG\"\"\"F+!\"\"%\"xG,&*,
,*F-F+%&gammaGF+F*F,F+F,F+,&F1F,F*F+F+F-F,,(%\"aGF+F*F+F1F,F,-F'6$F*F-
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{PARA 0 "" 0 "" {TEXT -1 6 "(24 b)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "DR2:=diff(y(n-1,x),x)=map(factor,subs(gamma=c+n,subs(
n=n-1,collect(subs(RE,op(2,DR1)),y))));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$DR2G/--&%\"DG6#\"\"#6#%\"yG6$,&%\"nG\"\"\"F1!\"\"%\"xG,&*(%\"
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{TEXT -1 6 "(26 a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "DR1:=
subs(alpha=a-n,gamma=c-n,diffruledown(hyperterm([alpha+n],[gamma+n],x,
k),k,y(n,x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DR1G/-%%diffG6$-%
\"yG6$%\"nG%\"xGF-,&*(,(%\"cG!\"\"\"\"\"F3F-F3F3F-F2F)F3F3*(,&F1F3F3F2
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gamma=c-n,diffruleup(hyperterm([alpha+n],[gamma+n],x,k),k,y(n,x)));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DR2G/--&%\"DG6#\"\"#6#%\"yG6$,&%\"
nG\"\"\"F1!\"\"%\"xG*(,&%\"aGF1F1F2F1,&%\"cGF1F1F2F2-F-6$F0F3F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "3.4.1: DDE(1,0,0)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diffrule([1,0,0],n);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7$/-%%diffG6$-%\"yG6$%\"nG%\"xGF,,&**,(%\"aG\"\"
\"*&F,F1%\"bGF1F1%\"cG!\"\"F1F,F5,&F,F1F1F5F5F(F1F5**,&F4F1F0F5F1F,F5F
6F5-F)6$,&F+F1F1F5F,F1F5/--&%\"DG6#\"\"#6#F)F:,&*(F9F1,&F0F1F1F5F1F,F5
F5*(F,F5FFF1F(F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "DDE(1,1,0)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diffrule([1,1,0],n);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$/-%%diffG6$-%\"yG6$%\"nG%\"xGF,,&*,
,L*&F,\"\"\",&%\"aGF1F+!\"\"F1F1**\"\"$F1F,F1,&%\"bGF1F+F4F1F+F1F4*(\"
\"#F1F,F1F+F1F1*(F,F1F7F1F2F1F4*(F6F1F,F1)F+F:F1F4*&F,F1F7F1F1*(F,F1F2
F1%\"cGF1F1*&F,F1)F2F:F1F4*&F,F1F@F1F4*&F,F1)F7F:F1F4**F:F1F,F1F@F1F+F
1F1**F6F1F,F1F2F1F+F1F4*(F,F1F7F1F@F1F1*$F=F1F4*&F7F1F+F1F4*&F2F1F+F1F
4*$)F@F:F1F4*&F7F1F2F1F4*&F2F1F@F1F1*(F:F1F@F1F+F1F1*&F7F1F@F1F1F1,&F,
F1F1F4F4F,F4,*F@F1F1F1F8F4F3F4F4F(F1F4*.,&F8F4F@F1F1,&F@F1F3F4F1-F)6$,
&F+F1F1F4F,F1FRF4F,F4FSF4F4/--&%\"DG6#F:6#F)FX,&*,,&F8F1F1F4F1,&F3F1F1
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "DDE(1,1,2)" }}}{EXCHG {PARA 0 "> \+
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{XPPMATH 20 "6#7$/-%%diffG6$-%\"yG6$%\"nG%\"xGF,,&*,,D*$),&%\"cG\"\"\"
*&\"\"#F4F+F4!\"\"F6F4F7*&\"\"$F4F3F4F4*&F,F4F2F4F7*(\"\"%F4F2F4F+F4F7
*(F,F4,&%\"bGF4F+F7F4F2F4F4**F6F4F,F4F2F4F+F4F4*(F,F4,&%\"aGF4F+F7F4F2
F4F4F6F7F,F4*(F<F4F,F4F+F4F7*(F9F4F,F4)F+F6F4F4*(F,F4F>F4F+F4F4*(F,F4F
BF4F+F4F4*(F,F4F>F4FBF4F7*&F,F4FBF4F7*&F,F4F>F4F7*&F<F4FFF4F7F4F(F4,&F
3F4F6F7F7,&F,F4F4F7F7F,F7F7**,&F3F4F4F7F4FNF7F,F7-F)6$,&F+F4F4F7F,F4F7
/--&%\"DG6#F66#F)FR,&*,FQF4,&F?F4F4F7F4,&FCF4F4F7F4FMF7FNF7F7*4F,F4Fgn
F4FhnF4,(F4F4F3F7F?F4F4,(F3F7F4F4FCF4F4F(F4FNF7FM!\"#FPF7F4" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "DDE(1,0,1)" }}}{EXCHG {PARA 0 "> \+
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{XPPMATH 20 "6#7$/-%%diffG6$-%\"yG6$%\"nG%\"xGF,,&**,(%\"cG!\"\"\"\"\"
F2*&F,F2%\"bGF2F2F2,&F,F2F2F1F1F,F1F(F2F1**,&F0F2F2F1F2F5F1F,F1-F)6$,&
F+F2F2F1F,F2F1/--&%\"DG6#\"\"#6#F)F9,&*(,&%\"aGF2F2F1F2F8F2F5F1F1*,FEF
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%\"nG%\"xGF,,&*.%\"bG\"\"\",.*(\"\"#F0F,F0F+F0F0F,!\"\"%\"aGF4*&F,F0,&
F/F0F+F0F0F4*&F,F0,&F5F0F+F4F0F0%\"cGF0F0F(F0,&F,F0F0F4F4F,F4,(F5F0F0F
4F/F4F4F4*.F/F0,&F:F0F5F4F0-F)6$,&F+F0F0F4F,F0F;F4F,F4F<F4F0/--&%\"DG6
#F36#F)F@,&*.F?F0,&F5F0F0F4F0,.*&F,F0F5F0F0F0F0F:F4F,F4F/F0*&F,F0F/F0F
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{PARA 0 "" 0 "" {TEXT -1 11 "DDE(0,0,-1)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "diffrule([0,0,-1],n);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$/-%%diffG6$-%\"yG6$%\"nG%\"xGF,,&*(,(%\"aG!\"\"%\"cG\"\"\"%\"b
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F1F5F1F1/--&%\"DG6#\"\"#6#F)F:,&*(F9F3F2F3F,F1F1*(F2F3F(F3F,F1F3" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "DDE(1,1,1)" }}}{EXCHG {PARA 0 "> \+
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{XPPMATH 20 "6#7$/-%%diffG6$-%\"yG6$%\"nG%\"xGF,,&**,.%\"cG!\"\"F,F1*(
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