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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart; gc( ); " }}
}{EXCHG {PARA 18 "" 0 "" {TEXT -1 13 "Ekhad (V5.03)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "interface(verboseproc=1); ever_load:=  NU
LL :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 22 "Routines pr\351paratoires" }}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 28 "Cr\351ation de la liste savelib" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 891 "unprotect (savelib) : ssavelib:= NULL:  \n
savelib:= proc() local filename, values, library, pathname, f; \ngloba
l savelibname, ssavelib;\n  if not assigned(savelibname) then savelibn
ame := op(1, [libname]) fi;\n  filename := args[nargs];\n  if not type
(filename, \{symbol,string\}) then \n     ERROR(`last argument (filena
me) must be a symbol/string`) fi;\n  values := args[1 .. nargs - 1];\n
  if not type([values], list(\{symbol, string\})) then \n     ERROR(`a
rguments must be symbol/string`) fi;\n  if not type([savelibname], lis
t(string)) then \n     ERROR(`savelibname assigned improperly`) fi;\ns
savelib:= ssavelib, values;\n    for library in [savelibname] do\n    \+
    pathname := cat(library, '`/`', filename);\n        f := subs('T' \+
= (values, pathname), proc() save T end);\n        if traperror(f()) <
> lasterror then RETURN() fi\n    od;\n    ERROR(`unable to save `.fil
ename.` in`, savelibname)\nend :" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 4 "init" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "system(dater)
: fd:= fopen(\"c:/windows/date.txt\", READ):\ndate_string:= readline(f
d): fclose(fd):\nmessage = \"ehkad  ;  author : < Shalosh B. Ekhad  > \+
 ;  last modif (V5) : \".\n (substring(date_string, 22..35)):\n`ekhad/
init`:= subs(% , proc( ) option `Copyright Pierre L. Douillet`; \nunpr
otect(gamma);\nprint(convert(message, symbol)); \nprint(nprintf(\"  ty
pe  'ezra()'  for help \"))  end ): %();" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%hoehkad~~;~~author~:~<~Shalosh~B.~Ekhad~~>~~;~~last~modif~(V5)~
:~Sam~18/03/2000G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<~~type~~'ezra()
'~~for~help~G" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "xreadlib" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "xreadlib:= proc(qui::uneval
) option `Copyright Pierre L. Douillet` ; local qui2, ou ; \nqui2:= ev
al(qui,1); to 10 do ou:= searchtext('`/`', qui2); if ou = 0 then break
 fi;\nqui2:= substring(qui2, ou+1..-1): assign(qui2, readlib(qui)); od
; qui2 ;\n end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)xreadlibGR6#'%$q
uiG%'unevalG6$%%qui2G%#ouG6#%=Copyright~Pierre~L.~DouilletG6\"C%>8$-%%
evalG6$9$\"\"\"?(F/F7F7\"#5%%trueGC&>8%-%+searchtextG6$.%\"/GF2@$/F=\"
\"!%&breakG>F2-%*substringG6$F2;,&F=F7F7F7!\"\"-%'assignG6$F2-%(readli
bG6#F6F2F/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 32 "solve1:=readlib(`solve/linear`):" }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 0 21 137 "op2us:=proc(ope, n, N, expr) local gu,i: \+
gu:=0:\n for i from 0 to degree(ope,N) do gu:= gu+coeff(ope,N,i)*subs(
n=n+i, expr): od:\ngu: end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 104 "op2us:=proc(ope, n, N, expr) local i:  \n add( coeff(ope,N,i)*s
ubs(n=n+i, expr), i=0..degree(ope,N))\nend:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "rf:=proc(a,b):  (a+b-1)!/(a-1)!: end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nprint:= lprint@nprintf;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'nprintG-%\"@G6$%'lprintG%(nprintfG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ever_load:= ever_load, op2us
, solve1, rf, ''nprint'';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ever_l
oadG6&%&op2usG%'solve1G%#rfG.%'nprintG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 "Paperc : differential equation sa
tisfied by an integral" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "AZpapc:=proc(INTEGRAND,y,
x) local D,SDZ1,gu,ope1:\ngu:=AZc(INTEGRAND,y,x,D): ope1:=gu[1]: SDZ1:
=gu[2]:\npaperc(INTEGRAND,y,x,D,ope1,SDZ1):\nend:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 106 "goremc:=proc(D,ORDER) local i,gu: gu:=0:\nf
or i from 0 to ORDER do gu:=gu+(bpc.i)*D^i: od:\nRETURN(gu): end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "pashetc:=proc(p,D) local gu
,p1,mekh,i:\np1:=normal(p): mekh:=denom(p1): p1:=numer(expand(p1)): gu
:=0:\nfor i from 0 to degree(p1,D) do\n gu:=gu+factor(coeff(p1,D,i))*D
^i:\nod:\nRETURN(gu,mekh):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 148 "AZc:=proc(A,y,x,D) local ORDER,gu,KAMA: KAMA:=8:\nfo
r ORDER from 0 to KAMA do gu:=duisc(A,ORDER,y,x,D):\n if gu <> 0 then \+
 RETURN(gu): fi:\nod: 0: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1863 "duisc:= proc(A,ORDER,y,x,D)\nlocal KAMA,gorem,conj, yakhas,lu1
a,LU1,P,Q,R,S,j1,g,Q1,Q2,l1,l2,mumu,\nmekb1,fu,meka1,k1,gugu,eq,ia1,va
1,va2,degg,i,shad:\n \nKAMA:=10: gorem:=goremc(D,ORDER): conj:=gorem: \+
yakhas:=0:\n\nfor i from 0 to degree(conj,D) do\nyakhas:=yakhas+normal
(coeff(conj,D,i)*gzor(A,x,i)/A):\nyakhas:=normal(yakhas):\nod:\n \nlu1
a:=yakhas: LU1:=numer(yakhas): yakhas:=1/denom(yakhas):\nyakhas:=norma
l(diff(yakhas,y)/yakhas+diff(A,y)/A):\nP:=LU1: Q:=numer(yakhas): R:=de
nom(yakhas):\nj1:=0:\nwhile j1 <= KAMA do\ng:=gcd(R,Q-j1*diff(R,y)):\n
if g <> 1 then\nQ2:=(Q-j1*diff(R,y))/g:\nR:=normal(R/g):\nQ:=normal(Q2
+j1*diff(R,y)):\nP:=P*g^j1:\nj1:=-1:\nfi:\nj1:=j1+1:\nod:\nP:=expand(P
): R:=expand(R): Q:=expand(Q): Q1:=Q+diff(R,y): Q1:=expand(Q1):\nl1:=d
egree(R,y): l2:=degree(Q1,y): meka1:=coeff(R,y,l1): mekb1:=coeff(Q1,y,
l2):\nif l1-1 <>l2 \nthen\nk1:=degree(P,y)-max(l1-1,l2):\nelse\n mumu:
= -mekb1/meka1:\n  if type(mumu,integer) and mumu > 0 \n  then\n   k1:
=max(mumu, degree(P,y)-l2):\n  else\n   k1:=degree(P,y)-l2:\n  fi:\nfi
:\nfu:=0:\nif k1 < 0 then\nRETURN(0):\nfi:\nfor ia1 from 0 to k1 do\nf
u:=fu+apc.ia1*y^ia1:\nod:\ngugu:=Q1*fu+R*diff(fu,y)-P: gugu:=expand(gu
gu):\ndegg:=degree(gugu,y):\nfor ia1 from 0 to degg do\neq[ia1+1]:=coe
ff(gugu,y,ia1)=0:\nod:\nfor ia1 from 0 to k1 do\nva1[ia1+1]:=apc.ia1:
\nod:\nfor ia1 from 0 to ORDER do\nva2[ia1+1]:=bpc.ia1:\nod:\neq:=conv
ert(eq,set):\nva1:=convert(va1,set):\nva2:=convert(va2,set):\nva1:=va1
 union va2 :\nva1:=solve1(eq,va1):\nfu:=subs(va1,fu):\ngorem:=subs(va1
,gorem):\nif fu=0 and gorem=0 then\n RETURN(0):\nfi:\nfor ia1 from 0 t
o k1 do\ngorem:=subs(apc.ia1=1,gorem):\nod:\nfor ia1 from 0 to ORDER d
o\ngorem:=subs(bpc.ia1=1,gorem):\nod:\nfu:=normal(fu):\nshad:=pashetc(
gorem,D):\nS:=lu1a*R*fu*shad[2]/P:\nS:=subs(va1,S):\nS:=normal(S):\nS:
=factor(S):\nfor ia1 from 0 to k1 do\nS:=subs(apc.ia1=1,S):\nod:\nfor \+
ia1 from 0 to ORDER do\nS:=subs(bpc.ia1=1,S):\nod:\nshad[1],S:\nend:\n
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "bdokcertc:=proc(A,y,x,
D,ope,S) local gu,i: gu:=0:\nfor i from 0 to degree(ope,D) do\ngu:=gu+
coeff(ope,D,i)*simplify(gzor(A,x,i)/A): gu:=normal(gu):\nod:\n \ngu:=g
u/simplify(diff(S*A,y)/A): normal(gu);\nend:\n \n \n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 163 "op2usc:=proc(ope,D,x,INTEGRAND) local gu
,i: gu:=coeff(ope,D,0)*INTEGRAND:\nfor i from 1 to degree(ope,D) do gu
:=gu+coeff(ope,D,i)*diff(INTEGRAND,x$i): od:\ngu: end: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 934 "paperc:=proc(INTEGRAND,y,x,D,ope1,
SDZ1):\nlprint(`A PROOF OF A DIFFERENTIAL EQUATION SATISFIED BY AN INT
EGRAL`):\nlprint(``):\nlprint(`By Shalosh B. Ekhad, Temple University,
 ekhad@math.temple.edu`):\nlprint(``):\nlprint(`I will give a short pr
oof of the following result.`):\nlprint(``):\n \nlprint(`Theorem:Let`,
 F(x,y), `be given by`):\nlprint(``):\nprint(INTEGRAND):\nlprint(``):
\nlprint(`and let`, INTEGRAL(x),`be the integral of`, F(x,y),` with`):
\nlprint(`respect to`, y,`.`):\nlprint(``):\nlprint(INTEGRAL(x),` sati
sfies the following linear differential equation`):\nlprint(``):\nprin
t(op2usc(ope1,D,x,INTEGRAL(x))):\nprint(`=0.`):\nlprint(``):\nlprint(`
PROOF: We cleverly construct`, G(x,y),`:=`):\nlprint(``):\nprint(SDZ1*
INTEGRAND):\nlprint(`with the motive that`):\nlprint(``):\nprint(op2us
c(ope1,D,x,F(x,y))):\nlprint(`=`,diff(G(x,y),y), `(check!)`):\nlprint(
``):\nlprint(`and the theorem follows upon integrating with respect to
`, y,`.QED.`):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 "Paperd : linear recurrence sati
sfied by an integral" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "AZp
apd:=proc(INTEGRAND,x,n) local D,SDZ1, gu, ope1:\ngu:=AZd(INTEGRAND,x,
n,D): ope1:=gu[1]: SDZ1:=gu[2]:\npaperd(INTEGRAND,x,n,D,ope1,SDZ1):\ne
nd:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "goremd:=proc(N,OR
DER) local i, gu: gu:=0:\nfor i from 0 to ORDER do gu:=gu+(bpc.i)*N^i:
 od:\nRETURN(gu): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "
AZd:= proc(A,y,n,N) local ORDER,gu,KAMA: KAMA:=8:\n \nfor ORDER from 0
 to KAMA do gu:=duisd(A,ORDER,y,n,N):\n if gu<>0 then  RETURN(gu): fi:
\nod: 0:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1870 "duisd:=
 proc(A,ORDER,y,n,N)\nlocal gorem, conj, yakhas,lu1a,LU1,P,Q,R,S,j1,g,
Q1,Q2,l1,l2,mumu,\nmekb1,fu,meka1,k1,gugu,eq,ia1,va1,va2,degg,shad,KAM
A,i1:\nKAMA:=10: gorem:=goremd(N,ORDER): conj:=gorem: yakhas:=0:\n \nf
or i1 from 0 to degree(conj,N) do\n yakhas:=yakhas+coeff(conj,N,i1)*si
mplify(subs(n=n+i1,A)/A):\nod:\n \nlu1a:=yakhas: LU1:=numer(yakhas): y
akhas:=1/denom(yakhas):\nyakhas:=normal(diff(yakhas,y)/yakhas+diff(A,y
)/A):\nP:=LU1: Q:=numer(yakhas): R:=denom(yakhas): j1:=0:\nwhile j1 <=
 KAMA do\n g:=gcd(R,Q-j1*diff(R,y)):\n if g <> 1 then\n  Q2:=(Q-j1*dif
f(R,y))/g: R:=normal(R/g): Q:=normal(Q2+j1*diff(R,y)):\n  P:=P*g^j1: j
1:=-1:\n fi:\n j1:=j1+1:\nod:\nP:=expand(P): R:=expand(R): Q:=expand(Q
): Q1:=Q+diff(R,y): Q1:=expand(Q1):\nl1:=degree(R,y): l2:=degree(Q1,y)
:\nmeka1:=coeff(R,y,l1): mekb1:=coeff(Q1,y,l2):\nif l1-1 <>l2 \nthen\n
 k1:=degree(P,y)-max(l1-1,l2):\nelse\n mumu:= -mekb1/meka1:\n  if type
(mumu,integer) and mumu > 0 \n  then\n   k1:=max(mumu, degree(P,y)-l2)
:\n  else\n   k1:=degree(P,y)-l2:\n  fi:\nfi:\nfu:=0:\nif k1 < 0 then
\nRETURN(0):\nfi:\nfor ia1 from 0 to k1 do\nfu:=fu+apc.ia1*y^ia1:\nod:
\ngugu:=Q1*fu+R*diff(fu,y)-P: gugu:=expand(gugu):\ndegg:=degree(gugu,y
):\nfor ia1 from 0 to degg do\n eq[ia1+1]:=coeff(gugu,y,ia1)=0:\nod:\n
for ia1 from 0 to k1 do\n va1[ia1+1]:=apc.ia1:\nod:\nfor ia1 from 0 to
 ORDER do\n va2[ia1+1]:=bpc.ia1:\nod:\neq:=convert(eq,set): va1:=conve
rt(va1,set): va2:=convert(va2,set):\nva1:=va1 union va2 : va1:=solve1(
eq,va1): fu:=subs(va1,fu):\ngorem:=subs(va1,gorem):\nif fu=0 and gorem
=0 then RETURN(0): fi:\nfor ia1 from 0 to k1 do gorem:=subs(apc.ia1=1,
gorem): od:\nfor ia1 from 0 to ORDER do gorem:=subs(bpc.ia1=1,gorem): \+
od:\nfu:=normal(fu):\n \n \nshad:=pashetc(gorem,N):\nS:=lu1a*R*fu*shad
[2]/P: S:=subs(va1,S): S:=normal(S): S:=factor(S):\nfor ia1 from 0 to \+
k1 do S:=subs(apc.ia1=1,S): od:\nfor ia1 from 0 to ORDER do S:=subs(bp
c.ia1=1,S): od:\nRETURN(shad[1],S):\nend:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "bdok
certd:=proc(A,y,n,N,ope,S) local gu,i: gu:=0:\nfor i from 0 to degree(
ope,N) do\ngu:=gu+coeff(ope,N,i)*simplify( subs(n=n+i,A)/A): gu:=norma
l(gu): od:\ngu:=gu/simplify(diff(S*A,y)/A): normal(gu); \nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 926 "paperd:=proc(INTEGRAND,x,n,N,ope1,SDZ1):\nlprint(`
`):\nlprint(`A PROOF OF A LINEAR RECURRENCE SATISFIED BY AN INTEGRAL`)
:\nlprint(``):\nlprint(`By Shalosh B. Ekhad, Temple University, ekhad@
math.temple.edu`):\nlprint(``):\nlprint(`I will give a short proof of \+
the following result.`):\nlprint(``):\n \nlprint(`Theorem:Let`, F(n,x)
, `be given by`):\nlprint(``):\nprint(INTEGRAND):\nlprint(``):\nlprint
(`and let`, INTEGRAL(n),`be the integral of`, F(n,x),` with`):\nlprint
(`respect to`, x,`.`):\nlprint(``):\nlprint(INTEGRAL(n),` satisfies th
e following linear recurrence equation`):\nlprint(``):\nprint(op2us(op
e1,N,n,INTEGRAL(n))):\nprint(`=0.`):\nlprint(``):\nlprint(`PROOF: We c
leverly construct`, G(n,x),`:=`):\nlprint(``):\nprint(SDZ1*INTEGRAND):
\nlprint(`with the motive that`):\nlprint(``):\nprint(op2us(ope1,N,n,F
(n,x))):\nlprint(`=`,diff(G(n,x),x)):\nlprint(``):\nlprint(`and the th
eorem follows upon integrating with respect to`, x,`.QED.`):\nend:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "gzor
:=proc(f,x,n) local i,gu: gu:=f:\nfor i from 1 to n do gu:=diff(gu,x):
 od:\nRETURN(gu): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "
gzor1:=proc(a,D,x) local i,gu: gu:=0:\nfor i from 0 to degree(a,D) do
\ngu:=gu+diff(coeff(a,D,i),x)*D^i+coeff(a,D,i)*D^(i+1):\nod:\nend:" }}
}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 195 "pashetd:=proc(p,N) local gu
,p1,mekh,i:\np1:=normal(p): mekh:=denom(p1): p1:= expand(numer(p1)): g
u:=0:\nfor i from 0 to degree(p1,N) do\n gu:=gu+factor(coeff(p1,N,i))*
N^i:\nod:\nRETURN(gu,mekh):\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 398 "bdokct
o:=proc(SUMMAND1,ORDER,k,n,N) local mu,gu,i,G1,ope,lu,SUMMAND:\nSUMMAN
D:=convert(SUMMAND1,factorial): mu:=ct(SUMMAND,ORDER,k,n,N):\nif mu=0 \+
then  RETURN(0): fi:\nope:=mu[1]: G1:=mu[2]*SUMMAND: gu:=0:\nfor i fro
m 0 to degree(ope,N) do\n gu:=gu+coeff(ope,N,i)*simplify( subs(n=n+i,S
UMMAND)/SUMMAND): gu:=normal(gu): od:\nlu:=simplify(subs(k=k+1,G1)/SUM
MAND)-mu[2]: lu:=normal(lu): normal(gu/lu);\nend:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 387 "bdokct:=proc(SUMMAND1,ORDER,k,n,N) local mu
,gu,i,G1,ope,lu,SUMMAND:\nSUMMAND:=convert(SUMMAND1,factorial): mu:=ct
(SUMMAND,ORDER,k,n,N):\nif mu=0 then RETURN(0):fi: \nope:=mu[1]: G1:=m
u[2]*SUMMAND:gu:=0:\nfor i from 0 to degree(ope,N) do\n gu:=gu+coeff(o
pe,N,i)*simplify1(SUMMAND,n,i): gu:=normal(gu): od:\nlu:=subs(k=k+1,mu
[2])*simplify1(SUMMAND,k,1)-mu[2]: lu:=normal(lu): normal(gu/lu);\nend
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1107 "celine:=proc(f,ii,jj)\nlocal m,j,l,i,tt,yy,zz,rr,xx,ss,zy,\n b, \+
iii, k, n, r, zzz:\n \nm:='m':j:='j':l:='l':i:='i':\niii:=ii*(jj+1)+jj
:\nxx:=seq(b[i],i=0..iii):i:='i':\nss:=numer(simplify(expand(sum(sum(b
[i*(jj+1)+j]*f(n-j,k-i)/f(n,k), \n            i=0..ii),j=0..jj)))):\nt
t:=coeffs(collect(ss,k),k):\nyy:=solve1(\{tt\},\{xx\}):\nzz:=simplify(
sum(sum(b[l*(jj+1)+m]*F(n-m,k-l),l=0..ii),m=0..jj)): \ni:='i':j:='j':l
print(` ` ):lprint(` `):lprint(`The full  recurrence is`):\nlprint(` `
):\nzz:=numer(simplify(subs(yy,zz))):\nfor i from 0 to ii do for j fro
m 0 to jj do                \nzz:=collect(zz,F(n-j,k-i)) od od:\nlprin
t(zz,`==0`);zzz:=zz:\nl:='l':j:='j':m:='m':\nr:=subs(yy,simplify(expan
d(sum(sum(sum(b[l*(jj+1)+m],l=j+1..ii)*simplify(expand(f(n-m,k- j)/f(n
,k))),j=0..ii),m=0..jj)))): \n \nrr:=simplify(factor(simplify(r))):lpr
int(` ` ):lprint(` `):\nlprint(`The telescoped form is`);lprint(` `);
\nzy:=factor(subs(yy,sum(simplify(sum(b[l*(jj+1)+m],l=0..ii))*F(n-m,k)
,m=0..jj))):\nlprint(zy,`==G(n,k)-G(n,k-1)`); \n \nlprint(` `);\nlprin
t(`where G(n,k)=R(n,k)*F(n,k) and the rational function  R(n,k) is `):
\nlprint(` `);\nrr;\nend:\n \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "findrec" }}{EXCHG 
{PARA 0 "> " 0 "findrec" {MPLTEXT 1 0 1042 "`ekhad/findrec`:= proc(f,D
EGREE,ORDER,n,N) local ope,var,eq,a,i,j,kk, n0,kv,var1,eq1,mu:\n(2+DEG
REE)*(1+ORDER)+1 - nops(f); if % > 0 then\n### WARNING: sprintf now ou
tputs strings; nprintf can be used if names are wanted\nprint(sprintf(
`Insufficient data for order = %a, degree = %a. Please, retry with %a \+
more entries. \\n`, ORDER, DEGREE, %));\nERROR():\nfi:\nope:=0: var:=
\{\}:\nfor i from 0 to ORDER do for j from 0 to DEGREE do ope:=ope+a[i
,j]*n^j*N^i: var:=var union \{a[i,j]\}: od: od:\n  \neq:=\{\}:\nfor n0
 from 1 to (1+DEGREE)*(1+ORDER)+2 do eq1:=0:\n  for i from 0 to ORDER \+
do eq1:=eq1+subs(n=n0,coeff(ope,N,i))*op(n0+i,f): od:\n  eq:= eq union
 \{eq1\}:\nod:\nvar1:=solve1(eq,var):  \n\nkv:=\{\}: for i from 1 to n
ops(var1) do mu:=op(i,var1):\n if op(1,mu)=op(2,mu) then kv:= kv union
 \{op(1,mu)\}: fi: od:\nope:=subs(var1, ope):\n\nif nops(kv)>1 then  p
rint(` either DEGREE or ORDER are too high`):\n   print(`The output is
 not the minimal possible operator`): fi:\n \nfor i from 1 to nops(kv)
 do ope:=subs(op(i,kv)= 1, ope): od:\n \ncollect(ope,N); end: # findre
c:=\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "ekhad[findrec]:= \+
'readlib'(`ekhad/findrec`) : savelib(`ekhad/findrec`, `ekhad/findrec.m
`) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 26 "ctold (quelle utilit\351 ???)" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 188 "RootOf1:=proc(f,x) local kv,kvi,i:\nkv:=
[solve(f=0,x)]: kvi:=\{\}:\nfor i from 1 to nops(kv) do\n   if type(kv
[i],integer) and kv[i]>0 then\n   kvi:=kvi union \{kv[i]\}\n  fi:\nod:
\nRETURN(kvi):\nend:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 2822 "ct
old:=proc(SUMMAND,ORDER,k,n,N)\nlocal gu,i,ope,POL1,SUMMAND1,yakhas,P,
Q,R,j,res1,kv,hakhi,g,\nA1, B1,  P1, SDZ, SDZ1,\ndegg, eq, fu, gugu, i
1, ia1, k1, kvutsa, l1, l2, meka1, mekb1, mumu, \nva1, va2,ope1,denFAC
,ope1a:\n \nif nargs<>5 then\nERROR(`Syntax: ct(SUMMAND,ORDER,summatio
n_variable,auxiliary_var, Shift_n)`):\nfi:\n \n \nope:=0:\n \nfor i fr
om 0 to ORDER do\n ope:=ope+bpc.i*N^i:\nod:\n \ngu:=hafel(1,convert(SU
MMAND,factorial),ope,n,N):\nPOL1:=gu[1]:\nSUMMAND1:=gu[2]:\ndenFAC:=gu
[3]:\nyakhas:=simplify1(SUMMAND1,k,-1):\nyakhas:=normal(1/yakhas):\n \+
\nprint(`yakhas is`,yakhas):\nP1:=1:\nQ:=numer(yakhas):\nR:=denom(yakh
as):\nres1:=resultant(Q,subs(k=k+j,R),k):\nprint(`res1 is`,res1):\nkv:
=RootOf1(res1,j):\n \n \n print(`kv is`,kv):\nwhile nops(kv) > 0 do\n \+
hakhi:=max(op(kv)):\n g:=gcd(Q,subs(k=k+hakhi,R)):\n Q:=normal(Q/g):\n
 R:=normal(R/subs(k=k-hakhi,g)):\n P1:=P1*product(subs(k=k-i1,g),i1=0.
.hakhi-1):\n res1:=resultant(Q,subs(k=k+j,R),k):\n kv:=RootOf1(res1,j)
:\nod:\n \nP:=POL1*P1:\n \nA1:=subs(k=k+1,Q)+R:\nA1:=expand(A1):\nB1:=
subs(k=k+1,Q)-R:\nB1:=expand(B1):\n### WARNING: degree(0,x) now return
s -infinity\nl1:=degree(A1,k):\nif B1=0 then\nl2:=-100:\nelse\n### WAR
NING: degree(0,x) now returns -infinity\nl2:=degree(B1,k):\nfi:\nmeka1
:=coeff(A1,k,l1):\nmekb1:=coeff(B1,k,l2):\nif l1<=l2 \nthen\n### WARNI
NG: degree(0,x) now returns -infinity\nk1:=degree(P,k)-l2:\nelif l2=l1
-1 \nthen\n mumu:= (-2)*mekb1/meka1:\n  if type(mumu,integer) \n  then
\n### WARNING: degree(0,x) now returns -infinity\n   k1:=max(mumu, deg
ree(P,k)-l1+1):\n  else\n### WARNING: degree(0,x) now returns -infinit
y\n   k1:=degree(P,k)-l1+1:\n  fi:\nelif l2 < l1-1\nthen\n### WARNING:
 degree(0,x) now returns -infinity\n k1:= max( 0, degree(P,k)-l1+1 ):
\nfi:\nfu:=0:\n \n \nprint(`k1 is`,k1):\nif k1 < 0 then\nRETURN(0):\nf
i:\nif k1 >= 0 then\nfor ia1 from 0 to k1 do\nfu:=fu+apc.ia1*k^ia1:\no
d:\ngugu:=subs(k=k+1,Q)*fu-R*subs(k=k-1,fu)-P:\ngugu:=expand(gugu):\n#
## WARNING: degree(0,x) now returns -infinity\ndegg:=degree(gugu,k):\n
 \nfor ia1 from 0 to degg do\neq[ia1+1]:=coeff(gugu,k,ia1)=0:\nod:\nfo
r ia1 from 0 to k1 do\nva1[ia1+1]:=apc.ia1:\nod:\nfor ia1 from 0 to OR
DER do\nva2[ia1+1]:=bpc.ia1\nod:\neq:=convert(eq,set):\nva1:=convert(v
a1,set):\nva2:=convert(va2,set):\nva1:=va1 union va2:\nva1:=solve1(eq,
va1):\nkvutsa:=\{va1\}:\nfu:=subs(va1,fu):\nope:=subs(va1,ope):\nfi:\n
 \n \n \nif ope=0 or kvutsa=\{\} or fu=0 then\nRETURN(0):\nfi:\ngu:=\{
\}:\nfor i1 from 0 to k1 do\ngu:=gu union \{apc.i1=1\}:\nod:\nfor i1 f
rom 0 to ORDER do\ngu:=gu union \{bpc.i1=1\}:\nod:\nfu:=subs(gu,fu):\n
ope:=subs(gu,ope):\n \nope:=pashet(ope,N):\nope1:=ope*denom(ope):\n \n
SDZ:=denom(ope)*subs(k=k+1,Q)*fu/P1/denFAC :\nSDZ1:=subs(k=k-1,SDZ)*si
mplify1(convert(SUMMAND,factorial),k,-1):\nSDZ1:=simplify(SDZ1):\n \no
pe1a:=0:\n### WARNING: degree(0,x) now returns -infinity\nfor i from 0
 to degree(ope1,N) do\nope1a:=ope1a+sort(coeff(ope1,N,i)*N^i):\nod:\n \+
\nRETURN(ope1a,SDZ1):\nend:\n \n \n \n \n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "ct (sous \+
ensemble de zeil)" }}{EXCHG {PARA 0 "> " 0 "ct" {MPLTEXT 1 0 2360 "`ek
had/ct`:=proc(SUMMAND,ORDER,k,n,N)\nlocal gu,i,ope,POL1,SUMMAND1,yakha
s,P,Q,R,res1,hakhi,g, A1, B1,  P1, SDZ, SDZ1, degg, eq, fu, gugu, i1, \+
ia1, k1, kvutsa, l1, l2, meka1, mekb1, mumu, va1, va2, ope1, denFAC, o
pe1a:\n \nif nargs<>5 then \n  ERROR(\"Syntax: ct(SUMMAND,ORDER,sum_va
r,aux_var, Shift_aux_var)\"): \nfi: \nif tovu(convert(SUMMAND,factoria
l),k,n)=0 \n then ERROR(nprintf(\"The summand can be separated into f(
%a)g(%a)\", k,n)): \nfi:\n \nope:=add(bpc.i*N^i, i=0..ORDER):\n \ngu:=
hafel(1,convert(SUMMAND,factorial),ope,n,N): POL1:=gu[1]:\nSUMMAND1:=g
u[2]: denFAC:=gu[3]: yakhas:=simplify1(SUMMAND1,k,-1): yakhas:=normal(
1/yakhas):\n \nP1:=1: Q:=numer(yakhas): R:=denom(yakhas): res1:=findgQ
R(Q,R,k,100):\n \nwhile res1[2]<>0 do\n g:=res1[1]: hakhi:=res1[2]: Q:
=normal(Q/g): R:=normal(R/subs(k=k-hakhi,g)): i1:='i1':\n P1:=P1*produ
ct(subs(k=k-i1,g),i1=0..hakhi-1): res1:=findgQR(Q,R,k,100):\nod:\n \nP
:=POL1*P1: \nA1:=subs(k=k+1,Q)+R: A1:=expand(A1): B1:=subs(k=k+1,Q)-R:
 B1:=expand(B1):\nl1:=degree(A1,k):  if B1=0 then l2:=-100: else l2:=d
egree(B1,k): fi:\nmeka1:=coeff(A1,k,l1): mekb1:=coeff(B1,k,l2):\nif l1
<=l2 \n then k1:=degree(P,k)-l2:\nelif l2=l1-1 \n  then mumu:= (-2)*me
kb1/meka1:\n  if type(mumu,integer) \n   then k1:=max(mumu, degree(P,k
)-l1+1): \n   else k1:=degree(P,k)-l1+1: \n  fi:\nelif l2 < l1-1 \n th
en k1:= max( 0, degree(P,k)-l1+1 ):\nfi:\n\nfu:=0: \nif k1 < 0 then RE
TURN(0): fi:\nif k1 >= 0 then\n  for ia1 from 0 to k1 do fu:=fu+apc.ia
1*k^ia1: od:\n  gugu:=subs(k=k+1,Q)*fu-R*subs(k=k-1,fu)-P: gugu:=expan
d(gugu): degg:=degree(gugu,k):\n  for ia1 from 0 to degg do eq[ia1+1]:
=coeff(gugu,k,ia1)=0: od:\n  for ia1 from 0 to k1 do va1[ia1+1]:=apc.i
a1: od:\n  for ia1 from 0 to ORDER do va2[ia1+1]:=bpc.ia1 od:\n  eq:=c
onvert(eq,set): va1:=convert(va1,set): va2:=convert(va2,set): va1:=va1
 union va2: \n  va1:=solve1(eq,va1): kvutsa:=\{va1\}: fu:=subs(va1,fu)
: ope:=subs(va1,ope):\nfi:\n \nif ope=0 or kvutsa=\{\} or fu=0 then RE
TURN(0): fi:\ngu:=\{\}:\nfor i1 from 0 to k1 do gu:=gu union \{apc.i1=
1\}: od:\nfor i1 from 0 to ORDER do gu:=gu union \{bpc.i1=1\}: od:\nfu
:=subs(gu,fu): ope:=subs(gu,ope): ope:=pashet(ope,N): ope1:=ope*denom(
ope):\nSDZ:=denom(ope)*subs(k=k+1,Q)*fu/P1/denFAC :\nSDZ1:=subs(k=k-1,
SDZ)*simplify1(convert(SUMMAND,factorial),k,-1): SDZ1:=simplify(SDZ1):
\n \nope1a:= 0: for i from 0 to degree(ope1,N) do ope1a:=ope1a+sort(co
eff(ope1,N,i))*N^i: od: \nRETURN(ope1a,SDZ1):\nend:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "findgQR:=proc(Q,R
,k,L) local j,g:\nfor j from 1 to L do g:=gcd(Q, subs(k=k+j,R)): if de
gree(g,k)>0 then RETURN(g,j): fi: od:\n1,0: end:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 116 "tovu:=proc(SU,k,n) local gu: \ngu:=simplify1
(simplify1(SU,n,1),k,1): if gu=1 then RETURN(0): else RETURN(1): fi:\n
end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "pashet:=proc(p,N) \+
local i,gu1,gu,p1: \np1:=normal(p): gu1:=denom(p1): p1:=expand(numer(p
1)):\ngu:=0: for i from 0 to degree(p,N) do gu:=gu+factor(coeff(p1,N,i
))*N^i: od:\nRETURN(gu/gu1): end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 509 "simpli
fy1:=proc(bitu,k,a) local gu,gu1,i,khez,sp: sp:=1: gu:=bitu:\nif type(
gu,`*`) then\n for i from 1 to nops(gu) do gu1:=op(i,gu):\n  if type(g
u1,`^`) and type(op(2,gu1), integer) then\n     khez:=op(2,gu1): gu1:=
op(1,gu1): sp:=sp*simplify(subs(k=k+a,gu1)/gu1)^khez:\n  else \n     s
p:=sp*simplify(subs(k=k+a,gu1)/gu1):\n  fi:\n od:\nelif type(gu,`^`) a
nd type(op(2,gu), integer) then\n khez:=op(2,gu): gu1:=op(1,gu): sp:=s
p*simplify(subs(k=k+a,gu1)/gu1)^khez:  \nelse \n sp:=simplify(subs(k=k
+a,gu)/gu):\nfi: sp: end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
323 "hafel:=proc(POL,SUMMAND,ope,n,N) local i,FAC,degg,bi,rat,POL1,SUM
MAND1,zub:\ndegg:=degree(ope,N): FAC:=0:\nfor i from 0 to degg do bi:=
coeff(ope,N,i): rat:=simplify1(SUMMAND,n,i):\n FAC:=FAC+bi*subs(n=n+i,
POL)*rat: FAC:=normal(FAC):\nod:\nPOL1:=numer(FAC): zub:=denom(FAC): S
UMMAND1:=SUMMAND/zub: RETURN(POL1,SUMMAND1,zub):\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "ekhad[ct]:= 'readlib'(`ekhad/ct`) : \nsavelib(`ekhad
/ct`, tovu, hafel, findgQR, pashet, simplify1, `ekhad/ct.m`) : " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 4 "zeil" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 716 "`ekha
d/zeil`:= proc(SUMMAND,k,n,N, max_order, resh) local ORDER, MAXORDER, \+
gu, gu1, RESH:\nif nargs<4 or nargs>6 then ERROR(nprintf(\"syntax :  z
eil(SUMMAND,k,n,N) or zeil(SUMMAND,k,n,N,MAXORDER) or zeil(SUMMAND,k,n
,N,MAXORDER,param_list)\")): fi: \nif nargs>=5 then MAXORDER:=max_orde
r else MAXORDER:=6 fi;\nif nargs>=6 then RESH:=resh else RESH:=[] fi; \+
\nif simplify1(SUMMAND,n,1)=1 then\n ERROR(nprintf(\"Summand does not \+
depend on %a, hence the sum is identically constant\", n)); \nfi:\nfor
 ORDER from 0 to MAXORDER do\n gu1:=cttry(SUMMAND, ORDER, k, n, RESH, \+
N):\n if gu1=1 then gu:=ct(SUMMAND,ORDER,k,n,N): \n   if gu<>0 then RE
TURN(gu): fi: \nfi:\nod:\nERROR(nprintf(\"No recurrence of order <= %a
 was found\", MAXORDER)); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1994 "cttry:=proc(SU
MMAND,ORDER,k,n,resh,N)\nlocal gu,i,ope,POL1,SUMMAND1,yakhas,P,Q,R,res
1,hakhi,g,A1, B1,  P1, \ndegg, eq, fu, gugu, i1, ia1, k1, kvutsa, l1, \+
l2, meka1, mekb1, mumu, va1, va2,eqg:\n \nif nargs<>6 then\n ERROR(npr
intf(\"Syntax : cttry(term, ORDER, sum_variable, aux_var, para_list, S
hift)\")): fi:\n \nif tovu(convert(SUMMAND,factorial),k,n)=0 then\n ER
ROR(nprintf(\"The summand can be separated into f(%a)g(%a)\",k,n)): fi
:\n \nope:=0: for i from 0 to ORDER do  ope:=ope+bpc.i*N^i: od:\n \ngu
:=hafel(1,convert(SUMMAND,factorial),ope,n,N): POL1:=gu[1]:\nSUMMAND1:
=gu[2]: yakhas:=simplify1(SUMMAND1,k,-1): yakhas:=normal(1/yakhas):\n \+
\nP1:=1: Q:=numer(yakhas): R:=denom(yakhas): res1:=findgQR(Q,R,k,100):
\n \nwhile res1[2]<>0 do g:=res1[1]: hakhi:=res1[2]: Q:=normal(Q/g): R
:=normal(R/subs(k=k-hakhi,g)): i1:='i1':\n P1:=P1*product(subs(k=k-i1,
g),i1=0..hakhi-1): res1:=findgQR(Q,R,k,100): od:\n \nP:=POL1*P1: A1:=s
ubs(k=k+1,Q)+R: A1:=expand(A1):\nB1:=subs(k=k+1,Q)-R: B1:=expand(B1): \+
l1:=degree(A1,k):\nif B1=0 then l2:=-100: else l2:=degree(B1,k): fi:\n
meka1:=coeff(A1,k,l1): mekb1:=coeff(B1,k,l2):\nif l1<=l2 then k1:=degr
ee(P,k)-l2:\nelif l2=l1-1 then mumu:= (-2)*mekb1/meka1:\n  if type(mum
u,integer) then k1:=max(mumu, degree(P,k)-l1+1): else k1:=degree(P,k)-
l1+1: fi:\nelif l2 < l1-1 then k1:= max( 0, degree(P,k)-l1+1 ): \nfi:
\nfu:=0:\nif k1 < 0 then RETURN(0): fi:\nif k1 >= 0 then for ia1 from \+
0 to k1 do fu:=fu+apc.ia1*k^ia1: od:\ngugu:=subs(k=k+1,Q)*fu-R*subs(k=
k-1,fu)-P: gugu:=expand(gugu): degg:=degree(gugu,k):\nfor ia1 from 0 t
o degg do eq[ia1+1]:=coeff(gugu,k,ia1)=0: od:\nfor ia1 from 0 to k1 do
 va1[ia1+1]:=apc.ia1: od:\nfor ia1 from 0 to ORDER do va2[ia1+1]:=bpc.
ia1 od:\neq:=convert(eq,set): va1:=convert(va1,set): va2:=convert(va2,
set): va1:=va1 union va2: eqg:=subs(n=1/2,eq):\n \nfor i from 1 to nop
s(resh) do eqg:=subs(op(i,resh)=i^2+3,eqg): od:\n \nva1:=solve1(eq,va1
): kvutsa:=\{va1\}: fu:=subs(va1,fu): ope:=subs(va1,ope): \nfi:\n \nif
 ope=0 or kvutsa=\{\} or fu=0 then RETURN(0): else  RETURN(1): fi: \ne
nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 6 "papers" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
615 "`ekhad/zeilpap`:=proc(SUMMAND,k,n) local  SDZ1, gu, ope1, N, G:\n
gu:= zeil(SUMMAND,k,n,N): ope1:=gu[1]: SDZ1:=gu[2]:\nnprint(\"\\n THEO
REM : Let F be given by \",  n, k, SUMMAND):\nprint(F(n,k) = SUMMAND):
\nnprint(\"\\n Then SUM(%a) = Sum(F(%a, %a), %a) satisfies the linear \+
recurrence equation :\", n, n, k, k):\nprint( op2us(ope1,n,N,SUM(n))=0
);\nnprint(\"\\n PROOF : We cleverly found the operator %a and constru
ct\", ope1):\nprint( G(n,k) = SDZ1*SUMMAND):\nnprint(\"\\n with the mo
tive that \"):\nprint(op2us(ope1,n,N,F(n,k)) = G(n,k+1)- G(n,k)):\nnpr
int(\"\\n and the theorem follows upon summing with respect to %a.\\n \+
QED.\", k):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "ekhad[
zeilpap]:= 'readlib'(`ekhad/zeilpap`) : \nsavelib(`ekhad/zeilpap`, `ek
had/zeilpap.m`) :" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Zeillim" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 396 "`ekhad/zeillim`:=proc(SUMMA
ND,k,n,N,alpha,beta) local ope, CERT, lu, i, gu, lu1, lu2:\ngu:= Zeill
im(SUMMAND,k,n,N,alpha+1,beta+1): \nope:=gu[1]: CERT:=gu[2]: lu:=gu[3]
: \nlu1:=(normal@simplify@subs)(k=alpha,SUMMAND)+subs(k=n-beta,SUMMAND
): \nlu2:= normal( add(coeff(ope,N,i)*simplify(subs(n=n+i,lu1)), i=0..
degree(ope,N) )):\nope, CERT, normal(expand(normal(simplify(expand(nor
mal(lu+lu2)))))):\nend: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
397 "`ekhad/Zeillim`:=proc(SUMMAND,k, n, N, alpha, beta) local ope, CE
RT, lu, ii, i, gu:\ngu:=zeil(SUMMAND,k,n,N): ope:=gu[1]: CERT:=gu[2]:
\nlu:=simplify(subs(k=n-beta+1,CERT)*subs(k=n-beta+1,SUMMAND))\n   -si
mplify(subs(k=alpha,CERT)*subs(k=alpha,SUMMAND)):\n \nlu:=lu+ add(\n  \+
add(coeff(ope,N,i)*simplify(subs(\{n=n+i, k=n-beta+ii\}, SUMMAND)), ii
=1..i),\n  i=0..degree(ope,N) ):\n\ngu, normal(expand(lu)):\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "ekhad[zeillim]:= 'readlib'(
`ekhad/zeillim`): savelib(`ekhad/zeillim`, `ekhad/zeillim.m`) :\nekhad
[Zeillim]:= 'readlib'(`ekhad/Zeillim`): savelib(`ekhad/Zeillim`, `ekha
d/Zeillim.m`) :" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "ekhad[
zeil]:= 'readlib'(`ekhad/zeil`) : \nsavelib(`ekhad/zeil`, cttry, tovu,
 hafel, findgQR, pashet, simplify1, `ekhad/zeil.m`) :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 5 "ezra " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8202 "`ekhad/ezra`:=proc()\nif args=NUL
L then\nlprint(``): print(`This is EKHAD, One of the Maple packages ac
companying the book `):\nprint(` \"A=B\" `):\nprint(` (published by A.
K. Peters, Welesley, 1996) by Marko Petkovsek, Herb Wilf, and Doron Ze
ilberger.`):\nlprint(``):\n\n\nprint(`EKHAD`): lprint(``):\nprint(`A M
aple package for proving Hypergemetric (Binomial Coeff.) and other kin
ds of identities`):\nprint(`This version (Feb, 25, 1999) is much faste
r than the previous version,`);\nprint(`thanks to a SLIGHT (yet POWERF
UL) modification suggested by FREDERIC CHYZAK `):\n\nlprint(``): print
(`The penultimate version, Feb. 1997, corrected a subtle bug discovere
d by Helmut Prodinger`):\nlprint(``): print(`Previous versions benefit
ed from comments by Paula Cohen, Lyle Ramshaw, and Bob Sulanke.`):\n\n
print(`The most current version is available on WWW at: http://www.mat
h.temple.edu/~zeilberg .`):\nprint(`Information about the book, and ho
w to order it, can 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 a
cknowledged in future versions.`):\nlprint(``):\nprint(`For general he
lp, and a list of the available functions, type \"ezra();\". For speci
fic help type \"ezra(procedure_name)\" `):\nlprint(``): \n\n\n\nprint(
`For help with a specific procedure, type \"ezra(procedure_name);\"`):
\nprint(`Contains procedures: `): \nprint(`'findrec', 'ct', 'zeil', 'z
eilpap', 'zeillim', 'AZd', 'AZc', 'AZpapd', 'AZpapc', 'celine'`):\nRET
URN():\nfi:\n\nif nargs > 1 then ERROR() fi; \n \nif args[1]=`celine` \+
then\nprint(`celine(FUNCTION,ORDER_n,ORDER_k) applies Sister Celine's \+
technique`):\nprint(`It inputs a function (n,k)->FUNCTION, and the gue
ssed orders in`):\nprint(`n and k, ORDER_n,ORDER_k respectively, and o
utputs a partial`):\nprint(`k-free recurrence`):\nprint(`it  also find
s the telescoped form of the recurrence.`):   \nprint(` In input, do n
ot raise products of factorials to powers;  `):  \nprint(`instead rais
e each factorial to the power. `):\nprint(`For example:celine((n,k)->b
inomial(n,k),1,1);`):\nprint(`celine ((n,k)->n!*(2*k)!*(-2)^(n-k)/(k!^
3*(n-k)!),2,2);`):\nfi:\n \n \nif args[1]=`findrec` then\nprint(`findr
ec(f,DEG,ORDER,n,N) finds empirically an ordi. linear recurrence`):\np
rint(` with polynomial coeffs. The input is a sequence f given as a li
st`):\nprint(`STARTING at f[1],i.e. f[0] is not considered`):\nprint(`
 where DEG:=the maximal degree of the coefficients`):\nprint(`and ORDE
R:=the order of the recurrence.  The output is the operator`):\nprint(
` in n and N, where N is the forward unit shift: Nf(n):=f(n+1).`):\npr
int(`For example findrec([1,1,2,3,5,8,13,21,34],0,2,n,N) should yield`
):\nprint(`N^2-N-1 , and findrec([1,2,5,14,42,132,429],1,1,m,M) should
 yield`):\nprint(`(m+1)*M-(4*m-2). If there is not enough data, you wi
ll get an`):\nprint(`an error message. If there is no operator, you wo
uld get 0`):\nfi:\n \nif args[1]=`AZpapc` then\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`):\nfi:\n \nif args[1]=`AZpapd` then\npr
int(`AZpapd(INTEGRAND,x,n) inputs a hypergeometric integrand`):\nprint
(`in the continuous variable 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)),`):\nprint(`and outputs a paper with a proof of the
 linear recurrence equation`):\nprint(`that the definite integral w.r.
t. to y (which is a function of n)`):\nprint(`assuming that the integr
and (or rather it times the certificate`):\nprint(`vanishes at the end
points, or it is a  contour 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(`Ger
t Almkvist and Doron Zeilberger, \"The method of differentiating`):\np
rint(`under the integral sign\", J. Symbolic Computation 10(1990), 571
-591`):\nprint(`It could be used to establish the recurrences of the `
):\nprint(`classical orthogonal polynomials, when they are defined in \+
terms`):\nprint(`or their generating funtion`):\nprint(`For example AZ
papd(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,n) gives the`):\nprint(`the recurre
nce, and proof, satisfied by the Legendre polynomials`):\n \nfi:\n \n \+
\nif args[1]=`AZd` then\nprint(`AZd(A,x,n,N) finds a recurrence of ord
er 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`):\nprint(`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 meth
od of differentiating`):\nprint(`under the integral sign\", J. Symboli
c Computation 10(1990), 571-591`):\nprint(`A should not be a product o
f a function of x and a function of n.`):\nprint(``):\nprint(`AZd(A,x,
n,N) returns the expression seq. ope(n,N),cert(x,n)`):\nprint(`satisfy
ing ope(n,N)A(x,n)=diff(cert(x,n)*A(x,n),x).`):\nprint(`If no recurren
ce is found, it returns 0.`):\nprint(``):\nprint(`A verbose version is
 AZpapd(A,x,n), type ezra(AZpapd) for details.`):\nprint(``):\nprint(`
For example AZd(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,n,N) gives`):\nprint(`th
e diff.eq., and proof certificate, for the Legendre polct:ynomials.`):
\n \nfi:\n \n \nif args[1]=`AZc` then\nprint(`AZc(A,y,x,D) tries to fi
nds a linear diff.eq. of order <=8,`):\nprint(` phrased in terms of x,
 and diff.w.r.t x, D`):\nprint(`for the integrale of A(x,y) with respe
ct to x, whenever A(x,y) is`):\nprint(`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 algorithn of`):\nprint(`Gert Almkvist and Dor
on Zeilberger, \"The method of differentiating`):\nprint(`under the in
tegral sign\", J. Symbolic Computation 10(1990), 571-591`):\nprint(`AZ
c(A,y,x,D) returns the expression seq. ope(x,D),cert(x,n)`):\nprint(`s
atisfying ope(x,D)A(x,y)=diff(cert(x,y)*A(x,y),y).`):\nprint(`If no li
near diff. eq. of order<=8 is found, it returns 0`):\nprint(`see AZpap
c for a verbose vsersion`):\nprint(`For example AZc(1/sqrt(1-2*x*t+t^2
)/t^(n+1),t,x,D) gives the`):\nprint(`the diff.eq., and proof certific
ate for the Legendre polynomials.`):\n \nfi:\n \n \n if args[1]=`ct` t
hen\n \n print(` ct(SUMMAND,ORDER,k,n,N)`): \nprint(`This is a Maple i
nplementation of the algorithm described in Ch. 6`):\nprint(`of the bo
ok A=B, first proposed in : D. Zeilberger, \"The method of`):\nprint(`
of creative telescoping\", J.Symbolic Computation 11(1991) 195-204`):
\nprint(``):\nprint(`finds a recurrence for SUMMAND in the parameters \+
k and n,   `):\nprint(` of order ORDER, with N is the chosen symbol fo
r the shift in n.`):\n print(``):\n print(`SUMMAND should be a product
 of factorials and/or binomial coeffs`):\n print(`and/or rising factor
ials, where (a)_k is denoted by rf(a,k)`):\n print(`and/or powers of k
 and n, and, optionally, a polynomial factor`):\nprint(`The output con
sists of an operator ope(N,n) and a certificate R(n,k)`):\nprint(`with
 the properties that if we define G(n,k):=R(n,k)*SUMMAND then`):\nprin
t(`ope(N,n)SUMMAND(n,k)=G(n,k+1)-G(n,k)`):\nprint(`which is a routinel
y verifiable identity.`):\n print(`For example \"ct(binomial(n,k),1,k,
n,N);\" would yield the output`):\nprint(`  N-2, k/(k-n-1) `):\n fi:\n
 \nif member(args[1], \{zeil, zeilpap, zeillim\}) then `ekhad/ezra/`.(
args[1]) () fi: \n%; \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1537 "`ekhad/ezra/zeil`:=proc() ; \n nprint(\"Syntax : zeil(SUMMAND, k
, n, N, [MAXORDER], [param_list]) \"):\n nprint(\"Result : ope(N,n), R
(n,k)\"):\n nprint(\"Example: zeil(binomial(n,k),k,n,N) ===> N-2, k/(k
-n-1) \\n\");\n\nnprint(\"This is a Maple inplementation of the algori
thm ZEIL described in Ch. 6\"):\nnprint(\" of the book A=B, first prop
osed in : D. Zeilberger, \"\"The method of\"):\nnprint(\" of creative \+
telescoping\"\", J.Symbolic Computation 11(1991) 195-204 \\n\"):\n\nnp
rint(\"Warning: It assumes that the definite summation w.r.t. k extend
s over \"):\nnprint(\" all k where it is non-zero, and that it is zero
 for other k\"):\nnprint(\" For non-natural summation limits, use zeil
lim \\n\"):\n\nnprint(\"ZEIL finds a linear recurrence equation for SU
MMAND, \"):\nnprint(\" with polynomial coefficients of ORDER<=MAXORDER
, \"):\nnprint(\" where the default of MAXORDER is 6 in the parameter \+
n,\"):\nnprint(\" the shift operator in n being N.\"):\nnprint(\" The \+
recurrence is given as ope(N,n)SUMMAND=G(n,k+1)-G(n,k)\"):\nnprint(\" \+
where G(n,k):=R(n,k)*SUMMAND, and R(n,k) is the 2nd item of output. \\
n\"):\n\n nprint(\"SUMMAND should be a product of factorials and/or bi
nomial coeffs\"):\n nprint(\" and/or rising factorials, where (a)_k is
 denoted by rf(a,k)\"):\n nprint(\" and/or powers in k and n, and, opt
ionally, a polynomial factor. \\n\"):\n\n nprint(\"The last optional p
arameter, is the list of other parameters, if present. \"):\n nprint(
\" Giving them causes considerable speedup. For example\"):\n nprint(
\" zeil(binomial(n,k)*binomial(a,k)*binomial(b,k), k, n, N, 6, [a,b]) \+
\\n\"):\n\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "`ekhad/
ezra/zeilpap`:=proc() ;\nnprint(\"Syntax : zeilpap(SUMMAND,k,n)\"):\nn
print(\"\\n Just like zeil but writes a paper with the proof\"):\nend:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 587 "`ekhad/ezra/zeillim`:=
proc() ;\nnprint(\"Syntax : zeillim(SUMMAND,k,n,N,alpha,beta) \"):\nnp
rint(\"Result : recurrence operator, certificate, right hand side \"):
\nnprint(\"Example: zeillim(binomial(n,k),k,n,N,0,1) ===> N-2, k/(k-n-
1), 1 \"):\n\nnprint(\"\\nSimilar to zeil(SUMMAND,k,n,N) but outputs a
 recurrence for \"):\nnprint(\" the sum of SUMMAND from k=alpha to k=n
-beta.\"):\nnprint(\" To say it again : the upper limit of the sum wil
l be n-beta, not beta. \"):\nnprint(\"\\nIn our example, SUM(n):=2^n-1
 satisfies the recurrence \"):\nnprint(\" (N-2)SUM(n)=1, as certified \+
by R(n,k):= k/(k-n-1) \"):\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "ekhad[
ezra]:= 'readlib'(`ekhad/ezra`) : \nsavelib(`ekhad/ezra`, `ekhad/ezra/
zeil`, `ekhad/ezra/zeilpap`, `ekhad/ezra/zeillim`, `ekhad/ezra.m`) : \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "savelib('ekhad', `ekhad/init`, ever_load, `ekh
ad.m`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "subs(readlib= _
readlib, eval(ekhad)): sort(op(%), proc(a,b) lexorder(lhs(a) , lhs(b))
 end): 'elements' = vector(%, column);\n'ever_loaded' = vector(sort([e
ver_load]), column);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)elementsGL)
/%(ZeillimG-%)_readlibG6#%.ekhad/ZeillimG/%#ctG-F)6#%)ekhad/ctG/%%ezra
G-F)6#%+ekhad/ezraG/%(findrecG-F)6#%.ekhad/findrecG/%%zeilG-F)6#%+ekha
d/zeilG/%(zeillimG-F)6#%.ekhad/zeillimG/%(zeilpapG-F)6#%.ekhad/zeilpap
G%'columnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,ever_loadedGL&%'nprin
tG%&op2usG%#rfG%'solve1G%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Contr\364le" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "select(type, \{ssavelib\},
 procedure): \nliste:= sort(subs(`ekhad/ezra`=NULL, convert(%, list) )
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&listeG75%&cttryG%.ekhad/Zeill
imG%)ekhad/ctG%0ekhad/ezra/zeilG%3ekhad/ezra/zeillimG%3ekhad/ezra/zeil
papG%.ekhad/findrecG%+ekhad/initG%+ekhad/zeilG%.ekhad/zeillimG%.ekhad/
zeilpapG%(findgQRG%&hafelG%&op2usG%'pashetG%#rfG%*simplify1G%'solve1G%
%tovuG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "readlib(mmint): \+
lasterror:= 'lasterror': _eps:= '_eps' : jime := 'jime' : \nfor _kkkk \+
in liste do lprint(_kkkk); mmint(_kkkk); od;" }}{PARA 6 "" 1 "" {TEXT 
-1 5 "cttry" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QD--------------~cttry~
--------------6\"" }}{PARA 6 "" 1 "" {TEXT -1 61 "Names used as global
 names, but not declared:      : bpc, apc" }}{PARA 6 "" 1 "" {TEXT -1 
13 "ekhad/Zeillim" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QL--------------~
ekhad/Zeillim~--------------6\"" }}{PARA 6 "" 1 "" {TEXT -1 58 "Local \+
variables used before assigned :             : i, ii" }}{PARA 6 "" 1 "
" {TEXT -1 8 "ekhad/ct" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QG----------
----~ekhad/ct~--------------6\"" }}{PARA 6 "" 1 "" {TEXT -1 61 "Names \+
used as global names, but not declared:      : bpc, apc" }}{PARA 6 "" 
1 "" {TEXT -1 54 "Local variables used before assigned :             :
 i" }}{PARA 6 "" 1 "" {TEXT -1 15 "ekhad/ezra/zeil" }}{PARA 6 "" 1 "" 
{TEXT -1 18 "ekhad/ezra/zeillim" }}{PARA 6 "" 1 "" {TEXT -1 18 "ekhad/
ezra/zeilpap" }}{PARA 6 "" 1 "" {TEXT -1 13 "ekhad/findrec" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#QL--------------~ekhad/findrec~-------------
-6\"" }}{PARA 6 "" 1 "" {TEXT -1 142 "Names used as global names, but \+
not declared:      : `Insufficient data for order = %a, degree = %a. P
lease, retry with %a more entries. \\n`, " }}{PARA 6 "" 1 "" {TEXT -1 
49 "       : ` either DEGREE or ORDER are too high`, " }}{PARA 6 "" 1 
"" {TEXT -1 58 "       : `The output is not the minimal possible opera
tor`" }}{PARA 6 "" 1 "" {TEXT -1 55 "These local variables were never \+
used:             : kk" }}{PARA 6 "" 1 "" {TEXT -1 54 "Local variables
 used before assigned :             : a" }}{PARA 6 "" 1 "" {TEXT -1 
10 "ekhad/init" }}{PARA 6 "" 1 "" {TEXT -1 10 "ekhad/zeil" }}{PARA 6 "
" 1 "" {TEXT -1 13 "ekhad/zeillim" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q
L--------------~ekhad/zeillim~--------------6\"" }}{PARA 6 "" 1 "" 
{TEXT -1 54 "Local variables used before assigned :             : i" }
}{PARA 6 "" 1 "" {TEXT -1 13 "ekhad/zeilpap" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#QL--------------~ekhad/zeilpap~--------------6\"" }}
{PARA 6 "" 1 "" {TEXT -1 54 "Local variables used before assigned :   \+
          : N" }}{PARA 6 "" 1 "" {TEXT -1 7 "findgQR" }}{PARA 6 "" 1 "
" {TEXT -1 5 "hafel" }}{PARA 6 "" 1 "" {TEXT -1 5 "op2us" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#QD--------------~op2us~--------------6\"" }}
{PARA 6 "" 1 "" {TEXT -1 54 "Local variables used before assigned :   \+
          : i" }}{PARA 6 "" 1 "" {TEXT -1 6 "pashet" }}{PARA 6 "" 1 "
" {TEXT -1 2 "rf" }}{PARA 6 "" 1 "" {TEXT -1 9 "simplify1" }}{PARA 6 "
" 1 "" {TEXT -1 6 "solve1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QE-------
-------~solve1~--------------6\"" }}{PARA 6 "" 1 "" {TEXT -1 121 "Name
s used as global names, but not declared:      : `solve/linear/field`,
 `solve/linear/algfun`, `solve/linear/radfun`, " }}{PARA 6 "" 1 "" 
{TEXT -1 32 "       : `solve/linear/radnum`, " }}{PARA 6 "" 1 "" 
{TEXT -1 74 "       : `1st argument must be a list or set of polynomia
ls (equations)`, " }}{PARA 6 "" 1 "" {TEXT -1 54 "       : `solve/line
ar/float`, `solve/linear/algnum`, " }}{PARA 6 "" 1 "" {TEXT -1 74 "   \+
    : `2nd argument must be a list or set of names (unknowns)`, exact,
 " }}{PARA 6 "" 1 "" {TEXT -1 57 "       : `solve/linear/complex`, `so
lve/linear/integer`, " }}{PARA 6 "" 1 "" {TEXT -1 70 "       : `solve/
linear/polynom`, `solve/linear/polyalg`, `# equations`" }}{PARA 6 "" 
1 "" {TEXT -1 4 "tovu" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "39 2 46" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }