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{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "read `hsum.mpl`;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%ZCopyright~1998~~Wolfram~Koepf,~Konrad-Zuse-Zentrum~
BerlinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rec2poly(n*s(n+1
)-(n+20)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*L&%&deltaG6#
\"\"!\"\"\"%\"nGF(,&F)F(\"#>F(F(,&F)F(\"#=F(F(,&F)F(\"#<F(F(,&F)F(\"#;
F(F(,&F)F(\"#:F(F(,&F)F(\"#9F(F(,&F)F(\"#8F(F(,&F)F(\"#7F(F(,&F)F(\"#6
F(F(,&F)F(\"#5F(F(,&F)F(\"\"*F(F(,&F)F(\"\")F(F(,&F)F(\"\"(F(F(,&F)F(
\"\"'F(F(,&F)F(\"\"&F(F(,&F)F(\"\"%F(F(,&F)F(\"\"$F(F(,&F)F(\"\"#F(F(,
&F)F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rec2poly(n*s
(n+1)-(n+40)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*^p&%&delt
aG6#\"\"!\"\"\"%\"nGF(,&F)F(\"#RF(F(,&F)F(\"#QF(F(,&F)F(\"#PF(F(,&F)F(
\"#OF(F(,&F)F(\"#NF(F(,&F)F(\"#MF(F(,&F)F(\"#LF(F(,&F)F(\"#KF(F(,&F)F(
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\"#EF(F(,&F)F(\"#DF(F(,&F)F(\"#CF(F(,&F)F(\"#BF(F(,&F)F(\"#AF(F(,&F)F(
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\"#;F(F(,&F)F(\"#:F(F(,&F)F(\"#9F(F(,&F)F(\"#8F(F(,&F)F(\"#7F(F(,&F)F(
\"#6F(F(,&F)F(\"#5F(F(,&F)F(\"\"*F(F(,&F)F(\"\")F(F(,&F)F(\"\"(F(F(,&F
)F(\"\"'F(F(,&F)F(\"\"&F(F(,&F)F(\"\"%F(F(,&F)F(\"\"$F(F(,&F)F(\"\"#F(
F(,&F)F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gosper((-
1)^k/k!,k);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in gosper) no hype
rgeometric term antidifference exists" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "rec2hyper((n+2)*DN(n+2)-(n+1)*DN(n+1)-(n+1)*DN(n),DN(
n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 83 "rec:=\nsumrecursion(hyperterm([-n,a,a+1/2,b],[2*a,
(b-n+1)/2,(b-n)/2+1],1,k),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>%$recG/,(*,,(%\"nG\"\"\"\"\"#F*%\"bGF*F*,(F*F*F)F*F,!\"\"F*,&F,F.F)F*
F*,(F)F*%\"aGF+F*F*F*-%\"sG6#,&F)F*F+F*F*F**,,(F)F*F*F*F,F*F*F/F*,(F)F
*F1F*F*F*F*,*F*F*F1F+F)F*F,F.F*-F36#,&F)F*F*F*F*!\"#*,F<F*,&F)F*F,F*F*
F9F*,(F1F+F,F.F)F*F*-F36#F)F*F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "rec2hyper(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#<$*.,&%\"nG\"\"\"F'F'F',&F&F'%\"bGF'F',(%\"aG\"\"#F)!\"\"F&F'F',(F&
F'F'F'F)F'F-,&F)F-F&F'F-,&F&F'F+F,F-**F(F'F*F'F.F-F/F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "rec:=\nsumrecursion(hyperterm([-n,a
,a+1/2,b],[2*a+1,(b-n)/2,(b-n+1)/2],1,k),k,s(n));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$recG/,(**,(\"\"#\"\"\"%\"nGF*%\"bG!\"\"F*,(F*F*F+F*F
,F-F*,(F+F*%\"aGF)F)F*F*-%\"sG6#,&F+F*F)F*F*F***F.F*,(F+F)F0F)\"\"$F*F
*,*F)F*F0F)F+F*F,F-F*-F26#,&F+F*F*F*F*F-**F;F*F8F*,*F*F*F0F)F+F*F,F-F*
-F26#F+F*F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rec2hyp
er(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$**,&%\"nG\"\"\"F'F'
F',*F'F'%\"aG\"\"#F&F'%\"bG!\"\"F',(F'F'F&F'F+F,F,,(F&F'F)F*F'F'F,*&F(
F'F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "rec:=sumrecursion
(\nhyperterm([-n,a,a+1/2,b],[2*a+1,(b-n+1)/2,(b-n)/2+1],1,k),k,s(n));
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,(*.,(\"\"#\"\"\"%\"nGF*%
\"bGF*F*,(F,!\"\"F*F*F+F*F*,&F,F.F+F*F*,(F+F*%\"aGF)F)F*F*,2*$F+F)F**&
F,F*F+F*F)F+\"\"$F,F*F)F*F1F)*$F1F)!\"%*$F,F)F*F*-%\"sG6#,&F)F*F+F*F*F
**,,(F+F*F*F*F,F*F*F/F*,*F,F.F*F*F1F)F+F*F*,F*$F+F5F)*&F+F)F,F*\"\"%*&
F1F*F+F)F)F3\"#6F+\"#=*&F,F)F+F*F)F4\"#5*&F1F)F+F*!\")*&F1F*F+F*\"#7*(
F1F*F,F*F+F*FCF1FLF6F7F8F5*&F1F*F,F*FC\"\")F**$F1F5FJF,\"\"'*&F1F*F,F)
F)F*-F:6#,&F*F*F+F*F*F.*.FUF*,&F+F*F,F*F*F?F*,(F+F*F,F.F1F)F*,2F3F*F4F
)F+\"\"&FQF*F1F)F8F*F,F5F6F7F*-F:6#F+F*F*\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 56 "this calculation took 1 1/2 hours on a HP-UX worksta
tion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "TIME:=time(): rec2hyper(rec
,s(n)); time()-TIME;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$*2,*%\"aG\"
\"#%\"bG\"\"\"F)F)%\"nGF)F),&F)F)F*F)F),&F*F)F(F)F),(F*F)F(!\"\"F&F'F)
,(F&F'F(F)F*F)F.,&F(F.F*F)F.,(F*F)F)F)F(F)F.,(F)F)F*F)F&F'F.*.,*F&!\"#
F(F)F)F)F*F)F)F,F)F-F),(F&F5F(F)F*F)F.F0F.F1F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"(RL]&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
82 "rec:=sumrecursion(\nhyperterm([-n,a,b,-1/2-a-b-n],[-a-n,-b-n,a+b+1
/2],1,k),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,(*.,(%\"
nG\"\"\"\"\"#F*%\"bGF*F*,(F)F*F*F*F,F*F*,(%\"aGF*F)F*F+F*F*,(F)F*F/F*F
*F*F*,*F/F+F)F*F+F*F,F+F*-%\"sG6#,&F)F*F+F*F*F**.,&F)F*F*F*F*,(F)F*F*F
*F,F+F*,(F)F*F*F*F/F+F*,*F/F*F+F*F,F*F)F*F*,*F/F*F*F*F)F*F,F*F*-F36#F)
F*F**,F0F*F-F*,2*&F/F*F)F*\"\"%F/\"\"'*&F/F*F,F*FA\"\"&F**$F)F+F+F)FBF
,FB*&F)F*F,F*FAF*F:F*-F36#F7F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "rec2hyper(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#<$*.,(%\"nG\"\"\"F'F'%\"aG\"\"#F',*F(F'F'F'F&F'%\"bGF'F',(F&F'F'F'F
+F)F',*F(F)F+F)F&F'F'F'!\"\",(F&F'F(F'F'F'F.,(F&F'F'F'F+F'F.**,&F&F'F'
F'F'F*F'F/F.F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "rec:=su
mrecursion(\nhyperterm([-n,a,b,1/2-a-b-n],[-a-n,1-b-n,a+b+1/2],1,k),k,
s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,(*.,(%\"nG\"\"\"F*F
*%\"bGF*F*,&F)F*F+F*F*,(%\"aGF*F)F*\"\"#F*F*,(F)F*F.F*F*F*F*,*F.F/F+F/
F)F*F*F*F*-%\"sG6#,&F)F*F/F*F*F**.,&F)F*F*F*F*,&F+F/F)F*F*,(F)F*F*F*F.
F/F*,*F.F*F*F*F)F*F+F*F*,(F)F*F.F*F+F*F*-F36#F)F*F**,F,F*F0F*F:F*,2*&F
.F*F)F*F/F.F/*&F.F*F+F*F/F*F**$F)F/F*F)F/F+\"\"$*&F)F*F+F*F/F*-F36#F7F
*!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rec2hyper(rec
,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$*.,(%\"nG\"\"\"F'F'%\"aG
\"\"#F',&%\"bGF)F&F'F',(F&F'F(F'F+F'F',(F(F)F+F)F&F'!\"\",(F&F'F(F'F'F
'F.,&F&F'F+F'F.**,&F&F'F'F'F'F,F'F/F.F0F." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 83 "rec:=sumrecursion(\nhyperterm([-n,a,b,1/2-a-b-n],[1
-a-n,1-b-n,a+b-1/2],1,k),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 ">%$
recG/,(*.,(%\"bG\"\"\"%\"nGF)F)F)F),&F(F)F*F)F),(F*F)%\"aGF)F)F)F),&F*
F)F-F)F),(F*F)F(\"\"#F-F0F)-%\"sG6#,&F*F)F0F)F)F)*.,&F*F)F)F)F),&F*F)F
(F0F),&F*F)F-F0F),*F*F)F)F)F(F)F-F)F),(F*F)F(F)F-F)F)-F26#F*F)F)*,F+F)
F.F)F9F),0*$F*F0F)*&F-F)F*F)F0*&F*F)F(F)F0F*F)F(F0F-F0*&F(F)F-F)F0F)-F
26#F6F)!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rec2hyp
er(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*.,&%\"nG\"\"\"%\"bG
\"\"#F&,&F%F&%\"aGF(F&,(F%F&F'F&F*F&F&,&F'F&F%F&!\"\",&F%F&F*F&F-,*F%F
&F-F&F*F(F'F(F-**,&F%F&F&F&F&F+F&F,F-F.F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 83 "rec:=sumrecursion(\nhyperterm([-n,a,b,3/2-a-b-n],[1
-a-n,1-b-n,a+b+1/2],1,k),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 ">%$
recG/,(*0,(%\"bG\"\"\"%\"nGF)F)F)F),&F(F)F*F)F),(F*F)%\"aGF)F)F)F),&F*
F)F-F)F),(F*F)F(\"\"#F-F0F),*F*F0\"\"$F)F-F0F(F0F)-%\"sG6#,&F*F)F0F)F)
F)*0,&F*F)F)F)F),&F*F)F(F0F),&F*F)F-F0F),*F*F)F)F)F(F)F-F)F),(F*F)F(F)
F-F)F),*!\"\"F)F-F0F(F0F*F0F)-F46#F*F)F)*.F+F)F.F),0*$F*F0F)*&F-F)F*F)
F0*&F*F)F(F)F0F*F)F(F0F-F0*&F(F)F-F)F0F),*F-F0F)F)F(F0F*F0F)F;F)-F46#F
8F)!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rec2hyper(r
ec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*.,&%\"nG\"\"\"F&F&F&,(F%
F&%\"bGF&%\"aGF&F&,*!\"\"F&F)\"\"#F(F,F%F,F&,*F)F,F&F&F(F,F%F,F+,&F%F&
F)F&F+,&F(F&F%F&F+*2F'F&,&F%F&F(F,F&,&F%F&F)F,F&F*F&F-F+,*F%F&F+F&F)F,
F(F,F+F.F+F/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "rec:=sumr
ecursion(\nhyperterm([-n,a,b,3/2-a-b-n],[1-a-n,2-b-n,a+b-1/2],1,k),k,s
(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 ">%$recG/,(*.,&%\"bG\"\"\"%\"nGF
)F),(F(F)F*F)!\"\"F)F),(F*F)%\"aGF)F)F)F),&F*F)F.F)F),*F*F)F,F)F.\"\"#
F(F1F)-%\"sG6#,&F*F)F1F)F)F)*.,&F*F)F)F)F),(F*F)F(F1F,F)F),&F*F)F.F1F)
,(F*F)F(F)F.F)F),*F*F)F,F)F.F)F(F)F)-F36#F*F)F)*,F+F)F/F),0*&F.F)F*F)F
1*&F(F)F.F)F1F.F)*$F*F1F)F,F)F(F1*&F*F)F(F)F1F)F:F)-F36#F7F)!\"#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rec2hyper(rec,s(n));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "<$*.,*%\"nG\"\"\"!\"\"F&%\"aGF&%\"bGF&F
&,(F%F&F)\"\"#F'F&F&,&F%F&F(F+F&,(F)F&F%F&F'F&F',&F%F&F(F&F',*F%F&!\"#
F&F(F+F)F+F'**,&F%F&F&F&F&F$F&F-F'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "rec:=sumrecursion(\nhyperterm([-n,a,b,5/2-a-b-n],[2-a
-n,2-b-n,a+b-1/2],1,k),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 ">%$re
cG/,(*0,&%\"bG\"\"\"%\"nGF)F),(F(F)F*F)!\"\"F)F),&F*F)%\"aGF)F),(F*F)F
.F)F,F)F),*F.\"\"#F)F)F(F1F*F1F),*F*F)!\"#F)F(F1F.F1F)-%\"sG6#,&F*F)F1
F)F)F)*0,&F*F)F)F)F),(F*F)F(F1F,F)F),(F*F)F.F1F,F)F),*!\"$F)F.F1F(F1F*
F1F),(F*F)F(F)F.F)F),*F*F)F,F)F.F)F(F)F)-F56#F*F)F)*.F+F)F/F),*F,F)F.F
1F(F1F*F1F),2*&F.F)F*F)\"\"%F.F1*&F(F)F.F)FF*$F*F1F1F*F3F=F)F(F1*&F*F)
F(F)FFF)F>F)-F56#F9F)F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "rec2hyper(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*2,*%\"n
G\"\"\"!\"\"F&%\"aGF&%\"bGF&F&,*!\"$F&F(\"\"#F)F,F%F,F&,(F%F&F)F,F'F&F
&,(F%F&F(F,F'F&F&,(F)F&F%F&F'F&F',(F%F&F(F&F'F&F',*F%F&F+F&F(F,F)F,F',
*F'F&F(F,F)F,F%F,F'*.,&F%F&F&F&F&F$F&F*F&F/F'F0F'F2F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "recpoly(n*(n+1)*s(n+3)-2*n*(n+3)*s(
n+1)+(n+1)*(n+3)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%&d
eltaG6#\"\"!\"\"\"%\"nGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "rechyper((n+1+2*b)*(n+2*b)*s(n+2)-4*(n+b)*(n+1+b)*s(n),s(n));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$*&,&%\"nG\"\"\"%\"bGF(F(,&F'F(F)\"
\"#!\"\"F+,$F%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "rechy
per(\n(n+1+2*b)*(n+2+2*b)*(n+2*b)*s(n+3)-8*(n+2+b)*(n+1+b)*(n+b)*s(n),
s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%*(,&!\"\"\"\"\"*&%\"IGF'\"
\"$#F'\"\"#F'F',&%\"nGF'%\"bGF'F',&F.F'F/F,F&,$*&F-F'F0F&F,,$*(,&F'F'F
(F'F'F-F'F0F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "rechyper
((n+2+2*b)*(n+2*b)*(n+3+2*b)*(n+1+2*b)*s(n+4)-\n16*(n+3+b)*(n+2+b)*(n+
1+b)*(n+b)*s(n),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&,$*(%\"IG\"
\"\",&%\"bGF&%\"nGF&F&,&F)F&F(\"\"#!\"\"!\"#,$*&F'F&F*F,F-,$F$F+,$F/F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "with(share): readshare(
gfun,analysis);" }}{PARA 6 "" 1 "" {TEXT -1 70 "See ?share and ?share,
contents for information about the share library" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%gfunG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "w
ith(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7O%(LaplaceG%.algebraics
ubsG%.algeqtodiffeqG%&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+d
iffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%)invborelG
%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+listtolistG%.listtora
tpolyG%*listtorecG%-listtoseriesG%5listtoseries/LaplaceG%1listtoseries
/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1listtoseries/ogfG%4
listtoseries/revegfG%4listtoseries/revogfG%,maxdegcoeffG%*maxdegeqnG%,
maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG%*optionsgfG%/ratp
olytocoeffG%(rec*recG%(rec+recG%,rectodiffeqG%*rectoprocG%.seriestoalg
eqG%/seriestodiffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpo
lyG%,seriestorecG%/seriestoseriesG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 161 "diffeq:=a*(1-z)*z*diff(F(z),z$3)+(a+a^2+a*b+z-3*a*z-
3*a^2*z-2*z^2+z^3)* \ndiff(F(z),z$2)+ (-3*a^2-2*a^3+b-z-4*a*z-b*z+z^2+
3*a*z^2)*diff(F(z),z)+\n2*a^2*(z-1)*F(z);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%'diffeqG,***%\"aG\"\"\",&F(F(%\"zG!\"\"F(F*F(-%%diffG
6$-F-6$-F-6$-%\"FG6#F*F*F*F*F(F(*&,2F'F(*$F'\"\"#F(*&F'F(%\"bGF(F(F*F(
*&F'F(F*F(!\"$*&F'F9F*F(F=*$F*F9!\"#*$F*\"\"$F(F(F/F(F(*&,2F8F=*$F'FBF
@F;F(F*F+F<!\"%*&F;F(F*F(F+F?F(*&F'F(F*F9FBF(F1F(F(*(F'F9,&F*F(F+F(F(F
3F(F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rec:=map(factor,di
ffeqtorec(diffeq,F(z),u(n)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$re
cG,**(,&%\"nG\"\"\"%\"aG\"\"#F),&F(F)F*F)F)-%\"uG6#F(F)F)*&,2*$F*F+F+F
(\"\"$F)F)*&F(F)F*F)\"\"%F*F5*&%\"bGF)F(F)F)F7F)*$F(F+F+F)-F.6#,&F(F)F
)F)F)!\"\"*(,&F(F)F+F)F),4*&F*F)F(F+F)F4F5F(F<*&F*F+F(F)F3*$F*F3F+F2\"
\"'F*F3F7F<F<F)F)-F.6#F>F)F<*,F*F),&F(F)F3F)F)F>F),*F(F)F7F)F*F)F+F)F)
-F.6#FGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rechyper(rec
,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#**,&%\"nG\"\"\"%\"aG\"\"#
F',&F&F'F(F'F',(F&F'F(F'%\"bGF'!\"\",&F&F'F'F'F-" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 76 "rec2hyper(\n(-1+n)*(2+n)*s(n)+(1-11*n-2*n^2)*
s(n+1)+(2+n)*(8+n)*s(n+2),s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#
*&,&%\"nG\"\"\"\"\"#F'F',&\"\"(F'F&F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 82 "rec:=sumrecursion(\n(-1)^k*binomial(r-s-k,k)*binomi
al(r-2*k,n-k)/(r-n-k+1),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 ">%$r
ecG/,(**,&%\"nG\"\"\"\"\"#F)F),(F(F*\"\"$F)%\"rG!\"\"F),*F(F)%\"sGF)F)
F)F-F.F)-F06#F'F)F)*(,(F(F*F)F)F-F.F),.*$F(F*F*F(F**&F-F)F(F)!\"#*$F0F
*F.F-F8*&F-F)F0F)F)F)-F06#,&F(F)F)F)F)F)**-F06#F(F),(F(F*F-F.F.F)F),(F
-F.F(F)F.F)F),&F0F.F(F)F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "rec2hyper(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"<$,$**,(%\"rG!\"\"%\"nG\"\"\"F'F)F),(F(\"\"#F&F'F'F)F),(F(F)F&F'%\"sG
F)F',(F(F+F)F)F&F'F'F',$**,&F-F'F(F)F)F*F),&F(F)F)F)F'F.F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ratio(binomial(s,n)/(r-2*n+1
),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**,&%\"sG!\"\"%\"nG\"\"\"F(,(
F'\"\"#%\"rGF&F&F(F(,&F'F(F(F(F&,(F'F*F(F(F+F&F&F&" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 105 "rec:=sumrecursion(binomial(n,k)*pochhammer
(c,k)*pochhammer(m,n-k)*\nhyperterm([-k,a,b],[c,d],1,j),j,s(k));" }}
{PARA 12 "" 1 "" {XPPMATH 20 ">%$recG/,(*,,&%\"kG\"\"\"\"\"#F)F),*%\"m
GF)%\"nGF)F(!\"\"F.F)F),*F-F)!\"#F)F,F)F(F.F),(F(F)%\"dGF)F)F)F)-%\"sG
6#F'F)F.*,,&F-F)F(F.F),(F-F)F.F)F(F.F),&%\"cGF)F(F)F),,%\"aGF)%\"bGF)F
(F.F:F.F2F.F)-F46#F(F)F)**F8F)F+F),<F(!\"$*&F=F)F(F)F)*$F(F*F0F<F)*&F=
F)F<F)F)F.F)*&F(F)F2F)F0F2F0*&F:F)F2F)F.F:F0*&F<F)F(F)F)F=F)*&F(F)F:F)
F0F)-F46#,&F(F)F)F)F)F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "rec2hyper(rec,s(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rec2hyper((n+2)*H(n+2)-(2*n+
3)*H(n+1)+(n+1)*H(n),H(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "read `qsum.mpl`;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%OCopyright~1998,~~Harald~Boeing~&~Wolfram~K
oepfG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;Konrad-Zuse-Zentrum~BerlinG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Rogers" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "RE:=qsumrecursion((-1)^k*q^(k*(3*k-1)/2)/qpochhammer(
q,q,n+k)/qpochhammer(q,q,n-k),q,k,S(n));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%#REG/,**,%\"qG\"\"\",&!\"\"F))F(%\"nGF)F),&F)F)F,F)F),&)F(,$F-
\"\"#F+F(F)F)-%\"SG6#F-F)F)*&,.)F(,&F2F)F-F2F+*$F(\"\"%F)*$F(\"\"$F)*$
F(F2F))F(,&F-F=F)F)F+)F(,$F-F=F+F)-F46#,&F-F)F+F)F)F)*(F(F),*F<F)F0F)F
>F)F:F)F)-F46#,&F-F)!\"#F)F)F+*&F(\"\"&-F46#,&F-F)!\"$F)F)F)\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "qrecsolve(RE,q,S(n),return=q
hypergeometric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$*$-%,qpochhamm
erG6%%\"qGF)%\"nG!\"\"1\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 68 "F:=(-1)^k*q^(k*(3*k-1)/2)/qpochhammer(q,q,n+k)/qpochhammer(q,q
,n-k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rat:=qsimpcomb(su
bs(k=-k,F)/F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG)%\"qG%\"kG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "qsumrecursion((1+rat)/2*F
,q,k,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&)%\"qG%\"nG!\"\"
\"\"\"F+F+-%\"SG6#F)F+F+-F-6#,&F)F+F*F+F*\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 6 "(9.17)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "leftRE:=
qsumrecursion(q^(k^2+k)/qpochhammer(q,q,k)/qpochhammer(q,q,n-k),q,k,S(
n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'leftREG/,(*&,&!\"\"\"\"\")%
\"qG%\"nGF*F*-%\"SG6#F-F*F**&,*F,F*)F,,$F-\"\"#F*F*F*F+F)F*-F/6#,&F-F*
F)F*F*F**&F,F*-F/6#,&F-F*!\"#F*F*F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "rightRE:=qsumrecursion((-1)^k*q^(k*(5*k+3)/2)/qpochh
ammer(q,q,n+k+1)/qpochhammer(q,q,n-k),q,k,S(n));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(rightREG/,.*,%\"qG\"\"#,&!\"\"\"\"\")F(%\"nGF,F,,&F,
F,F-F,F,,&)F(,&F,F,F.F)F,F+F,F,-%\"SG6#F.F,F,*(,6)F(,$F.\"\"&F+*$F(\"
\"%F,*$F(F:F,)F(,&F<F,F.F)F+)F(,&F)F,F.F)F+)F(,&\"\"$F,F.F)F+F(F,*$F(F
)F,*$F(FDF,)F(,&F)F,F.F:F+F,F(F,-F46#,&F.F,F+F,F,F+*&,<*$F(\"\"*F,*$F(
\"\")F,*$F(\"\"(F)*$F(\"\"'F)F=F)F>F,F;F,FBF,FFF,)F(,&F)F,F.F<F,F@F,)F
(,&F,F,F.F<F,)F(,$F.FUF,F,-F46#,&F.F,!\"#F,F,F,*(F(F,,@)F(,$F.F<F,FVF,
F@F,FXF,F>F)FBF)F;F,*$F(\"#5F,)F(,&F:F,F.F)F,FPF)FNF,)F(,&FUF,F.F)F,F=
F,FTF)FRF)F,-F46#,&F.F,!\"$F,F,F+*(,2F1F,F=F,F;F,FFF,)F(,$F.F)F,F@F,FT
F,FRF,F,F(F:-F46#,&F.F,!\"%F,F,F,*&F(\"#7-F46#,&F.F,!\"&F,F,F+\"\"!" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "`recursion/compare`(leftRE
,rightRE,S(n));" }}{PARA 6 "" 1 "" {TEXT -1 42 "Warning: Different rec
ursion orders. (2,5)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;Recursions~a
re~compatible.G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "rightter
m:=(-1)^k*q^(k*(5*k+3)/2)/qpochhammer(q,q,n+k+1)/qpochhammer(q,q,n-k):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rat:=qsimpcomb(subs(k=-
k,rightterm)/rightterm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG*()
%\"qG%\"kG!\"#,&!\"\"\"\"\"*()F'%\"nGF,F&F,F'F,F,F,,&F&F+*&F'F,F.F,F,F
+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rightRE:=qsumrecursion
((1+rat)/2*rightterm,q,k,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(
rightREG/,(*&,&!\"\"\"\"\")%\"qG%\"nGF*F*-%\"SG6#F-F*F**&,*F,F*)F,,$F-
\"\"#F*F*F*F+F)F*-F/6#,&F-F*F)F*F*F**&F,F*-F/6#,&F-F*!\"#F*F*F)\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "`recursion/compare`(leftR
E,rightRE,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Recursions~are~i
dentical.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "version of q-Dixon
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "RE:=qsumrecursion((-1)^k*qbino
mial(n+b,n+k,q)*qbinomial(n+c,c+k,q)*qbinomial(b+c,b+k,q)*q^(k*(3*k-1)
/2),q,k,S(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#REG/,**,%\"qG\"\"
$,&!\"\"\"\"\")F(%\"nGF,F,,&F,F,F-F,F,,&)F(,$F.\"\"#F+F(F,F,-%\"SG6#F.
F,F+*(F(F,,J*$F(\"\"&F,*$F(\"\"%F,)F(,*F.F,F<F,%\"cGF,%\"bGF,F+)F(,(F.
F3F)F,F@F,F,*$F(F)F,)F(,*F)F,F.F,F?F,F@F,F+)F(,(F)F,F.F,F@F,F+)F(,&F.F
3F)F,F+)F(,(F)F,F.F,F?F,F+)F(,(F.F3F)F,F?F,F,)F(,&F3F,F.F)F+)F(,*F3F,F
.F3F?F,F@F,F+)F(,(F3F,F.F)F@F,F+)F(,(F3F,F.F)F?F,F+)F(,(F3F,F.F3F?F,F,
)F(,(F3F,F.F3F@F,F,)F(,*F.F<F?F,F@F,F3F,F,)F(,&F.F)F,F,F+)F(,*F.F<F,F,
F?F,F@F,F,)F(,(F.F<F?F,F@F,F,F,-F56#,&F.F,F+F,F,F+*(,&)F(,(F.F,F?F,F@F
,F+F(F,F,,BFAF,FFF+FLF,FJF+F9F,)F(,(F.F)F?F,F@F,F+)F(,&F3F,F.F3F,)F(,*
F3F,F.F)F?F,F@F,F+F;F,F=F+*$F(\"\"'F,)F(,*F.F)F,F,F?F,F@F,F+)F(,(F.F,F
<F,F?F,F+FXF,FVF,)F(,(F.F,F<F,F@F,F+F,-F56#,&F.F,!\"#F,F,F,*,,&FaoF+*$
F(F3F,F,F`oF,,&FhpF,)F(,&F.F,F?F,F+F,,&FhpF,)F(,&F.F,F@F,F+F,-F56#,&F.
F,!\"$F,F,F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "TIME:=
time():\nqrecsolve(RE,q,S(n),return=qhypergeometric);\ntime()-TIME;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$*&-%,qpochhammerG6%)%\"qG,(\"\"\"
F,%\"cGF,%\"bGF,F*%\"nGF,-F'6%F*F*F/!\"\"1\"\"!F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"'(Ra%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
89 "F:=(-1)^k*qbinomial(n+b,n+k,q)*qbinomial(n+c,c+k,q)*qbinomial(b+c,
b+k,q)*q^(k*(3*k-1)/2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "
rat:=qsimpcomb(subs(k=-k,F)/F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
ratG)%\"qG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "RE2:=qsu
mrecursion((1+rat)/2*F,q,k,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$RE2G/,&*&,&\"\"\"F))%\"qG%\"nG!\"\"F)-%\"SG6#F,F)F)*&,&F-F))F+,(F,F)
%\"cGF)%\"bGF)F)F)-F/6#,&F,F)F-F)F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "assume(n,integer);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "sum2qhyper(F,q,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#*0-%,qpochhammerG6%%\"qGF',&%#n|irG\"\"\"%\"bGF*F*-F%6%F'F'F)!\"#-F%6
%F'F'F+F.-F%6%F'F',&F)F*%\"cGF*F*-F%6%F'F'F4F.-F%6%F'F',&F+F*F4F*F*-%$
phiG6&7&)F',$F)!\"\")F',$F+F@)F',$F4F@F'7%)F',&F*F*F+F*)F',&F*F*F4F*)F
',&F*F*F)F*F'**F'F*)F'F+F*)F'F)F*)F'F4F*F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "sum2qhyper(subs(k=k-n,F),q,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
*2)!\"\",$%\"nGF%\"\"\"-%,qpochhammerG6%%\"qGF,,&%\"cGF(F'F%F%-F*6%F,F
,,$F'\"\"#F%-F*6%F,F,,&%\"bGF(F.F(F(-F*6%F,F,,&F6F(F'F%F%)*$F,#\"\"$F2
*$F'F2F()F,,$F'#F(F2F(-%$phiG6&7%)F,,&F'F%F6F%)F,,$F'!\"#)F,,&F.F%F'F%
7$)F,,(F(F(F'F%F.F()F,,(F(F(F'F%F6F(F,**F,F()F,F'F()F,F6F()F,F.F(F(" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "sum2qhyper(subs(k=k-n,F)/
(qfactorial(n+b+c,q)/qfactorial(n,q)/qfactorial(b,q)/qfactorial(c,q)),
q,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*:)!\"\",$%\"nGF%\"\"\"-%,qpo
chhammerG6%%\"qGF,,&%\"cGF(F'F%F%-F*6%F,F,,$F'\"\"#F%-F*6%F,F,,&%\"bGF
(F.F(F(-F*6%F,F,,&F6F(F'F%F%)*$F,#\"\"$F2*$F'F2F()F,,$F'#F(F2F(-F*6%F,
F,,(F'F(F.F(F6F(F%-F*6%F,F,F'F(-F*6%F,F,F6F(-F*6%F,F,F.F(-%$phiG6&7%)F
,,&F'F%F6F%)F,,$F'!\"#)F,,&F.F%F'F%7$)F,,(F(F(F'F%F.F()F,,(F(F(F'F%F6F
(F,**F,F()F,F'F()F,F6F()F,F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "F:=qphihyperterm([q^(-2*n),b,c],[q^(1-2*n)/b,q^(1-2*n
)/c],q,q^(2-n)/b/c,k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r
at:=qsimplify(subs(k=2*n-k,F)/F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$ratG*&)%\"qG%\"nG\"\"\")F'%\"kG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "rec:=qsumrecursion((1+rat)*F,q,k,S(n));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%$recG/,&*.,&*&)%\"qG,$%\"nG\"\"#\"\"\"%\"bGF/!
\"\"*$F+F.F/F/,&F)F1F+F/F/,&*&F*F/%\"cGF/F1F2F/F/,&F5F1F+F/F/,&*(F0F/)
F+F-F/F6F/F1F+F/F/-%\"SG6#F-F/F/*2,&F+F/*&F0F/F:F/F1F/F+F/,&F/F/F:F/F/
,&*(F*F/F0F/F6F/F1F+F/F/,&F2F/FCF1F/,&F+F/*&F6F/F:F/F1F/,&F*F1F+F/F/-F
<6#,&F-F/F1F/F/F1\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "q
recsolve(rec,q,S(n),return=qhypergeometric);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7#7$*8-%,qpochhammerG6%%\"cG%\"qG%\"nG\"\"\"-F'6%%\"bGF
*F+F,-F'6%*&F/F,F)F,*$F*\"\"#F+F,-F'6%*(F*F,F/F,F)F,F3F+F,-F'6%,$F*!\"
\"F*F+F,-F'6%F*F3F+F,-F'6%F2F*F+F;-F'6%*&F/F,F*F,F3F+F;-F'6%F/F3F+F;-F
'6%*&F)F,F*F,F3F+F;-F'6%F)F3F+F;1\"\"!F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "too complicated" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "qsu
mrecursion(F,q,k,S(n));" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, compu
tation interrupted" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }