{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "MS Serif" 1 10 128 0 0 1 0 0 0 
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1 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{PSTYLE "Normal
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1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 258 1 {CSTYLE "" 
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2 2 19 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 255 
0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart; gc( );  ker
nelopts(ASSERT=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 13 "Ekhad (V6.63)" }}}{EXCHG {PARA 
259 "" 0 "" {TEXT -1 35 "chargement par la commande ekhad6()" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "Cr\351ation de la liste des proce
dures savelib'ed" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Ne pas encombr
er la biblioth\350que principale" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 
"unprotect(savelib); savelib:= proc() error \"savelib is gone\" end :
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Biblioth\350que compl\351ment
aire" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ssavelib:= NULL:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 1063 "xsavelib:= proc() local filename, value
s, library, pathname, f;\nglobal ssavelib, libname; option `Copyright \+
Pld`;\nif assigned(libname) then library := op(1, eval([libname], 1))
\n else error \"the global variable `libname' must be assigned\"\nend \+
if;\nif nargs < 2 then error \"no names specified to save\" end if;\nf
ilename := args[nargs];\nvalues := args[1 .. nargs - 1] ;\nif not type
(filename, '\{string, symbol\}') \n   then error \"last argument (file
name) must be a string\" end if;\nif not searchtext(\".m\", filename, \+
-2 .. -1) = 1 \n  then error \"last argument (filename) must be a *.m \+
string\" end if;\n if not type([values], 'list(\{symbol, string\})') \+
\n  then  f := remove(type, [values], \{symbol,string\}); \n  error \"
all arguments must be symbols, cannot save these %1\", f\n end if;\n  \+
pathname := cat(library, \"/\", filename);\n  f := subs('T' = ( (op@ma
p)(convert, [values], symbol), pathname), \n    proc() save T end proc
);\n  try f(); ssavelib:= ssavelib, values; return;  \n  catch: print(
lasterror);   \n  end try;\nerror \"unable to save %1 in %2\", [values
], library\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# \+
tzx:=\"\": xsavelib('tzx', `ekhad6.m`); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "ever_load:=  NULL :" }}}}{EXCHG {PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$quiG7PQ$AZc6\"Q$AZdF'Q'AZpapcF'Q'AZpapdF'Q(RootOf1F'
Q(ZeillimF'Q*bdokcertcF'Q*bdokcertdF'Q'bdokctF'Q(bdokctoF'Q'celineF'Q#
ctF'Q&ctoldF'Q&cttryF'Q&duiscF'Q&duisdF'Q*evalapplyF'Q%ezraF'Q(findgQR
F'Q(findrecF'Q'goremcF'Q'goremdF'Q%gzorF'Q&gzor1F'Q&hafelF'Q&paperF'Q'
paper3F'Q'papercF'Q'paperdF'Q'pashetF'Q(pashetcF'Q(pashetdF'Q#rfF'Q*si
mplify1F'Q'solve1F'Q%tovuF'Q%yafeF'Q&yafecF'Q%zeilF'Q&zeil4F'Q&zeil5F'
Q&zeil6F'Q(zeillimF'Q(zeilpapF'Q)zeilpap3F'Q)zeilpap5F'" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 4 "Ezra" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 409 "`ezra`:=proc( qui ); if nargs <>0 then return ( `ezra/` || qu
i )() fi;\nprint(`EKHAD,  Version of Sept. 13, 2000: adapted to Maple \+
6 `):\nprint(`A Maple package for proving Hypergeometric and other kin
ds of identities`):\nprintf(`\\n`):\nprint(`For help with a specific p
rocedure, type \"ezra(procedure_name);\"`):\nprint(`Contains : version
, celine, findrec,ct,zeil,zeilpap,zeillim,AZd,AZc,AZpapd,AZpapc`):\nen
d: %();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%WEKHAD,~~Version~of~Sept.~
13,~2000:~adapted~to~Maple~6~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coA
~Maple~package~for~proving~Hypergeometric~and~other~kinds~of~identitie
sG" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%[oFor~help~with~a~specific~procedure,~type~\"ezra(procedure_name);\"G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\pContains~:~version,~celine,~fi
ndrec,ct,zeil,zeilpap,zeillim,AZd,AZc,AZpapd,AZpapcG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1373 "`ezra/version`:= proc(): print(`Version of Sept. 13, 2000: ada
pted to Maple 6 `):\nprint(`Also works on Maple 5 and below`):\nprint(
`In the penultimate Version of Feb 25, 1999 a suggestion`):\nprint(` o
f Frederic Chyzak was used, with considerable `):\nprint(`speed-up. We
 thank him SO MUCH!`):lprint(``):\nprint(`The penpenultimate version, \+
Feb. 1997,`):\nprint(` corrected a subtle bug discovered by Helmut Pro
dinger`):\nlprint(``):\nprint(`Previous versions benefited from commen
ts by Paula Cohen, `):\nprint(`Lyle Ramshaw, and Bob Sulanke.`):\nlpri
nt(``):\nprint(`This is EKHAD, One of the Maple packages`):\nprint(`ac
companying the book `):\nprint(` \"A=B\" `):\nprint(` (published by A.
K. Peters, Wellesley, 1996) `):\nprint( ` by Marko Petkovsek, Herb Wil
f, and Doron Zeilberger.`):\nlprint(``):\nprint(`The most current vers
ion is available on WWW at:`):\nprint(` http://www.math.temple.edu/~ze
ilberg .`):\nprint(`Information about the book, and how to order it, c
an be found in`):\nprint(`http://www.central.cis.upenn.edu/~wilf/AeqB.
html .`):\nprint(`Please report all bugs to: zeilberg@math.temple.edu \+
.`):\nprint(`All bugs or other comments used will be acknowledged in f
uture`):\nprint(`versions.`):\nlprint(``):\nprint(`For general help, a
nd a list of the available functions,`):\nprint(` type \"ezra();\". Fo
r specific help type \"ezra(procedure_name)\" `):\nlprint(``):\n end: \+
 ezra(version);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OVersion~of~Sept.~
13,~2000:~adapted~to~Maple~6~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@Al
so~works~on~Maple~5~and~belowG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%XIn
~the~penultimate~Version~of~Feb~25,~1999~a~suggestionG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%Q~of~Frederic~Chyzak~was~used,~with~considerable~
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@speed-up.~We~thank~him~SO~MUCH!
