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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Yaounde, March 23, 2005" 
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 91 "Wolfram Koepf: Computer Algebr
a Algorithms for Orthogonal Polynomials and Special Functions" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 27 "Computation of Power Series" }{TEXT 256 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Maple supports truncated power ser
ies" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s
eries(exp(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!
F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$?\"\"\"&-%\"OG6#F%F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "The following algorithm for the c
omputation of Formal Power Series is from\nKoepf, Wolfram: Power Serie
s in Computer Algebra, Journal of Symbolic Computation 13, 1992, 581-6
03" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "re
ad \"FPS.mpl\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OPackage~Formal~Po
wer~Series,~Maple~V~-~Maple~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TCo
pyright~1995,~Dominik~Gruntz,~University~of~BaselG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%^oCopyright~2002,~Detlef~M|gzller~&~Wolfram~Koepf,~U
niversity~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "FPS
(exp(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%*factorial
G6#%\"kG!\"\")%\"xGF*\"\"\"/F*;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "infolevel[FPS]:=5:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "FPS(exp(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS/
FPS:   looking for DE of degree   1" }}{PARA 6 "" 1 "" {TEXT -1 36 "FP
S/FPS:   DE of degree   1   found." }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS
/FPS:   DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#F'G6#%\"xG\"\"
\"-%\"FGF'!\"\"\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS:   RE = \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#,&%\"kG\"\"\"F)F)*&-F%6#F
(F)F'!\"\"" }}{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE:   RE is of
 hypergeometric type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE:
   Symmetry number m :=    1" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hyper
geomRE:   RE: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"kG\"\"\"F'F'
F'-%\"aG6#F%F'-F)6#F&" }}{PARA 6 "" 1 "" {TEXT -1 45 "FPS/hypergeomRE:
   RE valid for all k >=    0" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hype
rgeomRE:   a(0) =   1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%
*factorialG6#%\"kG!\"\")%\"xGF*\"\"\"/F*;\"\"!%)infinityG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "FPS(exp(x^2),x);" }}{PARA 6 "" 1 "
" {TEXT -1 39 "FPS/FPS:   looking for DE of degree   1" }}{PARA 6 "" 
1 "" {TEXT -1 36 "FPS/FPS:   DE of degree   1   found." }}{PARA 6 "" 
1 "" {TEXT -1 16 "FPS/FPS:   DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/,&-%#F'G6#%\"xG\"\"\"*(\"\"#F)F(F)-%\"FGF'F)!\"\"\"\"!" }}{PARA 6 "" 
1 "" {TEXT -1 16 "FPS/FPS:   RE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"aG6#,&%\"kG\"\"\"F)F),$*(\"\"#F)-F%6#,&F(F)F)!\"\"F)F'F0F)" }}
{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE:   RE is of hypergeometri
c type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE:   Symmetry nu
mber m :=    2" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE:   RE: \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"kG\"\"\"\"\"#F'F'-%\"aG6#F
%F',$*&F(F'-F*6#F&F'F'" }}{PARA 6 "" 1 "" {TEXT -1 46 "FPS/hypergeomRE
:   RE valid for all k >=    -1" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hy
pergeomRE:   a(0) =   1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&
-%*factorialG6#%\"kG!\"\")%\"xG,$*&\"\"#\"\"\"F*F1F1F1/F*;\"\"!%)infin
ityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "a Puiseux series" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FPS(exp
(sqrt(x)),x);" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS/FPS:   looking for D
E of degree   1" }}{PARA 6 "" 1 "" {TEXT -1 39 "FPS/FPS:   looking for
 DE of degree   2" }}{PARA 6 "" 1 "" {TEXT -1 36 "FPS/FPS:   DE of deg
ree   2   found." }}{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS:   DE = " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(\"\"%\"\"\"%\"xGF'-%$F''G6#F(F'F'
*&\"\"#F'-%#F'GF+F'F'-%\"FGF+!\"\"\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 
16 "FPS/FPS:   RE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#,&%\"
kG\"\"\"F)F),$*&#F)\"\"#F)*(-F%6#F(F)F'!\"\",&*&F-F)F(F)F)F)F)F1F)F)" 
}}{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE:   RE is of hypergeomet
ric type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE:   Symmetry \+
number m :=    1" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE:   RE
: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**\"\"#\"\"\",&%\"kGF'F'F'F',
&*&F&F'F)F'F'F'F'F'-%\"aG6#F(F'F'-F-6#F)" }}{PARA 6 "" 1 "" {TEXT -1 
46 "FPS/hypergeomRE:   RE modified to    k = 1/2*k" }}{PARA 6 "" 1 "" 
{TEXT -1 36 "FPS/hypergeomRE:   => f :=    exp(x)" }}{PARA 6 "" 1 "" 
{TEXT -1 48 "FPS/hypergeomRE:   RE is of hypergeometric type." }}
{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE:   Symmetry number m :=  \+
  2" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hypergeomRE:   RE: " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(,&%\"kG\"\"\"\"\"#F'F',&F&F'F'F'F'-%\"aG6
#F%F'-F+6#F&" }}{PARA 6 "" 1 "" {TEXT -1 45 "FPS/hypergeomRE:   RE val
id for all k >=    0" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE: \+
  a(0) =   1" }}{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE:   a(1) =
   1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$SumG6$*&-%*factorialG6#,$
*&\"\"#\"\"\"%\"kGF.F.!\"\")%\"xGF/F./F/;\"\"!%)infinityGF.-F%6$*&-F)6
#,&*&F-F.F/F.F.F.F.F0)F2,&F/F.#F.F-F.F.F3F." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 17 "FPS(arcsin(x),x);" }}{PARA 6 "" 1 "" {TEXT -1 39 "
FPS/FPS:   looking for DE of degree   1" }}{PARA 6 "" 1 "" {TEXT -1 
39 "FPS/FPS:   looking for DE of degree   2" }}{PARA 6 "" 1 "" {TEXT 
-1 36 "FPS/FPS:   DE of degree   2   found." }}{PARA 6 "" 1 "" {TEXT 
-1 16 "FPS/FPS:   DE = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"
\"!\"\"*$)%\"xG\"\"#F'F'F'-%$F''G6#F+F'F'*&F+F'-%#F'GF/F'F'\"\"!" }}
{PARA 6 "" 1 "" {TEXT -1 16 "FPS/FPS:   RE = " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"aG6#,&%\"kG\"\"\"\"\"#F)**F(F*-F%6#F(F),&F(F)F)F)!
\"\"F'F/" }}{PARA 6 "" 1 "" {TEXT -1 48 "FPS/hypergeomRE:   RE is of h
ypergeometric type." }}{PARA 6 "" 1 "" {TEXT -1 44 "FPS/hypergeomRE:  \+
 Symmetry number m :=    2" }}{PARA 6 "" 1 "" {TEXT -1 23 "FPS/hyperge
omRE:   RE: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(,&%\"kG\"\"\"F(F(
F(,&F'F(\"\"#F(F(-%\"aG6#F)F(!\"\",$*&)F'F*F(-F,6#F'F(F." }}{PARA 6 "
" 1 "" {TEXT -1 45 "FPS/hypergeomRE:   RE valid for all k >=    0" }}
{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE:   a(0) =   0" }}{PARA 6 
"" 1 "" {TEXT -1 44 "FPS/hypergeomRE:   a(2*j) = 0   for all j>0." }}
{PARA 6 "" 1 "" {TEXT -1 29 "FPS/hypergeomRE:   a(1) =   1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%*factorialG6#,$*&\"\"#\"\"\"%\"
kGF-F-F-)\"\"%,$F.!\"\"F--F(6#F.!\"#,&*&F,F-F.F-F-F-F-F2)%\"xGF6F-/F.;
\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolev
el[FPS]:=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "procedure to compu
te a holonomic differential equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "DE:=HolonomicDE(arcsin(x),F(x));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,&*(,&%\"xG\"\"\"F*!\"\"F*,&F)
F*F*F*F*-%%diffG6$-%\"FG6#F)-%\"$G6$F)\"\"#F*F**&F)F*-F.6$F0F)F*F*\"\"
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(DE,F(x));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,&%$_C1G\"\"\"*&-%#lnG6#
,&F'F**$,&F*!\"\"*$)F'\"\"#F*F*#F*F5F*F*%$_C2GF*F*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 37 "some final examples: a Laurent series" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "FPS(arcsin(x)^2/x^5,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%*factorialG6#%\"kG\"\"#)\"\"%F*
\"\"\"-F(6#,&F.F.*&F+F.F*F.F.!\"\",&F*F.F.F.F3)%\"xG,&*&F+F.F*F.F.\"\"
$F3F./F*;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "a co
mplicated example that cannot be found in Gradshteyn/Ryshik" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "FPS(exp
(arcsin(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$SumG6$**,&*&\"
\"%\"\"\")%\"kG\"\"#F+F+F+F+!\"\"-%(ProductG6$,&*&F*F+)%\"jGF.F+F+F+F+
/F6;\"\"!F-F+-%*factorialG6#,$*&F.F+F-F+F+F/)%\"xGF=F+/F-;F9%)infinity
GF+-F%6$*,-F16$,(F+F+*&F.F+%\"jGF+F+*&F.F+)FKF.F+F+/FKF8F+)F.F-F+-F;6#
,&*&F.F+F-F+F+F+F+F/,(*&F.F+F,F+F+*&F.F+F-F+F+F+F+F/)F@FRF+FAF+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and an asymptotic series" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "FPS((er
f(x)-1)*exp(x^2),x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%
#PiG#!\"\"\"\"#-%$SumG6$*,)F'%\"kG\"\"\"-%*factorialG6#,$*&F(F/F.F/F/F
/)\"\"%,$F.F'F/-F16#F.F')*&F/F/%\"xGF',&*&F(F/F.F/F/F/F/F//F.;\"\"!%)i
nfinityGF/F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Also covered are \+
holonomic special functions" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "FPS(LegendreP(n,x),x);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,&*.\"\"#\"\"\"%#PiG#F&F%-%&GAMMAG6#,&F(F&*&F%!\"\"%\"n
GF&F.F.-F*6#,$*&F%F.F/F&F&F.F/F.-%$SumG6$*,-%+pochhammerG6$,$*&F%F.F/F
&F.%\"kGF&-F96$,&*&F%F.F/F&F&F(F&F=F&-%*factorialG6#,$*&F%F&F=F&F&F.)
\"\"%F=F&)%\"xGFEF&/F=;\"\"!%)infinityGF&F&*,F%F&F'F(-F*6#F@F.-F*6#F;F
.-F56$*,-F96$F,F=F&-F96$,&F&F&*&F%F.F/F&F&F=F&FGF&-FC6#,&F&F&*&F%F&F=F
&F&F.)FJFinF&FKF&F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "FPS(
LegendreP(n,x),x=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*.)!
\"\"%\"kG\"\"\")\"\"#,$F)F(F*-%+pochhammerG6$,&%\"nGF*F*F*F)F*-F/6$,$F
2F(F)F*-%*factorialG6#F)!\"#),&%\"xGF*F*F(F)F*/F);\"\"!%)infinityG" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "HolonomicDE(LegendreP(n,x),
F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(,&\"\"\"!\"\"%\"xGF'F',
&F)F'F'F'F'-%%diffG6$-%\"FG6#F)-%\"$G6$F)\"\"#F'F'*(,&%\"nGF'F'F'F'F7F
'F.F'F(*(F4F'F)F'-F,6$F.F)F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Algebra of \+
Holonomic Functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(
gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7P%(LaplaceG%.algebraicsubsG
%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG%&borelG%.cauchyproduc
tG%.diffeq*diffeqG%.diffeq+diffeqG%,diffeqtableG%2diffeqtohomdiffeqG%,
diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprtodiffeqG%
)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+listtolistG
%.listtoratpolyG%*listtorecG%-listtoseriesG%,maxdegcoeffG%*maxdegeqnG%
,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG%*optionsgfG%,pol
todiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+recG%,rectodiffeq
G%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriestodiffeqG%2seriesto
hypergeomG%-seriestolistG%0seriestoratpolyG%,seriestorecG%/seriestoser
iesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We consider the function \+
sin(x)*exp(x):" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 36 "The differential equation of sin(x):" }{MPLTEXT 1 0 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "DE1:=diff(F(x),x$2)+F(x)=0;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE1G/,&-%%diffG6$-%\"FG6#%\"xG-
%\"$G6$F-\"\"#\"\"\"F*F2\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "
The differential equation of exp(x):" }{MPLTEXT 1 0 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "DE2:=diff(F(x),x)-F(x)=0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$DE2G/,&-%%diffG6$-%\"FG6#%\"xGF-\"\"\"F*!
\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`diffeq*diffeq
`(DE1,DE2,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%%diffG6$-%\"FG
6#%\"xG-%\"$G6$F*\"\"#\"\"\"*&F.F/-F%6$F'F*F/!\"\"*&F.F/F'F/F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "and the sum sin(x)+exp(x) satisfie
s" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "`di
ffeq+diffeq`(DE1,DE2,F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%%di
ffG6$-%\"FG6#%\"xG-%\"$G6$F*\"\"$\"\"\"-F%6$F'-F,6$F*\"\"#!\"\"-F%6$F'
F*F/F'F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now a more complicate
d example: exp(x)*Ai(x)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "DE1:=diff(F(x),x)-F(x)=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$DE1G/,&-%%diffG6$-%\"FG6#%\"xGF-\"\"\"F*!\"\"\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "DE2:=HolonomicDE(AiryAi(x
),F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE2G/,&-%%diffG6$-%\"FG
6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F-F2F*F2!\"\"\"\"!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "`diffeq*diffeq`(DE1,DE2,F(x));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*&,&%\"xG!\"\"\"\"\"F(F(-%\"FG6#F&F(F(-%%diffG
6$F)-%\"$G6$F&\"\"#F(*&F2F(-F-6$F)F&F(F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 34 "and the sum exp(x)+Ai(x) satisfies" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "`diffeq+diffeq`(DE1,DE2,F(x));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<$,**&,(\"\"\"F'%\"xG!\"\"*$)F(\"\"#F'F'F'-%\"FG6#
F(F'F'*&,&F*F)F(F'F'-%%diffG6$F-F(F'F'*&F(F'-F36$F--%\"$G6$F(F,F'F)*&,
&F(F'F'F)F'-F36$F--F96$F(\"\"$F'F'/---%#@@G6$%\"DGF,6#F.6#\"\"!&%#_CGF
J" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Similarly, HolonomicDE yield
s" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Hol
onomicDE(exp(x)+AiryAi(x),F(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,
**&,&%\"xG\"\"\"F(!\"\"F(-%%diffG6$-%\"FG6#F'-%\"$G6$F'\"\"$F(F(*&,(F(
F(F'F)*$)F'\"\"#F(F(F(F-F(F(*&F'F(-F+6$F--F16$F'F8F(F)*(F'F(F&F(-F+6$F
-F'F(F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 24 "Hypergeometric Functions" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(x*hypergeom([],[3/2],-x^2/
4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "hypergeom([a,b],[c],x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%*hypergeomG6%7$%\"aG%\"bG7#%\"cG%\"xG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sumtools[hyperterm]([a,b],[c
],x,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,-%+pochhammerG6$%\"aG%\"k
G\"\"\"-F%6$%\"bGF(F))%\"xGF(F)-F%6$%\"cGF(!\"\"-%*factorialG6#F(F2" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sum(sumtools[hyperterm]([a
,b],[c],x,k),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*hyp
ergeomG6%7$%\"aG%\"bG7#%\"cG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "hypergeom([a,b],[c],1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%*hypergeomG6%7$%\"aG%\"bG7#%\"cG\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "simplify(hypergeom([a,b],[c],1));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#**-%&GAMMAG6#%\"cG\"\"\"-F%6#,(F'F(%\"aG!\"\"%
\"bGF-F(-F%6#,&F'F(F,F-F--F%6#,&F'F(F.F-F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 42 "Identific
ation of Hypergeometric Functions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
20 "We are interested in" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"s:=Sum((-1)^k/(2*k+1)!*x^(2*k+1),k=0..infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sG-%$SumG6$*()!\"\"%\"kG\"\"\"-%*factorialG6#,&*&\"
\"#F,F+F,F,F,F,F*)%\"xGF0F,/F+;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "F:=k->(-1)^k/(2*k+1)!*x^(2*k+1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"FGj+6#%\"kG6\"6$%)operatorG%&arrowGF(*()!\"
\"9$\"\"\"-%*factorialG6#,&*&\"\"#F0F/F0F0F0F0F.)%\"xGF4F0F(F(F(6#\"\"
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r:=F(k+1)/F(k);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*.)!\"\",&%\"kG\"\"\"F*F*F*-%*fa
ctorialG6#,&*&\"\"#F*F)F*F*\"\"$F*F')%\"xGF.F*)F'F)F'-F,6#,&*&F0F*F)F*
F*F*F*F*)F3F7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(r
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&*&\"\"#\"\"\"%\"kGF(F(F'F(
!\"\",&*&F'F(F)F(F(\"\"$F(F*%\"xGF'F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 5 "Hence" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "s=F(0)*hypergeom([],[3/2],-x^2/4);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%$SumG6$*()!\"\"%\"kG\"\"\"-%*factorialG6#,&*&\"\"#
F+F*F+F+F+F+F))%\"xGF/F+/F*;\"\"!%)infinityG*&F3F+-%*hypergeomG6%7\"7#
#\"\"$F1,$*&\"\"%F)F3F1F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Th
e following procedure uses the given algorithm and gives therefore the
 hypergeometric form:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "sumtools[Sumtohyper](F(k),k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%\"xG\"\"\"-%*HypergeomG6%7\"7##\"\"$\"\"#,$*&\"\"%!
\"\"F$F-F1F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Another example" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "F:=bin
omial(n,k)*binomial(-n-1,k)*((1-x)/2)^k;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"FG*(-%)binomialG6$%\"nG%\"kG\"\"\"-F'6$,&F)!\"\"F+F/F*F+),&*
&\"\"#F/%\"xGF+F/#F+F3F+F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "Sum(F,k=0..n)=sumtools[Sumtohyper](F,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$*(-%)binomialG6$%\"nG%\"kG\"\"\"-F)6$,&F+!\"
\"F-F1F,F-),&*&\"\"#F1%\"xGF-F1#F-F5F-F,F-/F,;\"\"!F+-%*HypergeomG6%7$
,$F+F1,&F+F-F-F-7#F-F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Details
 of this algorithm and an implementation can be found in the book" }}
{PARA 0 "" 0 "" {TEXT -1 15 "Wolfram Koepf: " }{TEXT 258 24 "Hypergeom
etric Summation" }{TEXT -1 38 ", Vieweg, Braunschweig/Wiesbaden, 1998
" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 88 "Computation
 of Recurrence Equations for Hypergeometric Functions: Faasenmyer's Al
gorithm" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "How does one generate t
he result" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "Sum(binomial(n,k),k=0..n)=\nsum(binomial(n,k),k=0..n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$-%)binomialG6$%\"nG%\"kG/F+;\"\"!F
*)\"\"#F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We do the following \+
more complicated example with Maple:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sum(k*binomial(n,k),k=0..n)=\nsum(k
*binomial(n,k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*
&%\"kG\"\"\"-%)binomialG6$%\"nGF(F)/F(;\"\"!F-,$*(\"\"#!\"\")F3F-F)F-F
)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "F:=(n,k)->k*binomial
(n,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGj+6$%\"nG%\"kG6\"6$%)o
peratorG%&arrowGF)*&9%\"\"\"-%)binomialG6$9$F.F/F)F)F)6$\"\"!F5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ansatz:=sum(sum(a(j,i)*F(n+j
,k+i),i=0..1),j=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,*
*(-%\"aG6$\"\"!F*\"\"\"%\"kGF+-%)binomialG6$%\"nGF,F+F+*(-F(6$F*F+F+,&
F,F+F+F+F+-F.6$F0F4F+F+*(-F(6$F+F*F+F,F+-F.6$,&F0F+F+F+F,F+F+*(-F(6$F+
F+F+F4F+-F.6$F<F4F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "an
satz:=ansatz/F(n,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ansatzG*(,*
*(-%\"aG6$\"\"!F+\"\"\"%\"kGF,-%)binomialG6$%\"nGF-F,F,*(-F)6$F+F,F,,&
F-F,F,F,F,-F/6$F1F5F,F,*(-F)6$F,F+F,F-F,-F/6$,&F1F,F,F,F-F,F,*(-F)6$F,
F,F,F5F,-F/6$F=F5F,F,F,F-!\"\"F.FC" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "ansatz:=expand(ansatz);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'ansatzG,8-%\"aG6$\"\"!F)\"\"\"*(-F'6$F)F*F*,&%\"kGF*F*F*!\"\"
%\"nGF*F**(F/F*F,F*F.F0F0**F/F0F,F*F.F0F1F*F**&F,F*F.F0F0*(-F'6$F*F)F*
,(F1F*F/F0F*F*F0F1F*F**&F6F*F8F0F**(-F'6$F*F*F*F.F0F1F*F**&F;F*F.F0F**
*F/F0F;F*F.F0F1F*F**(F/F0F;F*F.F0F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "ansatz:=normal(ansatz);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%'ansatzG*(,@*&)%\"kG\"\"#\"\"\"-%\"aG6$\"\"!F/F+!\"\"*&F(F+-F-
6$F/F+F+F+*(F,F+F)F+%\"nGF+F+**F*F+F2F+F5F+F)F+F0*&-F-6$F+F+F+F)F+F0*(
F8F+F5F+F)F+F0*(-F-6$F+F/F+F5F+F)F+F+*&F<F+F)F+F+*&F,F+F)F+F+*&F)F+F2F
+F0*&F8F+)F5F*F+F+*&F2F+FBF+F+*&F2F+F5F+F+*(F*F+F8F+F5F+F+F8F+F+,(F5F+
F)F0F+F+F0F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ansatz:=n
umer(ansatz);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'ansatzG,@*&)%\"kG
\"\"#\"\"\"-%\"aG6$\"\"!F.F*!\"\"*&F'F*-F,6$F.F*F*F**(F+F*F(F*%\"nGF*F
***F)F*F1F*F4F*F(F*F/*&-F,6$F*F*F*F(F*F/*(F7F*F4F*F(F*F/*(-F,6$F*F.F*F
4F*F(F*F**&F;F*F(F*F**&F+F*F(F*F**&F(F*F1F*F/*&F7F*)F4F)F*F**&F1F*FAF*
F**&F1F*F4F*F**(F)F*F7F*F4F*F*F7F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "eqs:=\{coeffs(ansatz,k)\};" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$eqsG<%,,*(\"\"#\"\"\"-%\"aG6$F)F)F)%\"nGF)F)F*F)*&F*
F))F-F(F)F)*&-F+6$\"\"!F)F)F/F)F)*&F1F)F-F)F),2*&F*F)F-F)!\"\"*&-F+6$F
3F3F)F-F)F)*(F(F)F1F)F-F)F7F*F7F1F7*&-F+6$F)F3F)F-F)F)F=F)F9F),&F9F7F1
F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sol:=solve(eqs,\{seq(
seq(a(j,i),j=0..1),i=0..1)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$s
olG<&/-%\"aG6$\"\"\"\"\"!F+/-F(6$F+F+,$*(%\"nG!\"\",&F1F*F*F*F*-F(6$F*
F*F*F2/-F(6$F+F*F//F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "
re:=sum(sum(a(j,i)*f(n+j,k+i),i=0..1),j=0..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#reG,**&-%\"aG6$\"\"!F*\"\"\"-%\"fG6$%\"nG%\"kGF+F+*&
-F(6$F*F+F+-F-6$F/,&F0F+F+F+F+F+*&-F(6$F+F*F+-F-6$,&F/F+F+F+F0F+F+*&-F
(6$F+F+F+-F-6$F<F6F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r
e:=subs(sol,re);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#reG,(**%\"nG!\"
\",&F'\"\"\"F*F*F*-%\"aG6$F*F*F*-%\"fG6$F'%\"kGF*F(**F'F(F)F*F+F*-F/6$
F',&F1F*F*F*F*F(*&F+F*-F/6$F)F5F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "re:=numer(normal(re/a(1,1)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#reG,,*&-%\"fG6$%\"nG%\"kG\"\"\"F*F,!\"\"F'F-*&-F(6$F
*,&F+F,F,F,F,F*F,F-F/F-*&-F(6$,&F*F,F,F,F1F,F*F,F," }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 7 "s:='s':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "RE:=subs(\{seq(seq(f(n+j,k+i)=s(n+j),i=0..1),j=0..1)
\},re);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG,(*(\"\"#\"\"\"-%\"sG
6#%\"nGF(F,F(!\"\"*&F'F(F)F(F-*&-F*6#,&F,F(F(F(F(F,F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RE:=map(factor,collect(RE,s))=0;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#REG/,&*(\"\"#\"\"\",&%\"nGF)F)F)F)
-%\"sG6#F+F)!\"\"*&-F-6#F*F)F+F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 43 "Now we use the implementation from the book" }}{PARA 0 "
" 0 "" {TEXT -1 15 "Wolfram Koepf: " }{TEXT 257 24 "Hypergeometric Sum
mation" }{TEXT -1 38 ", Vieweg, Braunschweig/Wiesbaden, 1998" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "restart; read \"hsum9.mpl\";
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VPackage~\"Hypergeometric~Summati
on\",~Maple~V~-~Maple~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YCopyrigh
t~1998-2004,~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(k*binomial(n,k),k,s(n),1,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"nG\"\"\"-%\"sG6#,&F&F'F'F'F'F'
*(\"\"#F'-F)6#F&F'F+F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "fasenmyer(binomial(n,k)^2,k,s(n),1,1);" }}{PARA 8 "" 
1 "" {TEXT -1 72 "Error, (in kfreerec) No kfree recurrence equation of
 order (1,1) exists\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fa
senmyer(binomial(n,k)^2,k,s(n),2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/,&*&,&%\"nG\"\"\"\"\"#F(F(-%\"sG6#F&F(F(*(F)F(-F+6#,&F'F(F(F(F(,&*&
F)F(F'F(F(\"\"$F(F(!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "fasenmyer(binomial(n-k,k),k,s(n),2,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(-%\"sG6#,&%\"nG\"\"\"\"\"#F*F*-F&6#F)!\"\"-F&6#,&F)F
*F*F*F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "[seq(sum(bi
nomial(n-k,k),k=0..n),n=0..10)]; n:='n':" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7-\"\"\"F$\"\"#\"\"$\"\"&\"\")\"#8\"#@\"#M\"#b\"#*)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fasenmyer((-1)^k*binomial(n,k)^2,k,
s(n),2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&%\"nG\"\"\"\"\"#F
(F(-%\"sG6#F&F(F(*(\"\"%F(-F+6#F'F(,&F'F(F(F(F(F(\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fasenmyer(binomial(n,k)^3,k,s(n),2,
1);" }}{PARA 8 "" 1 "" {TEXT -1 72 "Error, (in kfreerec) No kfree recu
rrence equation of order (2,2) exists\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "fasenmyer(binomial(n,k)^3,k,s(n),3,1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/,**(,&*&\"\"$\"\"\"%\"nGF)F)\"\"%F)F)),&F*F)F(F
)\"\"#F)-%\"sG6#F-F)F)*(F.F),**&\"\"*F))F*F(F)F)*&\"#dF))F*F.F)F)*&\"$
;\"F)F*F)F)\"#uF)F)-F06#,&F*F)F.F)F)!\"\"*(,&*&F(F)F*F)F)\"\"&F)F),(*&
\"#:F)F9F)F)*&\"#bF)F*F)F)\"#[F)F)-F06#,&F*F)F)F)F)F@**\"\")F),&*&F(F)
F*F)F)\"\"(F)F))FMF.F)-F06#F*F)F@\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 22 "Zeilberger's Algorithm" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "sumrecursion(k*binomial(n,k),k,s(n));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&*(\"\"#\"\"\",&%\"nGF'F'F'F'-%\"sG6#F)F'F'*&-F+
6#F(F'F)F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sum
recursion((-1)^k*binomial(n,k)^2,k,s(n));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&,&%\"nG\"\"\"\"\"#F(F(-%\"sG6#F&F(F(*(\"\"%F(,&F'F
(F(F(F(-F+6#F'F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "
sumrecursion(binomial(n,k)^3,k,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/,(*(\"\")\"\"\"),&%\"nGF'F'F'\"\"#F'-%\"sG6#F*F'F'*&,(*&\"\"(F')F*
F+F'F'*&\"#@F'F*F'F'\"#;F'F'-F-6#F)F'F'*&),&F*F'F+F'F+F'-F-6#F;F'!\"\"
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "With Zeilberger's algori
thm, we can do more complicated examples." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The Ap\351ry numbers" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum(binomial(n,k)^
2*binomial(n+k,k)^2,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Sum
G6$*&)-%)binomialG6$%\"nG%\"kG\"\"#\"\"\")-F)6$,&F+F.F,F.F,F-F./F,;\"
\"!F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "satisfy the recurrence e
quation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "sumrecursion(binomial(n,k)^2*binomial(n+k,k)^2,k,A(n));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,(*&),&%\"nG\"\"\"F)F)\"\"$F)-%\"AG6#F(F)F)
*(,&*&\"\"#F)F(F)F)F*F)F),(*&\"#<F))F(F1F)F)*&\"#^F)F(F)F)\"#RF)F)-F,6
#F'F)!\"\"*&),&F(F)F1F)F*F)-F,6#F>F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 76 "The power sums of the binomial coefficients were worth \+
a paper in the 1980s:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "sumrecursion(binomial(n,k)^4,k,s(n));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,(*,\"\"%\"\"\",&*&F&F'%\"nGF'F'\"\"&F'F',&*&F&F'
F*F'F'\"\"$F'F',&F*F'F'F'F'-%\"sG6#F*F'F'**\"\"#F',&*&F4F'F*F'F'F.F'F'
,(*&F.F')F*F4F'F'*&\"\"*F'F*F'F'\"\"(F'F'-F16#F/F'F'*&),&F*F'F4F'F.F'-
F16#FAF'!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumre
cursion(binomial(n,k)^5,k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,
***\"#K\"\"\",(*&\"#bF')%\"nG\"\"#F'F'*&\"$`#F'F,F'F'\"$#HF'F'),&F,F'F
'F'\"\"%F'-%\"sG6#F,F'F'*&,0\"'[S^F'*&\"(qFo#F'F+F'F'*&\"(t?3#F')F,\"
\"$F'F'*&\"'V0!*F')F,F3F'F'*&\"'*z0#F')F,\"\"&F'F'*&\"&:%>F')F,\"\"'F'
F'*&\"(kq#=F'F,F'F'F'-F56#F2F'!\"\"*&,0*&\"'F3JF'F+F'F'*&\"'\\f?F'F>F'
F'*&\"&)\\vF'FBF'F'\"&?$zF'*&\"''eX#F'F,F'F'*&\"&`X\"F'FEF'F'*&\"%b6F'
FIF'F'F'-F56#,&F,F'F-F'F'FO*(,(*&F*F'F+F'F'*&\"$V\"F'F,F'F'\"#%*F'F'),
&F,F'F?F'F3F'-F56#FcoF'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "sumrecursion(binomial(n,k)^6,k,s(n));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/,**0\"#C\"\"\",&*&\"\"'F'%\"nGF'F'\"\"&F'F',&*&\"\"#F'
F+F'F'\"\"$F'F',&*&F*F'F+F'F'\"\"(F'F',**&\"#\"*F')F+F0F'F'*&\"$P'F')F
+F/F'F'*&\"%\"\\\"F'F+F'F'\"%n6F'F'),&F+F'F'F'F0F'-%\"sG6#F+F'F'*&,6\"
)SV$H#F'*&\"*o<J!GF'F:F'F'*&\"*2=uy$F'F7F'F'*&\"*MI]F$F')F+\"\"%F'F'*&
\"*Ln\"z=F')F+F,F'F'*&\"'\")Q:F')F+\"\"*F'F'*&\")-g`rF')F+F*F'F'*&\")$
)*>u\"F')F+F3F'F'*&\"('4iCF')F+\"\")F'F'*&\"*wy]?\"F'F+F'F'F'-FA6#F?F'
!\"\"*(,&F+F'F/F'F',4*&\"%eMF'FgnF'F'*&\"&dq&F'FZF'F'*&\"'b&3%F'FWF'F'
*&\"(hnl\"F'FPF'F'*&\"(6#eTF'FLF'F'*&\"(a+h'F'F7F'F'*&\"(gl\\'F'F:F'F'
*&\"(_#4OF'F+F'F'\"'S\"o)F'F'-FA6#F_oF'F]o**F_oF',**&F6F'F7F'F'*&\"$k$
F'F:F'F'*&\"$!\\F'F+F'F'\"$A#F'F'),&F+F'F0F'F,F'-FA6#F]qF'F'\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sumrecursion(binomial(n,k)^7
,k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*,\"$G\"\"\"\",4*&\"'@
xUF')%\"nG\"\")F'F'*&\"(![w(*F')F,\"\"(F'F'*&\"):JP(*F')F,\"\"'F'F'*&
\"*$)Q*=bF')F,\"\"&F'F'*&\"+9jqY>F')F,\"\"%F'F'*&\"+LpcvVF')F,\"\"$F'F
'*&\"+eCp>hF')F,\"\"#F'F'*&\"+_@9p[F'F,F'F'\"+?\"*Q(o\"F'F'),&F,F'FEF'
FEF'),&F,F'F'F'F5F'-%\"sG6#F,F'F'*&,D\"1'p\")*pq!Q>#F'*&\"+&yjUC$F')F,
\"#;F'F'*&\"0LojlTy)>F')F,\"#7F'F'*&\"1]3qsJ#G8\"F')F,\"#6F'F'*&\"-g8#
G1E\"F')F,\"#:F'F'*&\".9k]oRG#F')F,\"#9F'F'*&\"37&G#*Q\\Yiw\"F'F8F'F'*
&\"23]l)**33paF'FDF'F'*&\"3kCj,CK\"z9\"F'F@F'F'*&\"3OF!\\h0sPm\"F'F<F'
F'*&\"3![GBX-u5U\"F'F4F'F'*&\"2%))*fI#o.U))F'F0F'F'*&\"2pjER()z:k\"F')
F,\"\"*F'F'*&\"1ol^'=o4!\\F')F,\"#5F'F'*&\"2SaaE)*z5I%F'F+F'F'*&\"/X&[
w-1c#F')F,\"#8F'F'*&\"2_vsu#G82;F'F,F'F'F'-FN6#FJF'!\"\"*(,@*&\"0KOgmE
b7\"F'FDF'F'*&\"0G<cUk@)>F'F@F'F'*&\"0[%Qs;poBF'F<F'F'*&\")\">o.$F'F_o
F'F'*&\"0;?(fReK?F'F8F'F'*&\".()y&fi<hF'F_pF'F'*&\".t+^\"oR7F'FcpF'F'*
&\"07565A@H\"F'F4F'F'*&\"/SOWI^%='F'F0F'F'*&\"/$3DGE1C#F'F+F'F'\".G4M@
w7'F'*&\"-UnRz3=F'FgnF'F'*&\",0@\">(z\"F'FYF'F'*&\"+jl\"*)3\"F'FipF'F'
*&\"/s5BFc!)QF'F,F'F'F'),&F,F'FAF'FEF'-FN6#F`sF'F_q**,4\")!G4_%F'*&\"*
'4x)4#F'F,F'F'*&\"*Yvb@%F'FDF'F'*&\"*JEWy%F'F@F'F'*&\"*%H(fN$F'F<F'F'*
&\"*(*)3!\\\"F'F8F'F'*&\")VR\"4%F'F4F'F'*&\"(7ZN'F'F0F'F'*&F*F'F+F'F'F
'F_sF'),&F,F'F=F'F5F'-FN6#FftF'F'*(,@\"0%)f1P(e7nF'*&\"2)))pqn8#o\\\"F
'FDF'F'*&\"2![S#y7k2#HF'F@F'F'*&\"2Kv_5S[,)QF'F<F'F'*&\",6v#=#e\"F'F_o
F'F'*&\"2'*e%G!>xBr$F'F8F'F'*&\"10?U-iWk<F'F_pF'F'*&\"0#GA9W'=-%F'FcpF
'F'*&\"2(G*)4)QU!QEF'F4F'F'*&\"20N<<asWT\"F'F0F'F'*&\"1o/4-i$3v&F'F+F'
F'*&\"/>/V7)\\g'F'FgnF'F'*&\".#R?tv)Q(F'FYF'F'*&\"-z#>#QS]F'FipF'F'*&
\"1+)o=s`Yn%F'F,F'F'F'FIF'-FN6#FLF'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "sumrecursion(binomial(n,k)^8,k,s(n));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#/,,*4\"#;\"\"\",&*&\"\")F'%\"nGF'F'\"#8F'F',&*&F
*F'F+F'F'\"\"(F'F',&*&F*F'F+F'F'\"\"*F'F',&*&F*F'F+F'F'\"#6F'F',&F+F'
\"\"#F'F',:*&\"*g`P-\"F')F+F5F'F'*&\"+!3Lk=$F')F+\"#5F'F'*&\",=:hg\\%F
')F+F2F'F'*&\"-2qD3'z$F')F+F*F'F'*&\".]7+')38#F')F+F/F'F'*&\".A$H6+]$)
F')F+\"\"'F'F'*&\"/#QYb&oIBF')F+\"\"&F'F'*&\"/d%y#G%Rj%F')F+\"\"%F'F'*
&\"/er%eg:V'F')F+\"\"$F'F'*&\"/!4e[)oMfF')F+F7F'F'*&\"/_eaSywKF'F+F'F'
\".?<!ys,#)F'F'),&F+F'F'F'FPF'-%\"sG6#F+F'F'*(\"#7F',N*&\"7o`FkqerAsI6
F'FWF'F'*&\"21epj-'Q6`F')F+\"#<F'F'*&\"7[h+T=BzXc/6F'F>F'F'*&\"7!3'f-v
e3\"R<f$F'FOF'F'*&\"5uib__L$3JJ\"F')F+\"#9F'F'*&\"7)pKU>Id%4UDBF'FSF'F
'\"4+G(Q#z%Hn%e)F'*&\"5*G\\9S<'=f8`F')F+F,F'F'*&\"7B*)y#)\\Kh>>^KF'FEF
'F'*&\"0S?)Q(>?M$F')F+\"#>F'F'*&\"7P8$>?>%fy![3#F'FBF'F'*&\"5Wx5$['\\5
b#R)F'F+F'F'*&\"-+k[30GF')F+\"#@F'F'*&\"/+3)y[gS\"F')F+\"#?F'F'*&\"7*f
V9[qKpD:;%F'FHF'F'*&\"7-Kx/#G+)pUDVF'FKF'F'*&\"6ofh$Gh](4G)QF'FenF'F'*
&\"4u`3lo)4=PEF')F+\"#:F'F'*&\"3<f!Q6!o\"eB%F')F+F&F'F'*&\"6-W4&3C-9\"
fw\"F')F+F_oF'F'*&\"66%*32JjYR2&[F'F;F'F'*&\"1S^YJ%4$4]F')F+\"#=F'F'F'
-F\\o6#F6F'!\"\"**F7F',H*&\"3g&)4HuWc'G)F'FenF'F'*&\"4%)\\)zp(*GfA?F'F
WF'F'*&\"4#=`<ve<()QMF'FSF'F'*&\"0y35k_rq(F'F]pF'F'*&\"4Q(>+m#e#>DVF'F
OF'F'*&\"3y_SYZ@I>\"*F'FBF'F'*&\"3A>+K;\"QU_$F'F>F'F'*&\"-?*H!e1JF'Feo
F'F'*&\"+![C:'pF'F[sF'F'*&\"4Q))Q2XfJ2<%F'FKF'F'*&\"4^J!R`jN2`JF'FHF'F
'*&\"4kub7!R![Z*=F'FEF'F'*&\"3>g3LOP@\"4\"F'F;F'F'*&\"2;lus.quo#F'FfrF
'F'*&\"./h170]'F'FcrF'F'*&\"/g=0Q#HZ)F'F_rF'F'*&\"1)4l!p`H%>&F'FdpF'F'
\"2!o(=.b6X^#F'*&\"3C%*f7djQ5@F'F+F'F'F'),&F+F'FXF'FXF'-F\\o6#FhuF'F_s
**,:\",c,$eeaF'*&\"-7yY*o]$F'F+F'F'*&\".mCY+x,\"F'FenF'F'*&\".!Q%f%eg<
F'FWF'F'*&\".\\)fa1<?F'FSF'F'*&\".OdtXng\"F'FOF'F'*&\"-O(zC*z!*F'FKF'F
'*&\"-U!fQ,k$F'FHF'F'*&\"-XvIg95F'FEF'F'*&\",=b#ps=F'FBF'F'*&\"+?TIg?F
'F>F'F'*&F:F'F;F'F'F'FguF'),&F+F'FTF'F/F'-F\\o6#FdwF'F'**F*F'F6F',L*&
\".!or)3H2(F'FeqF'F'*&\"0?()Re&GcJF'FipF'F'*&\"1/^TV7m]mF'F[sF'F'*&\"2
3Q\"R#o?zz)F'FeoF'F'*&\"3;Nbl6**R%>)F'FcrF'F'*&\"4/%R\"*G&R\"*>r&F'F_r
F'F'*&\"5(f?7lu@(z\"4$F'F]pF'F'*&\"6F#4Q8QmFqI8F'FdpF'F'*&\"6SRaOSg\"p
;DYF'FfrF'F'*&\"7SP9#G#4&HD5J\"F'F;F'F'*&\"7TEw*y\"y:&4s/$F'F>F'F'*&\"
7s^m.JrV\"**y\"eF'FBF'F'*&\"7r2b0pn+aa3\"*F'FEF'F'*&\"8c&=xdwdFF.j6F'F
HF'F'*&\"8kU\\9D&pM\"[$*>\"F'FKF'F'*&\"7@t7`:r/%p`$)*F'FOF'F'*&\"7t\\k
![W%yp\"RE'F'FSF'F'*&\"7[5M*[C,G*3')HF'FWF'F'*&\"7qX-(*G%*QiW-5F'FenF'
F'*&\"6oA$)4cBN$=8@F'F+F'F'\"5+YH`9!31R5#F'F'-F\\o6#FjnF'F'\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Four different representations of \+
the Legendre polynomials:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 28 "(a) We consider the summand:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "legendre1:=binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre1G*(-%)binomialG6$%\"n
G%\"kG\"\"\"-F'6$,&F)!\"\"F+F/F*F+),&#F+\"\"#F+*&F3F/%\"xGF+F/F*F+" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "The sum" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "Sum(legendre1,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$SumG6$*(-%)binomialG6$%\"nG%\"kG\"\"\"-F(6$,&F*!\"\"F,F0F+F,)
,&#F,\"\"#F,*&F4F0%\"xGF,F0F+F,/F+;\"\"!F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 37 "has the hypergeometric representation" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "Sumtohyper(legendre1,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%*HypergeomG6%7$,&%\"nG\"\"\"F)F),$F(!\"\"7#F),&#F)\"
\"#F)*&F/F+%\"xGF)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satis
fies the recurrence equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sum
recursion(legendre1,k,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,
&%\"nG\"\"\"F(F(F(-%\"PG6#F'F(F(*(,&*&\"\"#F(F'F(F(\"\"$F(F(%\"xGF(-F*
6#F&F(!\"\"*&,&F'F(F/F(F(-F*6#F6F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "(b) We consider the summand:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "legendre2:=1/2^n*binomial(n,k)^2*(x-1)^(n-k)*(x+1)^k;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*legendre2G**)\"\"#%\"nG!\"\"-%)
binomialG6$F(%\"kGF'),&\"\"\"F)%\"xGF0,&F(F0F-F)F0),&F1F0F0F0F-F0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "The sum" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "Sum(legendre2,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$SumG6$**)\"\"#%\"nG!\"\"-%)binomialG6$F)%\"kGF(),&\"\"\"F*%\"
xGF1,&F)F1F.F*F1),&F2F1F1F1F.F1/F.;\"\"!F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 37 "has the hypergeometric representation" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "Sumtohyper(legendre2,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*()\"\"#%\"nG!\"\"),&\"\"\"F'%\"xGF*F&F*-%*HypergeomG6%
7$,$F&F'F07#F**&,&F+F*F*F*F*F)F'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 37 "and satisfies the recurrence equation" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "sumrecursion(legendre2,k,P(n));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"F(F(F(-%\"PG6#F'F(F(*(,&*&\"\"#F(F'F(
F(\"\"$F(F(%\"xGF(-F*6#F&F(!\"\"*&,&F'F(F/F(F(-F*6#F6F(F(\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(c) We consider the summand:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "legendre3:=1/2^n*(-1)^k*binomial(n,
k)*binomial(2*n-2*k,n)*x^(n-2*k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%*legendre3G*,)\"\"#%\"nG!\"\")F)%\"kG\"\"\"-%)binomialG6$F(F+F,-F.6$,
&*&F'F,F(F,F,*&F'F,F+F,F)F(F,)%\"xG,&F(F,*&F'F,F+F,F)F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 7 "The sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "Sum(legendre3,k=0..floor(n/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$SumG6$*,)\"\"#%\"nG!\"\")F*%\"kG\"\"\"-%)binomialG6$F)F,F--F/6$,&*
&F(F-F)F-F-*&F(F-F,F-F*F)F-)%\"xG,&F)F-*&F(F-F,F-F*F-/F,;\"\"!-%&floor
G6#,$*&F(F*F)F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "has the hype
rgeometric representation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Sumtoh
yper(legendre3,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,)\"\"#%\"nG!\"
\"-%&GAMMAG6#,&*&F%\"\"\"F&F-F-F-F-F--F)6#,&F&F-F-F-!\"#)%\"xGF&F--%*H
ypergeomG6%7$,$*&F%F'F&F-F',&*&F%F'F&F-F'#F-F%F-7#,&F&F'F<F-*&F-F-*$)F
3F%F-F'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satisfies the re
currence equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sumrecursion(l
egendre3,k,P(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"
\"F(F(F(-%\"PG6#F'F(F(*(,&*&\"\"#F(F'F(F(\"\"$F(F(%\"xGF(-F*6#F&F(!\"
\"*&,&F'F(F/F(F(-F*6#F6F(F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
28 "(d) We consider the summand:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 
"legendre4:=x^n*hyperterm([-n/2,(1-n)/2],[1],1-1/x^2,k);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%*legendre4G*,)%\"xG%\"nG\"\"\"-%+pochhammerG6$
,$*&\"\"#!\"\"F(F)F0%\"kGF)-F+6$,&*&F/F0F(F)F0#F)F/F)F1F)-%*factorialG
6#F1!\"#),&F)F)*&F)F)*$)F'F/F)F0F0F1F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "The sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Sum(legendre
4,k=0..floor(n/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,)%\"
xG%\"nG\"\"\"-%+pochhammerG6$,$*&\"\"#!\"\"F)F*F1%\"kGF*-F,6$,&*&F0F1F
)F*F1#F*F0F*F2F*-%*factorialG6#F2!\"#),&F*F**&F*F**$)F(F0F*F1F1F2F*/F2
;\"\"!-%&floorG6#,$*&F0F1F)F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
37 "has the hypergeometric representation" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "Sumtohyper(legendre4,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&)%\"xG%\"nG\"\"\"-%*HypergeomG6%7$,$*&\"\"#!\"\"F&F'F
/,&*&F.F/F&F'F/#F'F.F'7#F'*(,&F'F/F%F'F',&F%F'F'F'F'F%!\"#F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and satisfies the recurrence equat
ion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sumrecursion(legendre4,k,P(n
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG\"\"\"F(F(F(-%\"PG6
#F'F(F(*(,&*&\"\"#F(F'F(F(\"\"$F(F(%\"xGF(-F*6#F&F(!\"\"*&,&F'F(F/F(F(
-F*6#F6F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Proof of Clausen's formula by Cauc
hy product:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "summand:=j->
hyperterm([a,b],[a+b+1/2],1,j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(
summandGj+6#%\"jG6\"6$%)operatorG%&arrowGF(-%*hypertermG6&7$%\"aG%\"bG
7#,(F0\"\"\"F1F4#F4\"\"#F4F49$F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 40 "Closedform(summand(j)*summand(k-j),j,k);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%*HypertermG6&7%,$*&\"\"#\"\"\"%\"bGF*F*,$*
&F)F*%\"aGF*F*,&F.F*F+F*7$,&*&F)F*F+F*F**&F)F*F.F*F*,(F.F*F+F*#F*F)F*F
*%\"kG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Proof of Clausen's form
ula by differential equations:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 
"The left hand factor satisfies the differential equation" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "DE:=sumdiffeq(su
mmand(j)*x^j,j,C(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG/,(**\"
\"#\"\"\",&F)!\"\"%\"xGF)F)F,F)-%%diffG6$-%\"CG6#F,-%\"$G6$F,F(F)F)*&,
.*(F(F)F,F)%\"aGF)F)F)F+*&F(F)F9F)F+*&F(F)%\"bGF)F+*(F(F)F,F)F<F)F)*&F
(F)F,F)F)F)-F.6$F0F,F)F)**F(F)F0F)F<F)F9F)F)\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 64 "Therefore the left hand side satisfies the differe
ntial equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "with(gfun):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "LHS:=`diffeq*diffeq`(DE,DE,C(x));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$LHSG,**&,&*(\"\")\"\"\")%\"aG\"\"#F*%\"bGF*F**(F)F*)F.F-F*F,F
*F*F*-%\"CG6#%\"xGF*F**&,8*(\"\"'F*F4F*F,F*F**(\"\"%F*F4F*F0F*F**(F8F*
F4F*F.F*F***\"#;F*F.F*F,F*F4F*F**&F-F*F4F*F**(F:F*F4F*F+F*F**&F-F*F.F*
!\"\"*&F:F*F0F*FA*(F)F*F.F*F,F*FA*&F-F*F,F*FA*&F:F*F+F*FAF*-%%diffG6$F
1F4F*F**&,.*(F8F*)F4F-F*F,F*F**(F8F*FLF*F.F*F**&F8F*FLF*F**(F8F*F4F*F,
F*FA*(F8F*F4F*F.F*FA*&\"\"$F*F4F*FAF*-FG6$F1-%\"$G6$F4F-F*F**&,&*&F-F*
FLF*FA*&F-F*)F4FRF*F*F*-FG6$F1-FV6$F4FRF*F*" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 73 "On the other hand the right hand side satisfies the dif
ferential equation" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "RHS:=sumdiffeq(hyperterm([2*a,2*b,a+b],[2*a+2*b,a+b+1
/2],x,k),k,C(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RHSG/,**,\"\")
\"\"\"-%\"CG6#%\"xGF)%\"bGF)%\"aGF),&F/F)F.F)F)F)*(\"\"#F),8*(F2F)F-F)
)F.F2F)F)*(F2F)F-F))F/F2F)F)**F(F)F.F)F/F)F-F)F)F-F)*&F2F)F7F)!\"\"*&F
2F)F5F)F:F/F:F.F:*(\"\"$F)F-F)F.F)F)*(\"\"%F)F.F)F/F)F:*(F=F)F-F)F/F)F
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