G" }}{PARA 6 "" 1 "" {TEXT -1 2 "``" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%GThe~penpenultimate~version,~Feb.~1997,G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%W~corrected~a~subtle~bug~discovered~by~Helmut~Prodinge
rG" }}{PARA 6 "" 1 "" {TEXT -1 2 "``" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%enPrevious~versions~benefited~from~comments~by~Paula~Cohen,~G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%?Lyle~Ramshaw,~and~Bob~Sulanke.G" }}
{PARA 6 "" 1 "" {TEXT -1 2 "``" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ITh
is~is~EKHAD,~One~of~the~Maple~packagesG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%7accompanying~the~book~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(~
\"A=B\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%N~(published~by~A.K.~Pet
ers,~Wellesley,~1996)~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%V~by~Marko
~Petkovsek,~Herb~Wilf,~and~Doron~Zeilberger.G" }}{PARA 6 "" 1 "" 
{TEXT -1 2 "``" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%QThe~most~current~v
ersion~is~available~on~WWW~at:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%H~
http://www.math.temple.edu/|irzeilberg~.G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%[oInformation~about~the~book,~and~how~to~order~it,~can
~be~found~inG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Shttp://www.central.
cis.upenn.edu/|irwilf/AeqB.html~.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%VPlease~report~all~bugs~to:~zeilberg@math.temple.edu~.G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%inAll~bugs~or~other~comments~used~will~be~ackno
wledged~in~futureG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*versions.G" }}
{PARA 6 "" 1 "" {TEXT -1 2 "``" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YFo
r~general~help,~and~a~list~of~the~available~functions,G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%jn~type~\"ezra();\".~For~specific~help~type~\"e
zra(procedure_name)\"~G" }}{PARA 6 "" 1 "" {TEXT -1 2 "``" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 3 "AZc" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 922 "`ezra/AZc`:=pr
oc()\nprint(`AZc(A,y,x,D) tries to finds a linear diff.eq. of order <=
8,`):\nprint(` phrased in terms of x, and diff.w.r.t x, D`):\nprint(`f
or the integrale of A(x,y) with respect to x, whenever A(x,y) is`):\np
rint(`hypergeometric in (x,y),i.e.  A_x(x,y)/A(x,y) and A_y(x,y)/A(x,y
) are`):\nprint(`rational funtions of x and y. It follows the algorith
n of`):\nprint(`Gert Almkvist and Doron Zeilberger, \"The method of di
fferentiating`):\nprint(`under the integral sign\", J. Symbolic Comput
ation 10(1990), 571-591`):\nprint(`AZc(A,y,x,D) returns the expression
 seq. ope(x,D),cert(x,n)`):\nprint(`satisfying ope(x,D)A(x,y)=diff(cer
t(x,y)*A(x,y),y).`):\nprint(`If no linear diff. eq. of order<=8 is fou
nd, it returns 0`):\nprint(`see AZpapc for a verbose vsersion`):\nprin
t(`For example AZc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x,D) gives the`):\npr
int(`the diff.eq., and proof certificate for the Legendre polynomials.
`):\nend: %();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fnAZc(A,y,x,D)~trie
s~to~finds~a~linear~diff.eq.~of~order~<=8,G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%L~phrased~in~terms~of~x,~and~diff.w.r.t~x,~DG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\ofor~the~integrale~of~A(x,y)~with~r
espect~to~x,~whenever~A(x,y)~isG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%`
ohypergeometric~in~(x,y),i.e.~~A_x(x,y)/A(x,y)~and~A_y(x,y)/A(x,y)~are
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Zrational~funtions~of~x~and~y.~I
t~follows~the~algorithn~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]oGert
~Almkvist~and~Doron~Zeilberger,~\"The~method~of~differentiatingG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%^ounder~the~integral~sign\",~J.~Symbo
lic~Computation~10(1990),~571-591G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%fnAZc(A,y,x,D)~returns~the~expression~seq.~ope(x,D),cert(x,n)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%Tsatisfying~ope(x,D)A(x,y)=diff(cert(
x,y)*A(x,y),y).G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZIf~no~linear~dif
f.~eq.~of~order<=8~is~found,~it~returns~0G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%Bsee~AZpapc~for~a~verbose~vsersionG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%gnFor~example~AZc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x,D)
~gives~theG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\othe~diff.eq.,~and~p
roof~certificate~for~the~Legendre~polynomials.G" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 980 "`ezra/AZpapc`:=proc()\nprint(`AZpapc(INTEGRAN
D,y,x) inputs a hypergeometric integrand`):\nprint(`in the continuous \+
variables y and x (i.e. the logarithmic derivatives`):\nprint(`diff(IN
TEGRAND,x)/INTEGRAND and diff(INTEGRAND,y)/INTEGRAND are`):\nprint(`ra
tional functions in x and y)`):\nprint(`and outputs a paper with a pro
of of the linear differential equation`):\nprint(`that the definite in
tegral w.r.t. to y (which is a function of x)`):\nprint(`satisfies. It
 uses the method of`):\nprint(`Gert Almkvist and Doron Zeilberger, \"T
he method of differentiating`):\nprint(`under the integral sign\", J. \+
Symbolic Computation 10(1990), 571-591.`):\nlprint(``):\nprint(`It cou
ld be used to establish the diff. eq. of the `):\nprint(`classical ort
hogonal polynomials, when they are defined in terms`):\nprint(`or thei
r generating funtion.`):\nprint(`For example AZpapc(1/sqrt(1-2*x*t+t^2
)/t^(n+1),t,x) gives the`):\nprint(`the differential equation satisfie
d by the Legendre polynomials`):\nend: %();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%XAZpapc(INTEGRAND,y,x)~inputs~a~hypergeometric~integra
ndG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%`oin~the~continuous~variables~
y~and~x~(i.e.~the~logarithmic~derivativesG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%jndiff(INTEGRAND,x)/INTEGRAND~and~diff(INTEGRAND,y)/IN
TEGRAND~areG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?rational~functions~i
n~x~and~y)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oand~outputs~a~paper~
with~a~proof~of~the~linear~differential~equationG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\\othat~the~definite~integral~w.r.t.~to~y~(which~is~a~
function~of~x)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Asatisfies.~It~use
s~the~method~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]oGert~Almkvist~a
nd~Doron~Zeilberger,~\"The~method~of~differentiatingG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%_ounder~the~integral~sign\",~J.~Symbolic~Computat
ion~10(1990),~571-591.G" }}{PARA 6 "" 1 "" {TEXT -1 2 "``" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%TIt~could~be~used~to~establish~the~diff.~eq.~o
f~the~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[oclassical~orthogonal~pol
ynomials,~when~they~are~defined~in~termsG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%=or~their~generating~funtion.G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%hnFor~example~AZpapc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x)~
gives~theG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnthe~differential~equa
tion~satisfied~by~the~Legendre~polynomialsG" }}}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 3 "AZd" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1036 "`ezr
a/AZd`:=proc()\nprint(`AZd(A,x,n,N) finds a recurrence of order ORDER<
=8`):\nprint(`phrased in terms of n, and the shift in n, N`):\nprint(`
for the integrale of A(x,n) with respect to x, whenever A(x,n) is`):\n
print(`hypergeometric in (x,n),i.e. A(x,n+1)/A(x,n) and A'(x,n)/A(x,n)
 are`):\nprint(`rational funtions of x and n. It follows the algorithn
 of`):\nprint(`Gert Almkvist and Doron Zeilberger, \"The method of dif
ferentiating`):\nprint(`under the integral sign\", J. Symbolic Computa
tion 10(1990), 571-591`):\nprint(`A should not be a product of a funct
ion of x and a function of n.`):\nprint(``):\nprint(`AZd(A,x,n,N) retu
rns the expression seq. ope(n,N),cert(x,n)`):\nprint(`satisfying ope(n
,N)A(x,n)=diff(cert(x,n)*A(x,n),x).`):\nprint(`If no recurrence is fou
nd, it returns 0.`):\nprint(``):\nprint(`A verbose version is AZpapd(A
,x,n), type ezra(AZpapd) for details.`):\nprint(``):\nprint(`For examp
le AZd(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,n,N) gives`):\nprint(`the diff.eq
., and proof certificate, for the Legendre polct:ynomials.`):\nend: %(
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RAZd(A,x,n,N)~finds~a~recurrenc
e~of~order~ORDER<=8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Mphrased~in~t
erms~of~n,~and~the~shift~in~n,~NG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
\\ofor~the~integrale~of~A(x,n)~with~respect~to~x,~whenever~A(x,n)~isG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^ohypergeometric~in~(x,n),i.e.~A(
x,n+1)/A(x,n)~and~A'(x,n)/A(x,n)~areG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%Zrational~funtions~of~x~and~n.~It~follows~the~algorithn~ofG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%]oGert~Almkvist~and~Doron~Zeilberger,
~\"The~method~of~differentiatingG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
^ounder~the~integral~sign\",~J.~Symbolic~Computation~10(1990),~571-591
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\oA~should~not~be~a~product~of~
a~function~of~x~and~a~function~of~n.G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fnAZd(A,x,n,N)~returns~the~
expression~seq.~ope(n,N),cert(x,n)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%Tsatisfying~ope(n,N)A(x,n)=diff(cert(x,n)*A(x,n),x).G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%IIf~no~recurrence~is~found,~it~returns~0.G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%]oA~verbose~version~is~AZpapd(A,x,n),~type~ezra(AZpapd)~for~details.
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%YFor~example~AZd(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,n,N)~givesG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%`othe~diff.eq.,~and~proof~certificate
,~for~the~Legendre~polct:ynomials.G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1191 "`ezra/AZpapd`:=proc()\nprint(`AZpapd(INTEGRAND,x,n)
 inputs a hypergeometric integrand`):\nprint(`in the continuous variab
le x and the discrete variable n`):\nprint(`i.e. i.e. A(x,n+1)/A(x,n) \+
and A'(x,n)/A(x,n) are rational functions`):\nprint(`of (x,n)),`):\npr
int(`and outputs a paper with a proof of the linear recurrence equatio
n`):\nprint(`that the definite integral w.r.t. to y (which is a functi
on of n)`):\nprint(`assuming that the integrand (or rather it times th
e certificate`):\nprint(`vanishes at the endpoints, or it is a  contou
r integrals`):\nprint(`satisfies. It assumes the following: A(x,n) is \+
not a product of`):\nprint(`of a function of n and a function of x`):
\nprint(`It uses the method of`):\nprint(`Gert Almkvist and Doron Zeil
berger, \"The method of differentiating`):\nprint(`under the integral \+
sign\", J. Symbolic Computation 10(1990), 571-591`):\nprint(`It could \+
be used to establish the recurrences of the `):\nprint(`classical orth
ogonal polynomials, when they are defined in terms`):\nprint(`or their
 generating funtion`):\nprint(`For example AZpapd(1/sqrt(1-2*x*t+t^2)/
t^(n+1),t,n) gives the`):\nprint(`the recurrence, and proof, satisfied
 by the Legendre polynomials`):\nend: %();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%XAZpapd(INTEGRAND,x,n)~inputs~a~hypergeometric~integra
ndG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Yin~the~continuous~variable~x~
and~the~discrete~variable~nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oi.e
.~i.e.~A(x,n+1)/A(x,n)~and~A'(x,n)/A(x,n)~are~rational~functionsG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%+of~(x,n)),G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%]oand~outputs~a~paper~with~a~proof~of~the~linear~recur
rence~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\othat~the~defini
te~integral~w.r.t.~to~y~(which~is~a~function~of~n)G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%jnassuming~that~the~integrand~(or~rather~it~times~th
e~certificateG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Yvanishes~at~the~en
dpoints,~or~it~is~a~~contour~integralsG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%jnsatisfies.~It~assumes~the~following:~A(x,n)~is~not~a~product~
ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Gof~a~function~of~n~and~a~func
tion~of~xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6It~uses~the~method~ofG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]oGert~Almkvist~and~Doron~Zeilber
ger,~\"The~method~of~differentiatingG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%^ounder~the~integral~sign\",~J.~Symbolic~Computation~10(1990),~571
-591G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%VIt~could~be~used~to~establi
sh~the~recurrences~of~the~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[oclas
sical~orthogonal~polynomials,~when~they~are~defined~in~termsG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%<or~their~generating~funtionG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnFor~example~AZpapd(1/sqrt(1-2*x*t+t
^2)/t^(n+1),t,n)~gives~theG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[othe~
recurrence,~and~proof,~satisfied~by~the~Legendre~polynomialsG" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "creative telescoping" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1041 "`ezra/ct`:=proc()\n print(` ct(SU
MMAND,ORDER,k,n,N)`): \nprint(`This is a Maple inplementation of the a
lgorithm described in Ch. 6`):\nprint(`of the book A=B, first proposed
 in : D. Zeilberger, \"The method of`):\nprint(`of creative telescopin
g\", J.Symbolic Computation 11(1991) 195-204`): print(``):\nprint(`fin
ds a recurrence for SUMMAND in the parameters k and n,   `):\nprint(` \+
of order ORDER, with N is the chosen symbol for the shift in n.`):  pr
int(``):\n print(`SUMMAND should be a product of factorials and/or bin
omial coeffs`):\n print(`and/or rising factorials, where (a)_k is deno
ted by rf(a,k)`):\n print(`and/or powers of k and n, and, optionally, \+
a polynomial factor`):\nprint(`The output consists of an operator ope(
N,n) and a certificate R(n,k)`):\nprint(`with the properties that if w
e define G(n,k):=R(n,k)*SUMMAND then`):\nprint(`ope(N,n)SUMMAND(n,k)=G
(n,k+1)-G(n,k)`):\nprint(`which is a routinely verifiable identity.`):
\n print(`For example \"ct(binomial(n,k),1,k,n,N);\" would yield the o
utput`):\nprint(`  N-2, k/(k-n-1) `):\nend: %();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9~ct(SUMMAND,ORDER,k,n,N)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%]oThis~is~a~Maple~inplementation~of~the~algorithm~desc
ribed~in~Ch.~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]oof~the~book~A=B,
~first~proposed~in~:~D.~Zeilberger,~\"The~method~ofG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%\\oof~creative~telescoping\",~J.Symbolic~Computatio
n~11(1991)~195-204G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%gnfinds~a~recurrence~for~SUMMAND~in~the~par
ameters~k~and~n,~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[o~of~order~O
RDER,~with~N~is~the~chosen~symbol~for~the~shift~in~n.G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[oSUMMAND~
should~be~a~product~of~factorials~and/or~binomial~coeffsG" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%fnand/or~rising~factorials,~where~(a)_k~is~den
oted~by~rf(a,k)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inand/or~powers~o
f~k~and~n,~and,~optionally,~a~polynomial~factorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%_oThe~output~consists~of~an~operator~ope(N,n)~and~a~ce
rtificate~R(n,k)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\owith~the~prop
erties~that~if~we~define~G(n,k):=R(n,k)*SUMMAND~thenG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%Eope(N,n)SUMMAND(n,k)=G(n,k+1)-G(n,k)G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%Jwhich~is~a~routinely~verifiable~identity.G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnFor~example~\"ct(binomial(n,k),
1,k,n,N);\"~would~yield~the~outputG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%2~~N-2,~k/(k-n-1)~G" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ever_load:= ever_load, ez
ra, `ezra/version`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ever_loadG6$
%%ezraG%-ezra/versionG" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "celine" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "solve1:= readlib(`solve/linear`) ;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'solve1GR6\"6'%\"kG%$solG%%notzG%%eqnsG%%v
arsG6#%aoCopyright~(c)~1992~by~the~University~of~Waterloo.~All~rights~
reserved.GF&C.>8'&9\"6#\"\"\">8(&F36#\"\"#@$-%%typeG6$F1%%listG>F1<#-%
#opG6#F1@$-F=6$F7F?>F7<#-FC6#F7>F1-%$mapG6$R6#%\"eGF&F&F&@%-F=6$9$%\"=
G,&-%$lhsG6#FVF5-%$rhsGFen!\"\"FVF&F&F&F1@$4-F=6$F1-%$setG6#%*algebrai
cGYQhn1st~argument~must~be~a~list~or~set~of~polynomials~(equations)6\"
@$4-F=6$F7-F^o6#%%nameGYQW2nd~argument~must~be~a~list~or~set~of~names~
(unknowns)Fco?&8$F1%%trueG@$54-F=6$F^p-%(polynomG6$%)anythingGF72F5-%'
degreeG6$F^pF7O%%FAILG-%)userinfoG6&F5.%&solveG%,#~equationsG-%%nopsGF
D@9-F=6$F1-F^o6#-Ffp6$%)rationalGF7>8%-%5solve/linear/integerG6$F1F7-F
=6$F1-F^o6#-Ffp6$%(numericGF7>F`r-%3solve/linear/floatGFcr-F=6$F1-F^o6
#-Ffp6$-%(complexG6#FjrF7>F`r-%5solve/linear/complexGFcr-F=6$F1-F^o6#-
%(ratpolyG6#F^r>F`r-%5solve/linear/polynomGFcr-F=6$F1-F^o6#-Ffp6$%'alg
numGF7>F`r-%4solve/linear/algnumGFcr-F=6$F1-F^o6#-Ffp6$%'radnumGF7>F`r
-%4solve/linear/radnumGFcr-F=6$F1-F^o6#-F_t6#Fjt>F`r-%5solve/linear/po
lyalgGFcr-F=6$F1-F^o6#-F_t6#Fdu>F`rFfu-F=6$F1-F^o6#-%'algfunGF`t>F`r-%
4solve/linear/algfunGFcr-F=6$F1-F^o6#-%'radfunGF`t>F`r-%4solve/linear/
radfunGFcr-%(hastypeG6$F1%&floatGC%>F1-%(convertG6%-%&evalfGFDF^r.%&ex
actG>F`r-%-solve/linearG6%F1F7&F36#;\"\"$9#@$0F`r%%NULLG>F`r-Fdx6#F`r>
F`r-%3solve/linear/fieldGFcr@%/F_yF:OF`r>8&-%%subsG6$F`r&F36#F^y@'-%'m
emberG6$\"\"!F]zF&-Fez6$Fgz-FN6$%'normalGF]zF&F`rF&F&F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "Erreur ds celine quand facteur global" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1092 "celine:=proc(f,ii,jj) glo
bal solve1, gi, gj, gl, gm, zz, zzz, yy, yyy; local tt,rr,xx,ss,zy, b,
 iii, k, n, r:\niii:=ii*(jj+1)+jj: b:=table(); k:='k': n:='n';\nxx:=se
q(b[gi], gi=0..iii):\nss:=numer(simplify(expand(add(add(b[gi*(jj+1)+gj
]*f(n-gj,k-gi)/f(n,k), gi=0..ii),gj=0..jj)))):\ntt:=coeffs(collect(ss,
k),k): yy:=solve1(\{tt\},\{xx\}):  \nindets(map(rhs, yy)) intersect \{
xx\}; if nops(%) <> 0 then yyy:= map(z-> `=`(z,1), %); fi;\nzz:=simpli
fy(add(add(b[gl*(jj+1)+gm]*'F'(n-gm,k-gl), gl=0..ii), gm=0..jj)): zzz:
=zz;  \nprintf(\"The full  recurrence is \\n\"):\nnumer(simplify(subs(
yy,yyy,zz))): zz:= collect(%, [seq(seq( 'F'(n-gj,k-gi), gi=0..ii), gj=
0..jj)]); \nprint(zz=  0) ; \nadd(add(add(b[gl*(jj+1)+gm], gl=gj+1..ii
)*simplify(expand(f(n-gm,k-gj)/f(n,k))), gj=0..ii), gm=0..jj);\nr:=sub
s(yy, yyy, simplify(expand(%))): \nrr:=simplify(factor(simplify(r))):
\nprintf(\"The telescoped form is \\n\"); \nzy:=factor(subs(yy, yyy, a
dd(simplify(add(b[gl*(jj+1)+gm], gl=0..ii))*'F'(n-gm,k), gm=0..jj))):
\nprint(zy = 'G'(n,k)-'G'(n,k-1)); \nprintf (\"where G(n,k)=R(n,k)*F(n
,k) and \"):\nprint('R'(n,k)=rr); return zz; end:\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "
ezra/celine" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 731 "`ezra/celine
`:=proc() local fu;\nprint(`celine(FUNCTION,ORDER_n,ORDER_k) applies S
ister Celine's technique`):\nprint(`It inputs a function (n,k)->FUNCTI
ON, and the guessed orders in`):\nprint(`n and k, ORDER_n,ORDER_k resp
ectively, and outputs a partial`):\nprint(`k-free recurrence`):\nprint
(`it  also finds the telescoped form of the recurrence.`):   \nprint(`
 In input, do not raise products of factorials to powers;  `):  \nprin
t(`instead raise each factorial to the power. `):\nprint(`For example:
celine((n,k)->binomial(n,k),1,1);`):\nprintf(\"\\nceline ((n,k)->n!*(2
*k)!*(-2)^(n-k)/(k!^3*(n-k)!),2,2);    ==> \\n\\n\"):\nfu:= (n,k)->n!*
(2*k)!*(-2)^(n-k)/(k!^3*(n-k)!); celine (fu,1,2): (ASSERT@simplify@sub
s)(F=fu, %=0);\nend:  ezra(celine);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%]oceline(FUNCTION,ORDER_n,ORDER_k)~applies~Sister~Celine's~technique
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnIt~inputs~a~function~(n,k)->FU
NCTION,~and~the~guessed~orders~inG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%gnn~and~k,~ORDER_n,ORDER_k~respectively,~and~outputs~a~partialG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%2k-free~recurrenceG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%Vit~~also~finds~the~telescoped~form~of~the~recurren
ce.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fn~In~input,~do~not~raise~pro
ducts~of~factorials~to~powers;~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
Linstead~raise~each~factorial~to~the~power.~G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%NFor~example:celine((n,k)->binomial(n,k),1,1);G" }}
{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 63 "celine ((
n,k)->n!*(2*k)!*(-2)^(n-k)/(k!^3*(n-k)!),2,2);    ==> " }}{PARA 6 "" 
1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 24 "The full  recurrence
 is " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&-%\"FG6$%\"nG%\"kG\"\"\"F
)F+F+*&,&F)\"\"%\"\"#!\"\"F+-F'6$,&F)F+F+F0F*F+F+*&,&F)!\"%F/F+F+-F'6$
F3,&F*F+F+F0F+F+*&,&F)F.F.F0F+-F'6$,&F)F+F/F0F*F+F+*&,&\"\")F+*&FAF+F)
F+F0F+-F'6$F>F9F+F+\"\"!" }}{PARA 6 "" 1 "" {TEXT -1 23 "The telescope
d form is " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*&-%\"FG6$%\"nG%\"k
G\"\"\"F*F,F,*(\"\"%F,-F(6$,&F*F,\"\"#!\"\"F+F,F*F,F3*&F.F,F/F,F,F,F*F
3,&-%\"GGF)F,-F76$F*,&F+F,F,F3F3" }}{PARA 6 "" 1 "" {TEXT -1 31 "where
 G(n,k)=R(n,k)*F(n,k) and " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6
$%\"nG%\"kG*&*&,&\"\"\"F,*&\"\"#F,F(F,F,F,,&F'F,F(!\"\"F,F,*$)F'F.F,F0
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ever_load:= ever_load,
 solve1, celine, `ezra/celine`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*
ever_loadG6'%%ezraG%-ezra/versionG%'solve1G%'celineG%,ezra/celineG" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 7 "findrec" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 790 "fi
ndrec:=proc(fff,DEGREE,ORDER,n,N) global solve1, gi, gj;\nlocal ope, v
ar, eq, a, kv, var1:\nif (1+DEGREE)*(1+ORDER)+2+ORDER>nops(fff) then\n
error sprintf( \"Insufficient data for a recurrence of order %a  and d
egree %a \", ORDER, DEGREE):\nfi:  a:= table(); \nvar:= \{seq(seq(a[gi
,gj], gi=0..ORDER), gj=0..DEGREE)\};\nope:= add(add(a[gi,gj]*n^gj*N^gi
 , gi=0..ORDER), gj=0..DEGREE);\n\neq:= \{seq(add(subs(n=gn, coeff(ope
,N,gi))*op(gn+gi,fff), gi= 0.. ORDER), gn = 1.. (1+DEGREE)*(1+ORDER)+2
)\}:\nvar1:=solve1(eq,var):\nkv:= indets(map(rhs, var1)) intersect var
 ;  ope:=subs(var1,ope):\nif nops(kv)>1 then  printf(\"either DEGREE o
r ORDER are too high\"):\n   printf(\"\\nThe output is not the minimal
 possible operator\"):\nfi:\nif kv = \{\} then return ope\n else retur
n subs(map(z-> `=`(z,1), kv) , ope) \nfi ;\nend:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "ezra
/findrec" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 833 "`ezra/findrec`:
=proc()\nprint(`findrec(f,DEG,ORDER,n,N) finds empirically an ordi. li
near recurrence`):\nprint(` with polynomial coeffs. The input is a seq
uence f given as a list`):\nprint(`STARTING at f[1],i.e. f[0] is not c
onsidered`):\nprint(` where DEG:=the maximal degree of the coefficient
s`):\nprint(`and ORDER:=the order of the recurrence.  The output is th
e operator`):\nprint(` in n and N, where N is the forward unit shift: \+
Nf(n):=f(n+1).`):\nprint(`For example findrec([1,1,2,3,5,8,13,21,34],0
,2,n,N) yields N^2-N-1 `):\nprint(`and findrec([1,2,5,14,42,132,429],1
,1,m,M) yields (m+2)*M-(4*m+2). `):\nprint(`If there is not enough dat
a, you will get an error.`):\nprint(`If there is no operator, you woul
d get 0`):\n(print@sort@findrec)([1,1,2,3,5,8,13,21,34,55,89], 0,2,n,N
);\ncollect(findrec([1,2,5,14,42,132,429],1,1,m,M), M);\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ezra(findrec); " }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%`ofindrec(f,DEG,ORDER,n,N)~finds~empirically~a
n~ordi.~linear~recurrenceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]o~with
~polynomial~coeffs.~The~input~is~a~sequence~f~given~as~a~listG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%MSTARTING~at~f[1],i.e.~f[0]~is~not~co
nsideredG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%S~where~DEG:=the~maximal
~degree~of~the~coefficientsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oand
~ORDER:=the~order~of~the~recurrence.~~The~output~is~the~operatorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%in~in~n~and~N,~where~N~is~the~forward
~unit~shift:~Nf(n):=f(n+1).G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oFor
~example~findrec([1,1,2,3,5,8,13,21,34],0,2,n,N)~yields~N^2-N-1~G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oand~findrec([1,2,5,14,42,132,429],1
,1,m,M)~yields~(m+2)*M-(4*m+2).~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
TIf~there~is~not~enough~data,~you~will~get~an~error.G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%IIf~there~is~no~operator,~you~would~get~0G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"NG\"\"#\"\"\"!\"\"F&F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&\"\"#\"\"\"%\"mGF'F'%\"MGF'F'F&!
\"\"*&\"\"%F'F(F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "ever_load:= ever_load, findrec, `ezra/findrec`;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ever_loadG6)%%ezraG%-ezra/versionG%
'solve1G%'celineG%,ezra/celineG%(findrecG%-ezra/findrecG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 36 "zeil, ct_proc, ct_try and the sequel" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 703 "zeil:=proc(SUMMAND,k,n,N,MAXORDER_,resh_)  \ngloba
l `ezra/zeil`, simplify1, ct_proc; local resh, MAXORDER, ORDER,gu,gu1:
\nif nargs <4 or nargs>6 then `ezra/zeil`(); error \"wrong syntax\" ; \+
fi:\nif nargs=4 then MAXORDER:=6: resh:=[]\n elif nargs=5 then MAXORDE
R:=MAXORDER_: resh:=[]\n elif nargs=6 then MAXORDER:=MAXORDER_: resh:=
resh_\nfi; \nif simplify1(SUMMAND,n,1)=1 then\n error sprintf(\"Summan
d does not depend on %a hence the sum is identically constant\", n):\n
fi:\nfor ORDER from 1 to MAXORDER do\n gu1:=ct_proc(SUMMAND,ORDER,k,n,
N,resh):\n if gu1=true then gu:=ct_proc(SUMMAND,ORDER,k,n,N):\n    if \+
gu<> false then return gu: fi:\n fi:\nod:\nerror sprintf(\"No recurren
ce of order<= %a was found\", MAXORDER):\nend:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 223 "simplify1:=  proc(expr, k, a)\n if type(expr,
 `*`) then map(procname, args)\n elif type(expr, `^`) and type(op(2, e
xpr), integer) then applyop(procname, 1, args)\n else simplify(subs(k \+
= k + a, expr)/expr)\n end if; %\nend proc:" }}{PARA 0 "" 0 "" 
{MPLTEXT 0 21 80 "essais := simplify1(n,n,1), simplify1(n^2,n,1), simp
lify1(f(n,z)^2*(n+3)^4,n,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "B
oolean : true => variables s\351parables" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "split_k_n:=proc(SU,k,n) global simplify1; \nsimplify1
(simplify1(SU,n,1),k,1):\nevalb(%=1);end:" }}{PARA 0 "" 0 "" {MPLTEXT 
0 21 69 "[n+k, (n^2+n)/(k^2), n*(k+n)-n*k]: essais:= %=map(split_k_n, \+
%,n,k); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Fait agir l'op\351rat
eur d'\351volution" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "action_ope:=
proc(POL,SUMMAND,ope,n,N) global simplify1, gi;\nlocal POL1,zub:\nadd(
coeff(ope,N,gi)*subs(n=n+gi,POL)*simplify1(SUMMAND,n,gi), gi=0..degree
(ope,N)); \nPOL1, zub:= (numer, denom)(%):\nPOL1,SUMMAND/zub,zub:\nend
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{MPLTEXT 0 21 145 "r
1,r2,r3:= action_ope(1, convert(binomial(n,k), factorial), add(C[i]*N^
i, i=0..4),n,N):\nessais:= collect(r1, [seq(C[i],i=0..4)], factor), r2
, r3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Cherche les facteurs \"q
ui vont se produire\" entre Q(k) et R(j+k)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 166 "findgQR:=proc(Q,R,k,L) local j,g:\nfor j from 1 to L
 do\n g:=gcd(Q,subs(k=k+j,R)): if degree(g,k)>0 then RETURN(g,j): fi:
\nod:\n1,0:\nend: # essais:= %((k+3)*(k+2),k,k,5);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 24 "Jolis polyn\364mes d'action" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "pashet:=proc(p,N) local nu, de:\nnu, de:= (numer, de
nom)( normal(p)):\nreturn collect(nu, N, factor)/de :\nend:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Le coeur de la chose" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 2217 "ct_proc:=proc(SUMMAND,ORDER,k,n,N,resh_)\ng
lobal action_ope, findgQR, pashet, simplify1, solve1, split_k_n, gi, g
j, apc, bpc;\nlocal A1, B1, P1, POL1, Q, R, SDZ, SDZ1, SUMMAND1, denFA
C, eq, fu, gg, gugu, jj, k1, kvutsa, l1, l2, meka1, mekb1, mumu, ope, \+
ope1, resh, va1 ; \n \nif nargs<5 or nargs>6 then\nerror sprintf(\"Syn
tax : cttry(term,ORDER,sum_variable,aux_var,Shift,para_list) \") :\nfi
:\nif nargs=5 then resh:= [] else resh:= resh_; fi;\nif split_k_n(conv
ert(SUMMAND,factorial),k,n) then\nerror sprintf(\"The summand can be s
eparated into f(%a)g(%a),k,n \"): \nfi:\n \nope:= add(bpc[gi]*N^gi, gi
=0..ORDER);\nPOL1, SUMMAND1, denFAC:=action_ope(1,convert(SUMMAND,fact
orial),ope,n,N):\nQ,R:= (numer, denom) (normal(1/simplify1(SUMMAND1,k,
-1))):\n\nP1:=1: gg, jj:= findgQR(Q,R,k,100):\nwhile jj <>0 do\n Q:=no
rmal(Q/gg):\n R:=normal(R/subs(k=k-jj,gg)):\n POL1:=POL1*mul(subs(k=k-
gj,gg),gj=0..jj-1):\n gg,jj:=findgQR(Q,R,k,100):\nod:\nPOL1:= POL1*P1;
\n\nA1:=expand(subs(k=k+1,Q)+R): B1:=expand(subs(k=k+1,Q)-R):\nl1:=deg
ree(A1,k): \nif B1=0 then l2:=-100: else l2:=degree(B1,k): fi:\nmeka1:
=coeff(A1,k,l1): mekb1:=coeff(B1,k,l2):\nif l1<=l2 \nthen\nk1:=degree(
POL1,k)-l2:\nelif l2=l1-1 \nthen\n mumu:= (-2)*mekb1/meka1:\n  if type
(mumu,integer) \n  then\n   k1:=max(mumu, degree(POL1,k)-l1+1):\n  els
e\n   k1:=degree(POL1,k)-l1+1:\n  fi:\nelif l2 < l1-1\nthen\n k1:= max
( 0, degree(POL1,k)-l1+1 ):\nfi:\n \nif k1 < 0 then return false : fi:
\n\nfu:= add(apc[gi]*k^gi, gi=0..k1);\n\ngugu:= expand(subs(k=k+1,Q)*f
u-R*subs(k=k-1,fu)-POL1):\neq:= \{seq(coeff(gugu,k,gi), gi=0..degree(g
ugu,k))\};:\nva1:=\{seq(apc[gi], gi=0..k1)\} union \{seq(bpc[gi], gi=0
..ORDER)\}:\n\n####  why ???????????\n\{seq(op(gi,resh)=gi^2+3, gi=1..
nops(resh))\}; # eqg:=subs(n=1/2,%, eq): \n \nva1:=solve1(eq,va1):\nkv
utsa:=\{va1\}: fu:= subs(va1,fu):\nope:=subs(va1,ope):\n \nif ope=0 or
 kvutsa=\{\} or fu=0 then return false \nelif nargs=6 then return true
 fi:\n\n\{seq(apc[gi]=1, gi=0..k1)\} union \{seq(bpc[gi]=1, gi=0..ORDE
R)\}:\nfu:=subs(%,fu); ope:=subs(%%,ope);\n \nope:=pashet(ope,N):\nope
1:=ope*denom(ope):\n \nSDZ:=denom(ope)*subs(k=k+1,Q)*fu/P1/denFAC :\nS
DZ1:=subs(k=k-1,SDZ)*simplify1(convert(SUMMAND,factorial),k,-1):\nSDZ1
:=simplify(SDZ1):\n\nadd(sort(coeff(ope1,N,gi))*N^gi, gi=0..degree(ope
1,N) ); \nreturn %,SDZ1:\nend:" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
22 "zeil, zeillim, zeilpap" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1023 "`ezra/zeil`:= proc();\nprintf (\"Syntax : zeil(SUMMAND,k,n,N,MAX
ORDER,param_list) ; \\nMAXORDER default is 6 ; params 5 and 6 are opti
onal \\n\\n\");\n printf(\"Requires : SUMMAND should be a product of f
actorials and/or binomial coeffs \\n\"):\n printf(\"and/or rising fact
orials, where (a)_k is denoted by rf(a,k) \\n\"):\n printf(\"and/or po
wers in k and n, and, optionally, a polynomial factor. \\n\"):\n print
f(\"The last optional parameter, is the list of other parameters,\"):
\n printf(\"if present. \\nGiving them causes considerable speedup. \\
nFor example\"):\n printf(\" zeil(binomial(n,k)*binomial(a,k)*binomial
(b,k),k,n,N,6,[a,b]) \\n\\n\"):\nprintf(\"Ensures : zeil finds a linea
r recurrence equation for SUMMAND \\n\");\nprintf(\"with polynomial co
efficients of ORDER<=MAXORDER in the parameter n, the shift operator i
n n being N \\n\"):\nprintf(\"The output is ope(N,n),R(n,k) where : op
e(N,n)SUMMAND=G(n,k+1)-G(n,k) and G(n,k)=R(n,k)*SUMMAND \\n\\n\"):\npr
intf(\"Example : zeil(binomial(n,k),k,n,N) yields to ( N-2, k/(k-n-1) \+
)  \\n\");\nend:  ezra(zeil);" }}{PARA 6 "" 1 "" {TEXT -1 51 "Syntax :
 zeil(SUMMAND,k,n,N,MAXORDER,param_list) ; " }}{PARA 6 "" 1 "" {TEXT 
-1 52 "MAXORDER default is 6 ; params 5 and 6 are optional " }}{PARA 
6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 76 "Requires : SUMM
AND should be a product of factorials and/or binomial coeffs " }}
{PARA 6 "" 1 "" {TEXT -1 60 "and/or rising factorials, where (a)_k is \+
denoted by rf(a,k) " }}{PARA 6 "" 1 "" {TEXT -1 64 "and/or powers in k
 and n, and, optionally, a polynomial factor. " }}{PARA 6 "" 1 "" 
{TEXT -1 73 "The last optional parameter, is the list of other paramet
ers,if present. " }}{PARA 6 "" 1 "" {TEXT -1 41 "Giving them causes co
nsiderable speedup. " }}{PARA 6 "" 1 "" {TEXT -1 74 "For example zeil(
binomial(n,k)*binomial(a,k)*binomial(b,k),k,n,N,6,[a,b]) " }}{PARA 6 "
" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 62 "Ensures : zeil fin
ds a linear recurrence equation for SUMMAND " }}{PARA 6 "" 1 "" {TEXT 
-1 100 "with polynomial coefficients of ORDER<=MAXORDER in the paramet
er n, the shift operator in n being N " }}{PARA 6 "" 1 "" {TEXT -1 96 
"The output is ope(N,n),R(n,k) where : ope(N,n)SUMMAND=G(n,k+1)-G(n,k)
 and G(n,k)=R(n,k)*SUMMAND " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 
6 "" 1 "" {TEXT -1 66 "Example : zeil(binomial(n,k),k,n,N) yields to (
 N-2, k/(k-n-1) )  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 597 "`ezra/zeillim`:=proc()\n print(` zeillim(SUMMAND,k
,n,N,alpha,beta) `):\nprint(`Similar to zeil(SUMMAND,k,n,N) but output
s a recurrence for `):\nprint(` the sum of SUMMAND from k=alpha to k=n
-beta .`):\nprint(`Outputs the recurrence operator, certificate and ri
ght hand side.`):\nprint(`Note carefully: The upper limit of the sum w
ill be n-beta, not beta. `):\nprint(`For example, \"zeillim(binomial(n
,k),k,n,N,0,1);\" gives output of `):\nprint(` N-2, k/(k-n-1),1 `):\np
rint(` which means that SUM(n):=2^n-1 satisfies the recurrence `):\npr
int(` (N-2)SUM(n)=1, as certified by R(n,k):=k/(k-n-1) `):\nend: %();
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D~zeillim(SUMMAND,k,n,N,alpha,bet
a)~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gnSimilar~to~zeil(SUMMAND,k,n
,N)~but~outputs~a~recurrence~for~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%O~the~sum~of~SUMMAND~from~k=alpha~to~k=n-beta~.G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\\oOutputs~the~recurrence~operator,~certificate~and~ri
ght~hand~side.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%`oNote~carefully:~
The~upper~limit~of~the~sum~will~be~n-beta,~not~beta.~G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%\\oFor~example,~\"zeillim(binomial(n,k),k,n,N,0,1
);\"~gives~output~of~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3~N-2,~k/(k
-n-1),1~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Z~which~means~that~SUM(n
):=2^n-1~satisfies~the~recurrence~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%S~(N-2)SUM(n)=1,~as~certified~by~R(n,k):=k/(k-n-1)~G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "`ezra/zeilpap`:=proc()\nprint(` ze
ilpap(SUMMAND,k,n) or zeilpap(SUMMAND,k,n,NAME,REF)`):\nprint(`Just li
ke zeil but writes a paper with the proof`):\nprint(`NAME and REF are \+
optional name and reference`):\nprint(`Warning: It assumes that the de
finite summation w.r.t. k`):\nprint(`extends over all k where it is no
n-zero, and that it is zero`):\nprint(`for other k`):\nprint(`For non-
natural summation limits, use zeillim`): \nend: %();" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%W~zeilpap(SUMMAND,k,n)~or~zeilpap(SUMMAND,k,n,NAME,
REF)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%QJust~like~zeil~but~writes~a
~paper~with~the~proofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%MNAME~and~R
EF~are~optional~name~and~referenceG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%YWarning:~It~assumes~that~the~definite~summation~w.r.t.~kG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%gnextends~over~all~k~where~it~is~non-zero,~
and~that~it~is~zeroG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,for~other~kG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%NFor~non-natural~summation~limits
,~use~zeillimG" }}}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 100 "yafe:=pr
oc(ope,N,n,SUM) global gi:\n add(coeff(ope,N,gi)*subs(n=n+gi,SUM), gi=
0..degree(ope,N));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
144 "ever_load:= ever_load, zeil, simplify1, split_k_n, action_ope, fi
ndgQR, pashet, ct_proc, `ezra/zeil`, `ezra/zeilpap` , `ezra/zeillim`, \+
ekhad6  ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*ever_loadG64%%ezraG%-e
zra/versionG%'solve1G%'celineG%,ezra/celineG%(findrecG%-ezra/findrecG%
%zeilG%*simplify1G%*split_k_nG%+action_opeG%(findgQRG%'pashetG%(ct_pro
cG%*ezra/zeilG%-ezra/zeilpapG%-ezra/zeillimG%'ekhad6G" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "ekhad6:= proc() print( ['celine', '
findrec', 'zeil', 'action_ope'] ) end; " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'ekhad6GR6\"F&F&F&-%&printG6#7&.%'celineG.%(findrecG.%%zeilG.%
+action_opeGF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 125 "proc(z) searchtext(\"ezra\", convert(z,string)); e
valb(%=0); end; select(%, [ever_load]);\nsubs(solve1=NULL, %): map(xmi
nt, %) :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"zG6\"F&F&C$-%+search
textG6$Q%ezraF&-%(convertG6$9$%'stringG-%&evalbG6#/%\"%G\"\"!F&F&F&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-%'solve1G%'celineG%(findrecG%%zeilG
%*simplify1G%*split_k_nG%+action_opeG%(findgQRG%'pashetG%(ct_procG%'ek
had6G" }}{PARA 6 "" 1 "" {TEXT -1 6 "celine" }}{PARA 6 "" 1 "" {TEXT 
-1 7 "findrec" }}{PARA 6 "" 1 "" {TEXT -1 4 "zeil" }}{PARA 6 "" 1 "" 
{TEXT -1 9 "simplify1" }}{PARA 6 "" 1 "" {TEXT -1 9 "split_k_n" }}
{PARA 6 "" 1 "" {TEXT -1 10 "action_ope" }}{PARA 6 "" 1 "" {TEXT -1 7 
"findgQR" }}{PARA 6 "" 1 "" {TEXT -1 6 "pashet" }}{PARA 6 "" 1 "" 
{TEXT -1 7 "ct_proc" }}{PARA 6 "" 1 "" {TEXT -1 6 "ekhad6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xsavelib(ever_load, `ekhad6.m`);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{MARK "14 8 0" 10 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }