{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 0 0 255 1 0 2 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 1 14 0 0 128 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{PSTYLE "No rmal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 3 " -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 128 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 36 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 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 3 0 3 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "restart; ekhad6(); w ith(pldx): kernelopts(ASSERT=true): " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7&%'celineG%(findrecG%%zeilG%+action_opeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\ppldx~v6-13~~;~~author~:~~~;~~la st~modif~(V6)~:~Lun~27/05/2002G" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 18 "Ekhad_test (V6.63)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 35 "charg ement par la commande ekhad6()" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Contr\364le des procedures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "liste_ekhad := [action_ope, celine, ct_proc, ekhad, ezra, `ezra /celine`, `ezra/zeil`, findgQR, pashet, simplify1, solve1, split_k_n, \+ try_celine, yafe, zeil];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,liste_e khadG71%+action_opeG%'celineG%(ct_procG%&ekhadG%%ezraG%,ezra/celineG%* ezra/zeilG%(findgQRG%'pashetG%*simplify1G%'solve1G%*split_k_nG%+try_ce lineG%%yafeG%%zeilG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "zen _col:= proc(uu, kk); nops(uu): \n(`kernel/transpose`@matrix)(kk, iquo( %,kk)+1, [op(uu), '``'$modp(-%,iquo(%,kk)+1)]); \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "'ekhad'= zen_col(liste_ekhad,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&ekhadG-%'matrixG6#7(7%%+action_op eG%*ezra/zeilG%+try_celineG7%%'celineG%(findgQRG%%yafeG7%%(ct_procG%'p ashetG%%zeilG7%F$%*simplify1G%!G7%%%ezraG%'solve1GF77%%,ezra/celineG%* split_k_nGF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "map(xmint, \+ liste_ekhad):" }}{PARA 6 "" 1 "" {TEXT -1 10 "action_ope" }}{PARA 6 " " 1 "" {TEXT -1 6 "celine" }}{PARA 6 "" 1 "" {TEXT -1 7 "ct_proc" }} {PARA 6 "" 1 "" {TEXT -1 5 "ekhad" }}{PARA 6 "" 1 "" {TEXT -1 4 "ezra " }}{PARA 6 "" 1 "" {TEXT -1 13 "`ezra/celine`" }}{PARA 6 "" 1 "" {TEXT -1 11 "`ezra/zeil`" }}{PARA 6 "" 1 "" {TEXT -1 7 "findgQR" }} {PARA 6 "" 1 "" {TEXT -1 6 "pashet" }}{PARA 6 "" 1 "" {TEXT -1 9 "simp lify1" }}{PARA 6 "" 1 "" {TEXT -1 6 "solve1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QE--------------~solve1~--------------6\"" }}{PARA 6 " " 1 "" {TEXT -1 114 "Names used as global names, but not declared: \+ : `solve/linear/algnum`, `# equations`, `solve/linear/radnum`, " }} {PARA 6 "" 1 "" {TEXT -1 77 " : `solve/linear/algfun`, `solve/li near/radfun`, `solve/linear/field`, " }}{PARA 6 "" 1 "" {TEXT -1 57 " \+ : `solve/linear/integer`, `solve/linear/complex`, " }}{PARA 6 " " 1 "" {TEXT -1 73 " : `solve/linear/polynom`, `solve/linear/pol yalg`, `solve/linear`, " }}{PARA 6 "" 1 "" {TEXT -1 29 " : `solv e/linear/float`" }}{PARA 6 "" 1 "" {TEXT -1 9 "split_k_n" }}{PARA 6 " " 1 "" {TEXT -1 10 "try_celine" }}{PARA 6 "" 1 "" {TEXT -1 4 "yafe" }} {PARA 6 "" 1 "" {TEXT -1 4 "zeil" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "GAMMA" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "guru2:= GAMMA(n+1/2)= (2*n)! /(n!)/2^(2*n)*GAMMA(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&guru2G /-%&GAMMAG6#,&%\"nG\"\"\"#F+\"\"#F+*&*&-%*factorialG6#,$F*F-F+-%%sqrtG 6#%#PiGF+F+*&-F16#F*F+)F-F3F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "guru3:= GAMMA(n+1/3)= (3*n)!/(n!)/3^(3*n)*GAMMA(1/3)* GAMMA(2/3)/GAMMA(n+2/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&guru3G/ -%&GAMMAG6#,&%\"nG\"\"\"#F+\"\"$F+,$*&*(-%*factorialG6#,$F*F-F+%#PiGF+ -%%sqrtG6#F-F+F+*(-F26#F*F+)F-F4F+-F'6#,&F*F+#\"\"#F-F+F+!\"\"F@" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "GAMMA(n+1/2)*(2^n)^2/(GAMMA (n)*n*sqrt(Pi)):\n%=convert((xcombipo@subs)(guru2, %), binomial): \ngu ru:= (op@solve)(subs(GAMMA(n)=n!/n, %), \{GAMMA(n+1/2)\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%guruG/-%&GAMMAG6#,&%\"nG\"\"\"#F+\"\"#F+* &*(-%)binomialG6$,$F*F-F*F+-%*factorialG6#F*F+-%%sqrtG6#%#PiGF+F+*$))F -F*F-F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`union`(seq (\{guru,guru2,guru3\}, n=0..5)): map(evalb,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "refact:= proc(y) eval(subs(GAMMA=(z-> (z-1)!), y)) end;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'refactGR6#%\"yG6\"F(F(-%%evalG6#-%%subsG6$/%& GAMMAGR6#%\"zGF(6$%)operatorG%&arrowGF(-%*factorialG6#,&9$\"\"\"F " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "Exemples pour zeil (oper, fac)" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Polyn\364mes orthogonaux" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "cp := proc (k, n) options o perator, arrow; 1/4*2^(-2*k+n+a)*GAMMA(-k+n-1/2+1/2*a)*GAMMA(1/2*a)*GA MMA(n+1)*(-1)^k/(GAMMA(k+1)*GAMMA(n+1-2*k)*sqrt(Pi)*GAMMA(n-1+a)) end \+ proc:\nfu:= unapply(x^(n-2*k)*cp(k,n), n,k);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "oper, fac:= zeil(fu(n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF),$*&*.)%\" xG,&9$\"\"\"*&\"\"#F49%F4!\"\"F4)F6,(F7!\"#F3F4%\"aGF4F4-%&GAMMAG6#,*F 7F8F3F4#F4F6F8*&#F4F6F4F6#,$F6#,&F3F4F4F4F4)F8F7F4F4 **-F>6#,&F7F4F4F4F4-F>6#,(F3F4F4F4*&F6F4F7F4F8F4-%%sqrtG6#%#PiGF4-F>6# ,(F3F4F4F8F6 $%%operG%$facG6$,*%\"nG\"\"\"F*F**(,(%\"aGF**&\"\"#F*F)F*F*F*F*F*%\"xG F*%\"NGF*!\"\"*&)F1F/F*,&F-F*F)F*F*F*,$*&**,&F)F*F*F*F*)F0F/F*,*%\"kGF /*&F/F*F)F*F2F*F*F-F2F*F " 0 " " {MPLTEXT 1 0 140 "res1:= yafe(oper,N,n,fu(n,k)): \nsimplify(fac*fu(n ,k)): gun:= unapply(%,n,k): res2:= gun(n,k+1)-gun(n,k):\nASSERT( simpl ify(res1/res2-1) = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "resulta t sur les sommes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "yafe(oper,N,n,F (n)); ASSERT(%=(n+1)*F(n)-(2*n+a+1)*x*F(n+1)+(n+a)*F(n+2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&%\"nG\"\"\"F'F'F'-%\"FG6#F&F'F'*(,(%\" aGF'*&\"\"#F'F&F'F'F'F'F'%\"xGF'-F)6#F%F'!\"\"*&,&F-F'F&F'F'-F)6#,&F&F 'F/F'F'F'" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "binomial" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fu:= binomial; oper, fac:= zeil(fu( n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuG%)binomialG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%%operG%$facG6$,&!\"#\"\"\"%\"NGF** &%\"kGF*,(%\"nG!\"\"F*F0F-F*F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " test" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "res1:= yafe(oper,N,n,fu(n, k)): \nsimplify(fac*fu(n,k)): gun:= unapply(%,n,k): res2:= gun(n,k+1)- gun(n,k):\nASSERT( (simplify@convert)(res1/res2-1, GAMMA) = 0);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "utilisation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "deb:=1: ran:= 1..deb-1+degree(oper,N):\n\{yafe(oper, N,n, F(n)), seq(F(n)= add(fu(n,k),k=0..n), n=ran)\};\nSum=rsolve(%, F (n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"FG6#\"\"\"\"\"#,&-F&6# %\"nG!\"#-F&6#,&F-F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$SumG) \"\"#%\"nG" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "binomial carr\351 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fu:= binomial^2; oper, f ac:= zeil(fu(n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuG*$) %)binomialG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%%operG%$ facG6$,(*&,&%\"nG\"\"\"F,F,F,%\"NGF,F,*&\"\"%F,F+F,!\"\"\"\"#F0*&*&,(F +!\"$\"\"$F0*&F1F,%\"kGF,F,F,)F8F1F,F,*$),(F+F0F,F0F8F,F1F,F0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "test" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "res1:= yafe(oper,N,n,fu(n,k)): \nsimplify(fac*fu(n,k)): gun:= unapply(%,n,k): res2:= gun(n,k+1)-gun(n,k):\nASSERT( (simplify@conver t)(res1/res2-1, GAMMA) = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "u tilisation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "deb:=1: ran:= 1..deb -1+degree(oper,N):\n\{yafe(oper, N,n, F(n)), seq(F(n)= add(fu(n,k),k=0 ..n), n=ran)\}:\nrsolve(%, F(n)): Sum(fu(n,k),k=0..n)= (simplify@conve rt)(subs(guru2, %), binomial);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ SumG6$*$)-%)binomialG6$%\"nG%\"kG\"\"#\"\"\"/F-;\"\"!F,-F*6$,$F,F.F," }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Monthly" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 100 "fu:= unapply(2^k*n/(n-k)*binomial(n-k,2*k),n, k); # fun:= eval(fu): \noper, fac:= zeil(fu(n,k),k,n,N);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&* ()\"\"#9%\"\"\"9$F2-%)binomialG6$,&F3F2F1!\"\",$F1F0F2F2F7F8F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%%operG%$facG6$,*!\"#\"\"\"%\"NGF** &\"\"#F*)F+F-F*!\"\"*$)F+\"\"$F*F*,$*&*(%\"kGF*,&F6F-F*F/F*,&%\"nGF/F6 F*F*F**(,(F9F/F*F/*&F2F*F6F*F*F*,(F9F/F-F/*&F2F*F6F*F*F*,(F9F/*&F2F*F6 F*F*F2F/F*F/F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "test" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "res1:= yafe(oper,N,n,fu(n,k)): \nsimplify( convert(fac*fu(n,k), GAMMA)): gun:= unapply(%,n,k); res2:= gun(n,k+1)- gun(n,k):\nASSERT( (simplify@convert)(res1/res2-1, GAMMA) = 0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gunGR6$%\"nG%\"kG6\"6$%)operatorG%& arrowGF),$*&*()\"\"#9%\"\"\"9$F3-%&GAMMAG6#,(F4F3F3F3F2!\"\"F3F3*&-F66 #,(F4F3*&\"\"$F3F2F3F9\"\"%F3F3-F66#,&F2F1F3F9F3F9F9F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "utilisation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "deb:=1: ran:= 1..deb-1+degree(oper,N):\n\{yafe(oper, N,n, F(n)), seq(F(n)= add(fu(n,k),k=0..n/3), n=ran)\};" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "Res(n)=rsolve(%, F(n)); # seq(%, n=1..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<&,*-%\"FG6#%\"nG!\"#-F&6#,&F(\"\"\"F- F-F-*&\"\"#F--F&6#,&F(F-F/F-F-!\"\"-F&6#,&F(F-\"\"$F-F-/-F&6#F7\"\"%/- F&6#F/F-/-F&6#F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ResG6#%\"nG, ()\"\"#F'#\"\"\"F**&F+F,)^#!\"\"F'F,F,*&F+F,)^#F,F'F,F," }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "6.4.2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "fu:= unapply((-1)^k*binomial(n,k)/binomial(x+k,k),n, k);\ns:= unapply('add'(fu(n,k),k=0..n), n); \noper, fac:= zeil(fu(n,k ),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6\"6$% )operatorG%&arrowGF)*&*&)!\"\"9%\"\"\"-%)binomialG6$9$F1F2F2-F46$,&%\" xGF2F1F2F1F0F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6#%\"nG6 \"6$%)operatorG%&arrowGF(-%$addG6$*&*&)!\"\"%\"kG\"\"\"-%)binomialG6$9 $F3F4F4-F66$,&%\"xGF4F3F4F3F2/F3;\"\"!F8F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%%operG%$facG6$,(*&,(%\"xG\"\"\"%\"nGF,F,F,F,%\"NGF, F,F+!\"\"F-F/*&*&%\"kGF,,&F+F,F2F,F,F,,(F-F/F,F/F2F,F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "test" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "res1:= yafe(oper,N,n,fu(n,k)): \nsimplify(fac*fu(n,k)): gun:= una pply(%,n,k): res2:= gun(n,k+1)-gun(n,k):\nASSERT( (simplify@convert)(r es1/res2-1, GAMMA) = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "utili sation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "deb:=1: ran:= 1..deb-1+de gree(oper,N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "\{yafe(oper, N,n, \+ F(n)), seq(F(n)= add(fu(n,k),k=0..n), n=1..1)\};\nSum=rsolve(%, F(n)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ranG;\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"FG6#\"\"\",&F(F(*&F(F(,&%\"xGF(F(F(!\"\"F-,&* &,&F,F-%\"nGF-F(-F&6#F1F(F(*&,(F,F(F1F(F(F(F(-F&6#,&F1F(F(F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%$SumG*&%\"xG\"\"\",&%\"nGF'F&F'!\"\" " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 5 "6.4.3" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 126 "fu:= unapply((-1)^k*binomial(n+1,k)*binomial( 2*n-2*k+1,n),n,k); \ns:= n-> add(fu(n,k),k=0..n);\noper, fac:= zeil(fu (n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6 \"6$%)operatorG%&arrowGF)*()!\"\"9%\"\"\"-%)binomialG6$,&9$F1F1F1F0F1- F36$,(F6\"\"#*&F:F1F0F1F/F1F1F6F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%$addG6$-%#fuG6$9$%\"kG /F3;\"\"!F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%%operG%$facG6 $,(*&,&%\"nG\"\"\"\"\"#F,F,%\"NGF,F,F+!\"\"F-F/,$*&**,(F+!\"$\"\"'F/*& F-F,%\"kGF,F,F,F7F,,(F+F/F,F/F7F,F,,(F+!\"#\"\"$F/*&F-F,F7F,F,F,F,*(,( F+F/F-F/F7F,F,,&F+F,F,F,F,,(F+F/F-F/*&F-F,F7F,F,F,F/F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "test" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "res1:= yafe(oper,N,n,fu(n,k)): \nsimplify(fac*fu(n,k)): gun:= una pply(%,n,k): res2:= gun(n,k+1)-gun(n,k):\nASSERT( (simplify@convert)(r es1/res2-1, GAMMA) = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "utili sation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "deb:=1: ran:= 1..deb-1+d egree(ope,N):\n\{yafe(oper, N,n, F(n)), seq(F(n)= s(n), n=ran)\}; rsol ve(%, F(n))=[ seq(s(n),n=0..5)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<& ,&*&,&%\"nG!\"\"\"\"#F(\"\"\"-%\"FG6#F'F*F**&,&F'F*F)F*F*-F,6#,&F'F*F* F*F*F*/-F,6#F)F*/-F,6#F*F*/-F,6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"7(F$F$F$F$F$F$" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "D ixon 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "fu:= unapply((-1) ^k*binomial(2*n,k)^3,n,k);\ns:= unapply('add'(fu(n,k),k=0..2*n), n); \+ \noper, fac:= zeil(fu(n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#fuGR6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&)!\"\"9%\"\"\")-%)binom ialG6$,$9$\"\"#F0\"\"$F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" sGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%$addG6$*&)!\"\"%\"kG\"\"\")-%)bi nomialG6$,$9$\"\"#F2\"\"$F3/F2;\"\"!F8F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6$%%operG%$facG6$,&*&),&%\"nG\"\"\"F-F-\"\"#F-%\"NGF-! \"#*(\"\"'F-,&F,\"\"$F.F-F-,&F,F4F-F-F-!\"\"*&*&)%\"kGF4F-,J*$)F:F.F- \"$Z\"*&\"%%3#F-)F,F.F-F-*&\"$[%F-)F,\"\"&F-F-*(\"%86F-F,F-F:F-F6*&\"# [F-F9F-F6*&\"$2#F-F:F-F6*(\"$C'F-)F,\"\"%F-F:F-F6*(\"%K>F-)F,F4F-F:F-F 6*(\"\"*F-)F:FOF-F,F-F-\"$;\"F-*(\"#!*F-FAF-F9F-F6*(\"$K\"F-F,F-F9F-F6 *(\"$[$F-FRF-F=F-F-*(\"$#zF-FAF-F=F-F-*&\"$%yF-F,F-F-*&\"%GFF-FRF-F-*( \"%9AF-FAF-F:F-F6*(\"$%fF-F,F-F=F-F-*&\"%g " 0 "" {MPLTEXT 1 0 157 "res1:= yafe(op er,N,n,fu(n,k)): \nsimplify(fac*fu(n,k)): gun:= unapply(%,n,k): res2:= gun(n,k+1)-gun(n,k):\nASSERT( (simplify@convert)(res1/res2-1, GAMMA) \+ = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "utilisation" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 200 "deb:=1: ran:= 1..deb-1+degree(ope,N):\n\{ya fe(oper, N,n, F(n)), seq(F(n)= s(n), n=ran)\}: \nsubs(guru3, rsolve(%, F(n))): (simplify@convert)(%, binomial): \nsubs(add=Sum, eval(s))(n)= convert(%, factorial);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*& )!\"\"%\"kG\"\"\")-%)binomialG6$,$%\"nG\"\"#F*\"\"$F+/F*;\"\"!F0*&*&-% *factorialG6#,$F1F3F+)F)F1F+F+*$)-F:6#F1F3F+F)" }}}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 9 "Dixon 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "dk:= (-1)^k*binomial(a+b,a+k)*binomial(b+c,b+k)*binomial(c+a,c+k); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "exo:= unapply(Sum(dk, k=-a..a), a,b,c);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "val:= unapply( (a+b+c)!/ a!/b!/c!, a,b,c); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dkG**)!\"\"% \"kG\"\"\"-%)binomialG6$,&%\"aGF)%\"bGF),&F.F)F(F)F)-F+6$,&F/F)%\"cGF) ,&F/F)F(F)F)-F+6$,&F4F)F.F),&F4F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$exoGR6%%\"aG%\"bG%\"cG6\"6$%)operatorG%&arrowGF*-%$SumG6$**)! \"\"%\"kG\"\"\"-%)binomialG6$,&9$F59%F5,&F:F5F4F5F5-F76$,&9&F5F:F5,&F@ F5F4F5F5-F76$,&F;F5F@F5,&F;F5F4F5F5/F4;,$F:F3F:F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$valGR6%%\"aG%\"bG%\"cG6\"6$%)operatorG%&arrowGF **&-%*factorialG6#,(9$\"\"\"9%F49&F4F4*(-F06#F3F4-F06#F5F4-F06#F6F4!\" \"F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fu:= unapply(dk ,a,k);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "oper, fac:= zeil(fu(n,k), k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"aG%\"kG6\"6$%)o peratorG%&arrowGF)**)!\"\"9%\"\"\"-%)binomialG6$,&9$F1%\"bGF1,&F6F1F0F 1F1-F36$,&F7F1%\"cGF1,&F7F1F0F1F1-F36$,&F6$%%operG%$facG6$,,*&,&%\"nG\"\"#F,\" \"\"F-%\"NGF-F-*&F,F-F+F-!\"\"F,F0*&F,F-%\"cGF-F0*&F,F-%\"bGF-F0*&*&,& F2F-%\"kGF-F-,&F4F-F8F-F-F-,(F+F0F-F0F8F-F0" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 4 "test" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "res1:= yafe( oper,N,n,fu(n,k)): \nsimplify(fac*fu(n,k)): gun:= unapply(%,n,k): res2 := gun(n,k+1)-gun(n,k):\nASSERT( (simplify@convert)(res1/res2-1, GAMMA ) = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "conclure" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "deb:=0: ran:= deb..deb-1+degree(ope,N);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "\{yafe(oper, N,n, F(n)), seq(F(n)= \+ (value@exo)(n,b,c), n=ran)\}; rsolve(%, F(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ranG;\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/- %\"FG6#\"\"!-%)binomialG6$,&%\"bG\"\"\"%\"cGF.F-,&*&,*F-!\"#*&\"\"#F.F /F.!\"\"*&F5F.%\"nGF.F6F5F6F.-F&6#F8F.F.*&,&F8F5F5F.F.-F&6#,&F8F.F.F.F .F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%&GAMMAG6#,*%\"nG\"\"\"F)F)% \"bGF)%\"cGF)F)*(-F%6#,&F(F)F)F)F)-F%6#,&F*F)F)F)F)-F%6#,&F+F)F)F)F)! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(eval@subs)(Sum=add , (exo)(0,b,c));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)binomialG6$,&% \"bG\"\"\"%\"cGF(F'" }}}{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 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Exemples en vrac pour zeil (ope, fac)" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "6.5" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "fu:= unapply(binomial(n,2*k)*binomial(2*k,k)/2 ^(2*k),n,k); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "fun:= eval(fu): s: = unapply('add'(fu(n,k),k=0..2*n), n): \nope, fac:= zeil(fu(n,k),k,n,N );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6\"6$%)operat orG%&arrowGF)*&*&-%)binomialG6$9$,$9%\"\"#\"\"\"-F06$F3F4F6F6)F5F3!\" \"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%$opeG%$facG6$,(*&,&%\" nG!\"\"\"\"\"F,F-%\"NGF-F-*&\"\"#F-F+F-F-F-F-,$*&*$)%\"kGF0F-F-,(F+F-F -F-*&F0F-F5F-F,F,\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 " simplify(convert(fac*fu(n,k), GAMMA)): gun:= unapply(refact(%),n,k);\n nu,de,fa:= action_ope(1,fu(n,k), ope,n,N): \nres1:= simplify(convert(n u*de, GAMMA)):\ngun(n,k+1)-gun(n,k): res2:= simplify(convert(%, GAMMA) ): \nP(N)(f)=K(f), (factor@expand@refact)(res1), (evalb@simplify)(res1 /res2-1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gunGR6$%\"nG%\"kG6\" 6$%)operatorG%&arrowGF)*&*&)\"\"%,&\"\"\"F29%!\"\"F2-%*factorialG6#9$F 2F2*&)-F66#,&F3F2F2F4\"\"#F2-F66#,(F8F2F2F2*&F>F2F3F2F4F2F4F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%/--%\"PG6#%\"NG6#%\"fG-%\"KGF)*&*&-%*f actorialG6#%\"nG\"\"\",**$)F2\"\"#F3F3F2F3*(\"\"%F3F2F3%\"kGF3!\"\"*&F 7F3F:F3F;F3F3**)F9F:F3-F06#,&F2F3*&F7F3F:F3F;F3,(F2F3F3F3*&F7F3F:F3F;F 3)-F06#F:F7F3F;%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 " deb:=0: ran:= deb..deb-1+degree(ope,N):\n\{ collect(-action_ope(1,f(n) ,ope,n,N)[1]/2, f, factor), seq(f(n)= s(n), n=ran)\};\nrsolve(%, f(n)) : tmp:= convert((refact@simplify@subs)(guru2, %), binomial);\n[ seq(tm p=s(n),n=0..5)]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,&*&,&%\"nG!\" \"#\"\"\"\"\"#F(F*-%\"fG6#F'F*F**&,&F'#F*F+F1F*F*-F-6#,&F'F*F*F*F*F*/- F-6#\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tmpG*&)\"\"#,$%\"nG! \"\"\"\"\"-%)binomialG6$,$F)F'F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7(/\"\"\"F%F$/#\"\"$\"\"#F'/#\"\"&F)F+/#\"#N\"\")F./#\"#jF0F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Legendre" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "fu:= unapply(1/2^n*(-1)^k*b inomial(2*n-2*k,n-k)*binomial(n-k,k)*x^(n-2*k),n,k); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "fun:= eval(fu): s:= unapply('add'(fu(n,k),k=0..2 *n), n): \nope, fac:= zeil(fu(n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&**)!\"\" 9%\"\"\"-%)binomialG6$,&9$\"\"#*&F8F2F1F2F0,&F7F2F1F0F2-F46$F:F1F2)%\" xG,&F7F2*&F8F2F1F2F0F2F2)F8F7F0F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%$opeG%$facG6$,**&,&%\"nG!\"\"\"\"#F,\"\"\")%\"NGF-F.F.*(%\"xGF. ,&F+F-\"\"$F.F.F0F.F.F+F,F.F,,$*&**,&F+F.F.F.F.,(F+F-F.F.*&F-F.%\"kGF. F,F.F;F.)F2F-F.F.*&,(F+F.F.F.*&F-F.F;F.F,F.,(F+F.F-F.*&F-F.F;F.F,F.F,F -" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 257 "simplify(convert(fac*fu(n,k), GAMMA)): gun:= \+ unapply(refact(%),n,k);\nnu,de,fa:= action_ope(1,fu(n,k), ope,n,N): \n res1:= simplify(convert(nu*de, GAMMA)):\ngun(n,k+1)-gun(n,k): res2:= s implify(convert(%, GAMMA)): \nP(N)(f)=K(f), (evalb@simplify)(res1/res2 -1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gunGR6$%\"nG%\"kG6\"6$%)o peratorG%&arrowGF)*&*.)!\"\"9%\"\"\")\"\"%,&F2F2F1F0F2,&9$F2F2F2F2)%\" xG,(F7F2\"\"#F2*&F;F2F1F2F0F2-%*factorialG6#,(F7F2#F2F;F2F1F0F2)F;F7F2 F2*(-%%sqrtG6#%#PiGF2-F>6#F:F2-F>6#,&F1F2F2F0F2F0F)F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/--%\"PG6#%\"NG6#%\"fG-%\"KGF)%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "deb:=1: ran:= deb..deb-1+de gree(ope,N):\nreq:= collect(action_ope(1,f(n),ope,n,N)[1], f, factor); \n\{req , seq(f(n)= s(n), n=ran)\};\ntmp:= rsolve(%, f(n)): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$reqG,(*&,&%\"nG!\"\"\"\"#F)\"\"\"-%\"fG6# ,&F(F+F*F+F+F+*&,&F(F)F+F)F+-F-6#F(F+F+*(%\"xGF+,&F(F*\"\"$F+F+-F-6#,& F(F+F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&%\"nG!\"\"\"\" #F(\"\"\"-%\"fG6#,&F'F*F)F*F*F**&,&F'F(F*F(F*-F,6#F'F*F**(%\"xGF*,&F'F )\"\"$F*F*-F,6#,&F'F*F*F*F*F*/-F,6#F),&#F(F)F**&#F6F)F*)F4F)F*F*/-F,6# F*F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "seq(s(n)=orthopoly[ P](n,x),n=0..5): map(evalb,\{%\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "action_ope(1,P [n](x),ope,n,N)[1]: rel:= collect(subs(n=n-2, %), P, expand);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$relG,(*&%\"nG\"\"\"-&%\"PG6#F'6#%\" xGF(!\"\"*&,&F'F/F(F(F(-&F+6#,&F'F(\"\"#F/F-F(F(*&,&*&F.F(F'F(F6F.F/F( -&F+6#,&F'F(F(F/F-F(F(" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "S\351r ie g\351n\351ratrice" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "def_ sg:= Sum(P[n](x)*z^n,n = 0 .. infinity)= sg(z,x); xmanip_sommes();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'def_sgG/-%$SumG6$*&-&%\"PG6#%\"nG6# %\"xG\"\"\")%\"zGF.F1/F.;\"\"!%)infinityG-%#sgG6$F3F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rule1:= combine(z*diff(def_sg,z)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule1G/-%$SumG6$*(-&%\"PG6#%\"n G6#%\"xG\"\"\")%\"zGF.F1F.F1/F.;\"\"!%)infinityG*&F3F1-%%diffG6$-%#sgG 6$F3F0F3F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "(yjoli@xjoli @expand@xrecule)(z*rule1):\nrule2:= (op@solve)(%, \{Sum(P[n-1](x)*z^n* n,n = 0 .. infinity)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule2G/ -%$SumG6$*(-&%\"PG6#,&%\"nG\"\"\"F0!\"\"6#%\"xGF0)%\"zGF/F0F/F0/F/;\" \"!%)infinityG,(-F'6$*&F*F0F4F0F6F0-&F,6#F1F2F1*&)F5\"\"#F0-%%diffG6$- %#sgG6$F5F3F5F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "(yjol i@xjoli@expand@xrecule)(z*rule2):\nrule3:= (op@solve)(%, \{Sum(P[n-2]( x)*z^n*n,n = 0 .. infinity)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& rule3G/-%$SumG6$*(-&%\"PG6#,&%\"nG\"\"\"\"\"#!\"\"6#%\"xGF0)%\"zGF/F0F /F0/F/;\"\"!%)infinityG,*-F'6$*&F*F0F5F0F7F1*&F1F0-&F,6#!\"#F3F0F2*&F6 F0-&F,6#F2F3F0F2*&)F6\"\"$F0-%%diffG6$-%#sgG6$F6F4F6F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "(yjoli@xjoli@expand@xrecule)(z*def _sg):\nrule4:= (op@solve)(%, \{Sum(P[n-1](x)*z^n,n = 0 .. infinity)\}) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule4G/-%$SumG6$*&-&%\"PG6#,&% \"nG\"\"\"F0!\"\"6#%\"xGF0)%\"zGF/F0/F/;\"\"!%)infinityG,&-&F,6#F1F2F0 *&F5F0-%#sgG6$F5F3F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 " (yjoli@xjoli@expand@xrecule)(z*rule4):\nrule5:= (op@solve)(%, \{Sum(P[ n-2](x)*z^n,n = 0 .. infinity)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&rule5G/-%$SumG6$*&-&%\"PG6#,&%\"nG\"\"\"\"\"#!\"\"6#%\"xGF0)%\"zGF/ F0/F/;\"\"!%)infinityG,(-&F,6#!\"#F3F0*&F6F0-&F,6#F2F3F0F0*&)F6F1F0-%# sgG6$F6F4F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "map(u->Su m(u*z^n,n=0..infinity), rel): (yjoli@xjoli@expand)(%):\ncollect(%, Sum );\nsubs(rule1, rule2, rule3, rule4, rule5, P[-1]=0, P[-2]=0, %): \neq d:= collect(%, [S, diff, sg], normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&%\"xG\"\"\"-%$SumG6$*(-&%\"PG6#,&%\"nGF&F&!\"\"6#F%F&)%\"zGF 0F&F0F&/F0;\"\"!%)infinityGF&\"\"#-F(6$*(-&F-6#F0F2F&F3F&F0F&F5F1-F(6$ *(-&F-6#,&F0F&F9F1F2F&F3F&F0F&F5F1-F(6$*&FCF&F3F&F5F&*&F%F&-F(6$*&F+F& F3F&F5F&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqdG,&*&,(%\"zG!\"\"* $)F(\"\"$\"\"\"F)*(\"\"#F-%\"xGF-)F(F/F-F-F--%%diffG6$-%#sgG6$F(F0F(F- F-*&,&*&F0F-F(F-F-*$F1F-F)F-F5F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dso0:= subs(dsolve(eqd, sg(z,x)), sg(z,x)): " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "dso:= sg(z,x)= dso0/subs(z=0, dso0) /value(s(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dsoG/-%#sgG6$%\"zG %\"xG*&^#\"\"\"F-*$-%%sqrtG6#,(!\"\"F-*$)F)\"\"#F-F3*(F6F-F*F-F)F-F-F- F3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 23 "Exo 4.6.3b -- cf celine" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "fu:= unapply(binomial(x,k)*binomial(y,n-k), n,k ):\ns:= unapply( Sum(fu(n,k),k=0..n),n,x,y); ope, fac:= zeil(fu(n,k),k ,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6%%\"nG%\"xG%\"yG6\"6 $%)operatorG%&arrowGF*-%$SumG6$*&-%)binomialG6$9%%\"kG\"\"\"-F36$9&,&9 $F7F6!\"\"F7/F6;\"\"!F6$%$ opeG%$facG6$,*%\"xG!\"\"%\"yGF*%\"nG\"\"\"*&,&F,F-F-F-F-%\"NGF-F-*&*&% \"kGF-,(F+F*F,F-F3F*F-F-,(F,F-F-F-F3F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "simplify(convert(fac*fu(n,k), GAMMA)): gun:= unapply (convert(refact(%), binomial),n,k);\nnu,de,fa:= action_ope(1,fu(n,k), \+ ope,n,N): \nres1:= simplify(convert(nu*de, GAMMA)):\ngun(n,k+1)-gun(n, k): res2:= simplify(convert(%, GAMMA)): \nP(N)(f)=K(f), res2, (evalb@s implify)(res1/res2-1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gunGR6$ %\"nG%\"kG6\"6$%)operatorG%&arrowGF),$*(,(%\"xG\"\"\"9%!\"\"F1F1F1-%)b inomialG6$F0,&F2F1F1F3F1-F56$%\"yG,(9$F1F1F1F2F3F1F3F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/--%\"PG6#%\"NG6#%\"fG-%\"KGF),$*&*(,,*&%\"x G\"\"\"%\"nGF3F3F2F3*&F2F3%\"kGF3!\"\"*&F6F3%\"yGF3F7F6F7F3-%&GAMMAG6# ,&F2F3F3F3F3-F;6#,&F9F3F3F3F3F3**-F;6#,*F9F3F4F7F6F3F3F3F3-F;6#,(F2F3F 6F7F3F3F3-F;6#,(F4F3\"\"#F3F6F7F3-F;6#,&F6F3F3F3F3F7F7%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "deb:=1: ran:= deb..deb-1+deg ree(ope,N):\nreq:= collect(-action_ope(1,f(n),ope,n,N)[1], f, factor); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "solve(subs(n=m-1, req), f(m))/ f(m-1): map(Product,%,m=2..n);\n(value@select)(has,%,1/m)*remove(has,% ,1/m):\nstudent[changevar](m=n-p,%,p): subs((n-p=2..n)=(p=0..n-2), %)* value(s(1,x,y)):\n(refact@simplify@value)(%): convert(%, binomial);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$reqG,&*&,(%\"yG\"\"\"%\"nG!\"\"%\" xGF)F)-%\"fG6#F*F)F)*&,&F*F+F)F+F)-F.6#,&F*F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%(ProductG6$,*%\"yG\"\"\"%\"mG!\"\"F)F)%\"xGF)/ F*;\"\"#%\"nGF)-F%6$*&F)F)F*F+F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%)binomialG6$,&%\"yG\"\"\"%\"xGF(%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "S\351rie g\351n\351ratrice : cf le \247 de m\352me nom da ns exemples celine" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "2F1" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 110 "fu:= unapply((a+k-1)!/(a-1)! *(b+k-1)! /(b-1)!*(n- 1)!/(n+k-1)!/(k)!,n,k);\nfu(n,k+1)/fu(n,k): ratio= expand(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fuGR6$%\"nG%\"kG6\"6$%)operatorG%&a rrowGF)*&*(-%*factorialG6#,(9%\"\"\"F4!\"\"%\"aGF4F4-F06#,(%\"bGF4F3F4 F4F5F4-F06#,&9$F4F4F5F4F4**-F06#,&F6F4F4F5F4-F06#,&F:F4F4F5F4-F06#,(F> F4F3F4F4F5F4-F06#F3F4F5F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&ra tioG*&*&,&%\"kG\"\"\"%\"aGF)F),&%\"bGF)F(F)F)F)*&,&F(F)F)F)F),&%\"nGF) F(F)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "s:= unapply( Sum(fu(n,k),k=0..infinity),n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"sGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%$SumG6$*&*(-%*factorialG6#,(% \"aG\"\"\"%\"kGF6F6!\"\"F6-F26#,(%\"bGF6F7F6F6F8F6-F26#,&9$F6F6F8F6F6* *-F26#,&F5F6F6F8F6-F26#,&F " 0 "" {MPLTEXT 1 0 50 "refact(fu(n,k)/value(s(n))): fun:= unapply(%,n,k);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "ope, fac:= zeil(fun(n,k),k,n,N);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$funGR6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*& **-%*factorialG6#,(9%\"\"\"F4!\"\"%\"aGF4F4-F06#,(%\"bGF4F3F4F4F5F4-F0 6#,(9$F4F6F5F4F5F4-F06#,(F>F4F:F5F4F5F4F4*,-F06#,&F6F4F4F5F4-F06#,&F:F 4F4F5F4-F06#,(F>F4F3F4F4F5F4-F06#F3F4-F06#,*F6F5F:F5F>F4F4F5F4F5F)F)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%$opeG%$facG6$,&!\"\"\"\"\"%\"N GF**&%\"kGF*,(%\"aGF)%\"bGF)%\"nGF*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ope, fac:= zeil(fu(n,k),k,n,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%$opeG%$facG6$,&*(,&%\"nG\"\"\"%\"bG!\"\"F,,&F+F,%\" aGF.F,%\"NGF,F,*&,(F+F,F0F.F-F.F,F+F,F.*&%\"kGF,F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "simplify(convert(fac*fu(n,k), GAMMA)): g un:= unapply(refact(%),n,k);\nnu,de,fa:= action_ope(1,fu(n,k), ope,n,N ): \nres1:= simplify(convert(nu*de, GAMMA)):\ngun(n,k+1)-gun(n,k): res 2:= simplify(convert(%, GAMMA)): \nP(N)(f)=K(f), (factor@expand@refact )(res1), (evalb@simplify)(res1/res2-1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gunGR6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&*(-%*fa ctorialG6#9$\"\"\"-F06#,(9%F3F3!\"\"%\"aGF3F3-F06#,(%\"bGF3F7F3F3F8F3F 3**-F06#,&F7F3F3F8F3-F06#,&F9F3F3F8F3-F06#,&F=F3F3F8F3-F06#,(F2F3F7F3F 3F8F3F8F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/--%\"PG6#%\"NG6#%\"f G-%\"KGF),$*&*.-%*factorialG6#%\"nG\"\"\"-F16#,&%\"kGF4%\"aGF4F4-F16#, &%\"bGF4F8F4F4F9F4F=F4,**&F8F4F=F4!\"\"*&F9F4F=F4F@*&F9F4F8F4F@*&F8F4F 3F4F4F4F4*.F7F4F " 0 "" {MPLTEXT 1 0 152 "deb:=1: ran:= deb. .deb-1+degree(ope,N): \nreq:= collect(action_ope(1,f(n),ope,n,N)[1], f ):\n\{ req, seq(f(n)= value(s(n)), n=ran)\};\ntmp:= rsolve(%, f(n)); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"fG6#\"\"\"*&-%&GAMMAG6#,(% \"aG!\"\"%\"bGF/F(F(F(*&-F+6#,&F(F(F.F/F(-F+6#,&F(F(F0F/F(F/,&*&,(*$)% \"nG\"\"#F(F/*&F.F(F=F(F(*&F0F(F=F(F(F(-F&6#F=F(F(*&,*F;F(F?F/F@F/*&F. F(F0F(F(F(-F&6#,&F=F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tm pG*&*&-%&GAMMAG6#%\"nG\"\"\"-F(6#,(F*F+%\"aG!\"\"%\"bGF0F+F+*&-F(6#,&F *F+F1F0F+-F(6#,&F*F+F/F0F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Exemples pou r celine " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "try_celine(bino mial,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$funG6$%\"nG%\"kG-%)b inomialGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$gunG6$%\"nG%\"kG,$*& *&,&F'\"\"\"F(!\"\"F--%)binomialGF&F-F-F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$frfG,(-%\"FG6$%\"nG%\"kG\"\"\"-F'6$,&F)F+F+!\"\"F*F/ -F'6$F.,&F*F+F+F/F/%%trueG" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'trf_eqG,*-%\"FG6$%\"nG%\"kG\"\"\"*&\"\"#F +-F'6$,&F)F+F+!\"\"F*F+F1-%\"GGF(F1-F36$F),&F*F+F+F1F+%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6$%\"nG%\"kG,$*&,&F'\"\"\"F(!\" \"F,F'F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$trfG,&-%\"SG6#%\"nG\" \"\"*&\"\"#F*-F'6#,&F)F*F*!\"\"F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$SumG6$-%\"FG6$%\"nG%\"kG/F+;,$%)infinityG!\"\"F/)\"\"#F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "try_celine((n,k)->k*binomial (n,k),1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$funG6$%\"nG%\"kG*&F (\"\"\"-%)binomialGF&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$gunG6$% \"nG%\"kG*&*(,&F'\"\"\"F(!\"\"F,F(F,-%)binomialGF&F,F,F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$frfG,(*&,&%\"nG!\"\"\"\"\"F*F*-%\"FG6$F(% \"kGF*F**&-F,6$,&F(F*F*F)F.F*F(F*F**&-F,6$F2,&F.F*F*F)F*F(F*F*%%trueG " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/% 'trf_eqG,**&*&,&%\"nG\"\"\"F*!\"\"F*-%\"FG6$F)%\"kGF*F*F)F+F+*&\"\"#F* -F-6$F(F/F*F*-%\"GGF.F+-F56$F),&F/F*F*F+F*%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6$%\"nG%\"kG*&,&F'\"\"\"F(!\"\"F+F'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$trfG,&-%\"SG6#%\"nG\"\"\"*&*(\"\"#F*-F'6# ,&F)F*F*!\"\"F*F)F*F*F0F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Sum G6$-%\"FG6$%\"nG%\"kG/F+;,$%)infinityG!\"\"F/,$*&F*\"\"\")\"\"#F*F3#F3 F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "try_celine(binomial^2 ,2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$funG6$%\"nG%\"kG*$)-%)bi nomialGF&\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$gunG6$%\"n G%\"kG,$*&*&,0*$)F'\"\"$\"\"\"F/*(\"\")F0F(F0)F'\"\"#F0!\"\"*&F4F0F3F0 F5*(\"\"(F0F'F0)F(F4F0F0*(\"\"%F0F(F0F'F0F0*&F4F0)F(F/F0F5*&F4F0F9F0F5 F0)-%)binomialGF&F4F0F0*$F.F0F5F5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/ %$frfG,.*&-%\"FG6$%\"nG%\"kG\"\"\"F*F,F,*&,&F*!\"#F,F,F,-F(6$,&F*F,F,! \"\"F+F,F,*&F.F,-F(6$F2,&F+F,F,F3F,F,*&F2F,-F(6$,&F*F,\"\"#F3F+F,F,*&, &F " 0 "" {MPLTEXT 1 0 51 "try_celine((n,k)->(-1)^k*binomial(n,k)*x^k/k!,2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$funG6$%\"nG%\"kG*&*()!\"\"F(\"\" \"-%)binomialGF&F-)%\"xGF(F-F--%*factorialG6#F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$gunG6$%\"nG%\"kG*&*,%\"xG\"\"\",&F'F,F(!\"\"F,)F.F( F,-%)binomialGF&F,)F+F(F,F,*&)F'\"\"#F,-%*factorialG6#F(F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$frfG,**&-%\"FG6$%\"nG%\"kG\"\"\"F*F,F,*&, &F*!\"#F,F,F,-F(6$,&F*F,F,!\"\"F+F,F,*&%\"xGF,-F(6$F2,&F+F,F,F3F,F,*&F 2F,-F(6$,&F*F,\"\"#F3F+F,F,%%trueG" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%'trf_eqG,,-%\"FG6$%\"nG%\"kG\"\"\"*& ,&*&,&F)\"\"#F+!\"\"F+F)F1F1*&%\"xGF+F)F1F+F+-F'6$,&F)F+F+F1F*F+F+*&*& F6F+-F'6$,&F)F+F0F1F*F+F+F)F1F+-%\"GGF(F1-F=6$F),&F*F+F+F1F+%%trueG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6$%\"nG%\"kG*&*&%\"xG\"\"\",&F 'F,F(!\"\"F,F,*$)F'\"\"#F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$trf G,(-%\"SG6#%\"nG\"\"\"*&*&,(F)\"\"#F*!\"\"%\"xGF/F*-F'6#,&F)F*F*F/F*F* F)F/F/*&*&F3F*-F'6#,&F)F*F.F/F*F*F)F/F*" }}{PARA 6 "" 1 "" {TEXT -1 32 "no closed form available for Sum" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "try_celine((n,k)->binomial(n,k)*binomial(2*k,k)*(-2)^ (n-k),2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$funG6$%\"nG%\"kG*(- %)binomialGF&\"\"\"-F+6$,$F(\"\"#F(F,)!\"#,&F'F,F(!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$gunG6$%\"nG%\"kG*&**,**&F(\"\"\"F'F-\"\"#F' F-*&F.F-)F(F.F-!\"\"F(F1F--%)binomialGF&F--F36$,$F(F.F(F-)!\"#,&F'F-F( F1F-F-*$)F'F.F-F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$frfG,,*&-%\"FG 6$%\"nG%\"kG\"\"\"F*F,F,*&,&F*\"\"%\"\"#!\"\"F,-F(6$,&F*F,F,F1F+F,F,*& ,&F*!\"%F0F,F,-F(6$F4,&F+F,F,F1F,F,*&,&F*F/F/F1F,-F(6$,&F*F,F0F1F+F,F, *&,&\"\")F,*&FBF,F*F,F1F,-F(6$F?F:F,F,%%trueG" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'trf_eqG,*-%\"FG6$% \"nG%\"kG\"\"\"*&*(\"\"%F+,&F)F+F+!\"\"F+-F'6$,&F)F+\"\"#F0F*F+F+F)F0F 0-%\"GGF(F0-F66$F),&F*F+F+F0F+%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6$%\"nG%\"kG*&,**&F(\"\"\"F'F,\"\"#F'F,*&F-F,)F(F-F,!\"\"F(F 0F,*$)F'F-F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$trfG,&-%\"SG6#%\" nG\"\"\"*&*(\"\"%F*,&F)F*F*!\"\"F*-F'6#,&F)F*\"\"#F/F*F*F)F/F/" }} {PARA 6 "" 1 "" {TEXT -1 32 "no closed form available for Sum" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Exo 4.6.3b, page 72" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "fu:= unapply(binomial(x,k)*binomial(y,n-k), n,k):\ns:= unappl y( Sum(fu(n,k),k=0..n),n,x,y); try_celine(fu,2,1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"sGR6%%\"nG%\"xG%\"yG6\"6$%)operatorG%&arrowGF*-%$ SumG6$*&-%)binomialG6$9%%\"kG\"\"\"-F36$9&,&9$F7F6!\"\"F7/F6;\"\"!F " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "S\351rie g\351n\351ratrice" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "def_sg:= Sum(S(n)*z^n,n = 0 .. infinity)= sg(z); xman ip_sommes();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'def_sgG/-%$SumG6$*& -%\"SG6#%\"nG\"\"\")%\"zGF-F./F-;\"\"!%)infinityG-%#sgG6#F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rule1:= combine(z*diff(def_s g,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule1G/-%$SumG6$*(-%\"SG6 #%\"nG\"\"\")%\"zGF-F.F-F./F-;\"\"!%)infinityG*&F0F.-%%diffG6$-%#sgG6# F0F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "(yjoli@xjoli@exp and@xrecule)(z*rule1):\nrule2:= (op@solve)(%, \{Sum(S(n-1)*z^n*n,n = 0 .. infinity)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule2G/-%$SumG 6$*(-%\"SG6#,&%\"nG\"\"\"F/!\"\"F/)%\"zGF.F/F.F//F.;\"\"!%)infinityG,( -F'6$*&F*F/F1F/F3F/-F+6#F0F0*&-%%diffG6$-%#sgG6#F2F2F/)F2\"\"#F/F/" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "(yjoli@xjoli@expand@xrecul e)(z*rule2):\nrule3:= (op@solve)(%, \{Sum(S(n-2)*z^n*n,n = 0 .. infini ty)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule3G/-%$SumG6$*(-%\"SG 6#,&%\"nG\"\"\"\"\"#!\"\"F/)%\"zGF.F/F.F//F.;\"\"!%)infinityG,*-F'6$*& F*F/F2F/F4F0*&F0F/-F+6#!\"#F/F1*&F3F/-F+6#F1F/F1*&-%%diffG6$-%#sgG6#F3 F3F/)F3\"\"$F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "(yjoli@ xjoli@expand@xrecule)(z*def_sg):\nrule4:= (op@solve)(%, \{Sum(S(n-1)*z ^n,n = 0 .. infinity)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule4G /-%$SumG6$*&-%\"SG6#,&%\"nG\"\"\"F/!\"\"F/)%\"zGF.F//F.;\"\"!%)infinit yG,&-F+6#F0F/*&-%#sgG6#F2F/F2F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "(yjoli@xjoli@expand@xrecule)(z*rule4):\nrule5:= (op@s olve)(%, \{Sum(S(n-2)*z^n,n = 0 .. infinity)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rule5G/-%$SumG6$*&-%\"SG6#,&%\"nG\"\"\"\"\"#!\"\"F/) %\"zGF.F//F.;\"\"!%)infinityG,(-F+6#!\"#F/*&F3F/-F+6#F1F/F/*&-%#sgG6#F 3F/)F3F0F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "collect(tr f*n, S); map(u->Sum(u*z^n,n=0..infinity), %): (yjoli@xjoli@expand)(%): \ncollect(%, Sum);\nsubs(rule1, rule2, rule3, rule4, rule5, S(-1)=0, S (-2)=0, %): \neqd:= collect(%, [S, diff, sg], normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,*%\"nG\"\"\"%\"xG!\"\"\"\"#F)%\"yGF)F'-%\"SG6 #,&F&F'F*F)F'F'*&F&F'-F-6#F&F'F'*&,*F&F*F*F)F+F)F(F)F'-F-6#,&F&F'F'F)F 'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,-%$SumG6$*(-%\"SG6#%\"nG\"\" \")%\"zGF+F,F+F,/F+;\"\"!%)infinityGF,*&\"\"#F,-F%6$*(-F)6#,&F+F,F,!\" \"F,F-F,F+F,F/F,F,*&,(%\"xGF;F4F;%\"yGF;F,-F%6$*&F8F,F-F,F/F,F,-F%6$*( -F)6#,&F+F,F4F;F,F-F,F+F,F/F,*&F=F,-F%6$*&FFF,F-F,F/F,F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqdG,&*&,(%\"zG\"\"\"*$)F(\"\"$F)F)*&\"\"#F)) F(F.F)F)F)-%%diffG6$-%#sgG6#F(F(F)F)*&,**&%\"xGF)F/F)!\"\"*&F/F)%\"yGF )F:*&F9F)F(F)F:*&F(F)F " 0 "" {MPLTEXT 1 0 45 "eqd; dso0:= subs(dsolve(eqd, sg(z)), sg(z)); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,(%\"zG\"\"\"*$)F&\"\"$F'F'*&\"\"# F')F&F,F'F'F'-%%diffG6$-%#sgG6#F&F&F'F'*&,**&%\"xGF'F-F'!\"\"*&F-F'%\" yGF'F8*&F7F'F&F'F8*&F&F'F:F'F8F'F1F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dso0G*&%$_C1G\"\"\"),&%\"zGF'F'F',&%\"yGF'%\"xGF'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dso:= sg(z)= dso0/subs(z=0, \+ dso0)/value(s(0,x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dsoG/-%#s gG6#%\"zG),&F)\"\"\"F,F,,&%\"yGF,%\"xGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(s(n,-2,5)*z^n, n=0..5): %= (factor@value)(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&-F%6$*&-%)binomialG6$!\"#% \"kG\"\"\"-F,6$\"\"&,&%\"nGF0F/!\"\"F0/F/;\"\"!F5F0)%\"zGF5F0/F5;F9F3* $),&F;F0F0F0\"\"$F0" }}}{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 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 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 0 21 8524 "proc () \nif nops([args])=1 and op(1,[args])=`findrec` then\nprintf(\"findr ec(f,DEG,ORDER,n,N) finds empirically an ordi. linear recurrence\"):\n printf(\" with polynomial coeffs. The input is a sequence f given as a list\"):\nprintf(\"STARTING at f[1],i.e. f[0] is not considered\"):\n printf(\" where DEG:=the maximal degree of the coefficients\"):\nprint f(\"and ORDER:=the order of the recurrence. The output is the operato r\"):\nprintf(\" in n and N, where N is the forward unit shift: Nf(n): =f(n+1).\"):\nprintf(\"For example findrec([1,1,2,3,5,8,13,21,34],0,2, n,N) should yield\"):\nprintf(\"N^2-N-1 , and findrec([1,2,5,14,42,132 ,429],1,1,m,M) should yield\"):\nprintf(\"(m+1)*M-(4*m-2). If there is not enough data, you will get an\"):\nprintf(\"an error message. If t here is no operator, you would get 0\"):\nfi:\n \nif nops([args])=1 an d op(1,[args])=`AZpapc` then\nprintf(\"AZpapc(INTEGRAND,y,x) inputs a \+ hypergeometric integrand\"):\nprintf(\"in the continuous variables y a nd x (i.e. the logarithmic derivatives\"):\nprintf(\"diff(INTEGRAND,x) /INTEGRAND and diff(INTEGRAND,y)/INTEGRAND are\"):\nprintf(\"rational \+ functions in x and y)\"):\nprintf(\"and outputs a paper with a proof o f the linear differential equation\"):\nprintf(\"that the definite int egral w.r.t. to y (which is a function of x)\"):\nprintf(\"satisfies. \+ It uses the method of\"):\nprintf(\"Gert Almkvist and Doron Zeilberger , \"The method of differentiating\"):\nprintf(\"under the integral sig n\", J. Symbolic Computation 10(1990), 571-591.\"):\nprintf(\"\"):\npr intf(\"It could be used to establish the diff. eq. of the \"):\nprintf (\"classical orthogonal polynomials, when they are defined in terms\") :\nprintf(\"or their generating funtion.\"):\nprintf(\"For example AZp apc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x) gives the\"):\nprintf(\"the diffe rential equation satisfied by the Legendre polynomials\"):\n \nfi:\n \+ \n \n \nif nops([args])=1 and op(1,[args])=`AZpapd` then\nprintf(\"AZp apd(INTEGRAND,x,n) inputs a hypergeometric integrand\"):\nprintf(\"in \+ the continuous variable x and the discrete variable n\"):\nprintf(\"i. e. i.e. A(x,n+1)/A(x,n) and A'(x,n)/A(x,n) are rational functions\"): \nprintf(\"of (x,n)),\"):\nprintf(\"and outputs a paper with a proof o f the linear recurrence equation\"):\nprintf(\"that the definite integ ral w.r.t. to y (which is a function of n)\"):\nprintf(\"assuming that the integrand (or rather it times the certificate\"):\nprintf(\"vanis hes at the endpoints, or it is a contour integrals\"):\nprintf(\"sati sfies. It assumes the following: A(x,n) is not a product of\"):\nprint f(\"of a function of n and a function of x\"):\nprintf(\"It uses the m ethod of\"):\nprintf(\"Gert Almkvist and Doron Zeilberger, \"The metho d of differentiating\"):\nprintf(\"under the integral sign\", J. Symbo lic Computation 10(1990), 571-591\"):\nprintf(\"It could be used to es tablish the recurrences of the \"):\nprintf(\"classical orthogonal pol ynomials, when they are defined in terms\"):\nprintf(\"or their genera ting funtion\"):\nprintf(\"For example AZpapd(1/sqrt(1-2*x*t+t^2)/t^(n +1),t,n) gives the\"):\nprintf(\"the recurrence, and proof, satisfied \+ by the Legendre polynomials\"):\n \nfi:\n \n \nif nops([args])=1 and o p(1,[args])=`AZd` then\nprintf(\"AZd(A,x,n,N) finds a recurrence of or der ORDER<=8\"):\nprintf(\"phrased in terms of n, and the shift in n, \+ N\"):\nprintf(\"for the integrale of A(x,n) with respect to x, wheneve r A(x,n) is\"):\nprintf(\"hypergeometric in (x,n),i.e. A(x,n+1)/A(x,n) and A'(x,n)/A(x,n) are\"):\nprintf(\"rational funtions of x and n. It follows the algorithn of\"):\nprintf(\"Gert Almkvist and Doron Zeilbe rger, \"The method of differentiating\"):\nprintf(\"under the integral sign\", J. Symbolic Computation 10(1990), 571-591\"):\nprintf(\"A sho uld not be a product of a function of x and a function of n.\"):\nprin tf(\"\"):\nprintf(\"AZd(A,x,n,N) returns the expression seq. ope(n,N), cert(x,n)\"):\nprintf(\"satisfying ope(n,N)A(x,n)=diff(cert(x,n)*A(x,n ),x).\"):\nprintf(\"If no recurrence is found, it returns 0.\"):\nprin tf(\"\"):\nprintf(\"A verbose version is AZpapd(A,x,n), type ezra(AZpa pd) for details.\"):\nprintf(\"\"):\nprintf(\"For example AZd(1/sqrt(1 -2*x*t+t^2)/t^(n+1),t,n,N) gives\"):\nprintf(\"the diff.eq., and proof certificate, for the Legendre polct:ynomials.\"):\n \nfi:\n \n \nif n ops([args])=1 and op(1,[args])=`AZc` then\nprintf(\"AZc(A,y,x,D) tries to finds a linear diff.eq. of order <=8,\"):\nprintf(\" phrased in te rms of x, and diff.w.r.t x, D\"):\nprintf(\"for the integrale of A(x,y ) with respect to x, whenever A(x,y) is\"):\nprintf(\"hypergeometric i n (x,y),i.e. A_x(x,y)/A(x,y) and A_y(x,y)/A(x,y) are\"):\nprintf(\"ra tional funtions of x and y. It follows the algorithn of\"):\nprintf(\" Gert Almkvist and Doron Zeilberger, \"The method of differentiating\") :\nprintf(\"under the integral sign\", J. Symbolic Computation 10(1990 ), 571-591\"):\nprintf(\"AZc(A,y,x,D) returns the expression seq. ope( x,D),cert(x,n)\"):\nprintf(\"satisfying ope(x,D)A(x,y)=diff(cert(x,y)* A(x,y),y).\"):\nprintf(\"If no linear diff. eq. of order<=8 is found, \+ it returns 0\"):\nprintf(\"see AZpapc for a verbose vsersion\"):\nprin tf(\"For example AZc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x,D) gives the\"): \nprintf(\"the diff.eq., and proof certificate for the Legendre polyno mials.\"):\n \nfi:\n \n \n if nops([args])=1 and op(1,[args])=`ct` the n\n \n printf(\" ct(SUMMAND,ORDER,k,n,N)\"): \nprintf(\"This is a Mapl e inplementation of the algorithm described in Ch. 6\"):\nprintf(\"of \+ the book A=B, first proposed in : D. Zeilberger, \"The method of\"):\n printf(\"of creative telescoping\", J.Symbolic Computation 11(1991) 19 5-204\"):\nprintf(\"\"):\nprintf(\"finds a recurrence for SUMMAND in t he parameters k and n, \"):\nprintf(\" of order ORDER, with N is the chosen symbol for the shift in n.\"):\n printf(\"\"):\n printf(\"SUMM AND should be a product of factorials and/or binomial coeffs\"):\n pri ntf(\"and/or rising factorials, where (a)_k is denoted by rf(a,k)\"): \n printf(\"and/or powers of k and n, and, optionally, a polynomial fa ctor\"):\nprintf(\"The output consists of an operator ope(N,n) and a c ertificate R(n,k)\"):\nprintf(\"with the properties that if we define \+ G(n,k):=R(n,k)*SUMMAND then\"):\nprintf(\"ope(N,n)SUMMAND(n,k)=G(n,k+1 )-G(n,k)\"):\nprintf(\"which is a routinely verifiable identity.\"):\n printf(\"For example \"ct(binomial(n,k),1,k,n,N);\" would yield the o utput\"):\nprintf(\" N-2, k/(k-n-1) \"):\n fi:\n \nif nops([args])=1 \+ and op(1,[args])=`zeil` then\nprintf(\"The syntax is:\"):\n printf(\" \+ zeil(SUMMAND,k,n,N) or zeil(SUMMAND,k,n,N,MAXORDER) or \"):\n printf( \" zeil(SUMMAND,k,n,N,MAXORDER,parameter_list) \"):\n printf(\" finds \+ a linear recurrence equation for SUMMAND, with\"):\n printf(\" polynom ial coefficients\"):\n printf(\"of ORDER<=MAXORDER, where the default \+ of MAXORDER is 6\"):\n printf(\"in the parameter n, the shift operator in n being N\"):\nprintf(\"of the form ope(N,n)SUMMAND=G(n,k+1)-G(n,k )\"):\nprintf(\"where G(n,k):=R(n,k)*SUMMAND, and R(n,k) is the 2nd it em of output.\"):\nprintf(\"The output is ope(N,n),R(n,k) .\"):\n prin tf(\"For example zeil(binomial(n,k),k,n,N) would yield\"):\nprintf(\" \+ N-2, k/(k-n-1) \");\nprintf(\" in which N-2 is the \"ope\" operator, and k/(k-n-1) is R(n,k) \");\n printf(\"SUMMAND should be a product o f factorials and/or binomial coeffs\"):\n printf(\"and/or rising facto rials, where (a)_k is denoted by rf(a,k)\"):\n printf(\"and/or powers \+ in k and n, and, optionally, a polynomial factor.\"):\n printf(\"\"): \n printf(\"The last optional parameter, is the list of other paramete rs,\"):\n printf(\"if present. Giving them causes considerable speedup . For example\"):\n printf(\" zeil(binomial(n,k)*binomial(a,k)*binomia l(b,k),k,n,N,6,[a,b])\"):\n \nfi:\n \nif nops([args])=1 and op(1,[args ])=`zeilpap` then\nprintf(\" zeilpap(SUMMAND,k,n) or zeilpap(SUMMAND,k ,n,NAME,REF)\"):\nprintf(\"Just like zeil but writes a paper with the \+ proof\"):\nprintf(\"NAME and REF are optional name and reference\"):\n printf(\"Warning: It assumes that the definite summation w.r.t. k\"): \nprintf(\"extends over all k where it is non-zero, and that it is zer o\"):\nprintf(\"for other k\"):\nprintf(\"For non-natural summation li mits, use zeillim\"):\nfi:\n \nif nops([args])=1 and op(1,[args])=`zei llim` then\n printf(\" zeillim(SUMMAND,k,n,N,alpha,beta) \"):\nprintf( \"Similar to zeil(SUMMAND,k,n,N) but outputs a recurrence for \"):\npr intf(\" the sum of SUMMAND from k=alpha to k=n-beta .\"):\nprintf(\"Ou tputs the recurrence operator, certificate and right hand side.\"):\np rintf(\"Note carefully: The upper limit of the sum will be n-beta, not beta. \"):\nprintf(\"For example, \"zeillim(binomial(n,k),k,n,N,0,1); \" gives output of \"):\nprintf(\" N-2, k/(k-n-1),1 \"):\nprintf(\" wh ich means that SUM(n):=2^n-1 satisfies the recurrence \"):\nprintf(\" \+ (N-2)SUM(n)=1, as certified by R(n,k):=k/(k-n-1) \"):\nfi:\n \n \nend: " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 172 "yafec:=proc(ope,D,x,INT EGRAND)\nlocal gu,i:\n \n gu:=coeff(ope,D,0)*INTEGRAND:\n \n \nfor i f rom 1 to degree(ope,D) do\n gu:=gu+coeff(ope,D,i)*diff(INTEGRAND,x$i): \nod:\n \ngu:\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 38 "rf: =proc(a,b):\n (a+b-1)!/(a-1)!:\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 198 "RootOf1:=proc(f,x)\nlocal kv,kvi,i:\nkv:=[solve(f=0 ,x)]:\nkvi:=\{\}:\nfor i from 1 to nops(kv) do\n \n if type(kv[i],int eger) and kv[i]>0 then\n kvi:=kvi union \{kv[i]\}\n fi:\n \nod:\n \+ \nRETURN(kvi):\n \nend:\n \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " \n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 2397 "ct_qroc:=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, i1, ia1, k1 , kvutsa, l1, l2, meka1, mekb1, mumu, \nva1, va2,ope1,denFAC,ope1a:\n \+ \nif nargs<>5 then\nERROR(\"Syntax: ct(SUMMAND,ORDER,summation_variabl e,auxiliary_var, Shift_n)\"):\nfi:\n \nif split_k_n(convert(SUMMAND,fa ctorial),k,n) then\nERROR(\"The summand can be separated into f(`,k,`) g(`,n,`)\"): \nfi:\n \nope:=0:\n \nfor i from 0 to ORDER do\n ope:=ope +bpc[i]*N^i:\nod:\n \ngu:=action_ope(1,convert(SUMMAND,factorial),ope, n,N):\nPOL1:=gu[1]:\nSUMMAND1:=gu[2]:\ndenFAC:=gu[3]:\nyakhas:=simplif y1(SUMMAND1,k,-1):\nyakhas:=normal(1/yakhas):\n \nP1:=1:\nQ:=numer(yak has):\nR:=denom(yakhas):\n \n \nres1:=findgQR(Q,R,k,100):\n \nwhile re s1[2]<>0 do\n g:=res1[1]:\n hakhi:=res1[2]:\n Q:=normal(Q/g):\n R:=nor mal(R/subs(k=k-hakhi,g)):\n P1:=P1*product(subs(k=k-i1,g),i1=0..hakhi- 1):\n res1:=findgQR(Q,R,k,100):\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):\nl1:=d egree(A1,k):\nif B1=0 then\nl2:=-100:\nelse\nl2:=degree(B1,k):\nfi:\nm eka1:=coeff(A1,k,l1):\nmekb1:=coeff(B1,k,l2):\nif l1<=l2 \nthen\nk1:=d egree(P,k)-l2:\nelif l2=l1-1 \nthen\n mumu:= (-2)*mekb1/meka1:\n if t ype(mumu,integer) \n then\n k1:=max(mumu, degree(P,k)-l1+1):\n els e\n k1:=degree(P,k)-l1+1:\n fi:\nelif l2 < l1-1\nthen\n k1:= max( 0 , degree(P,k)-l1+1 ):\nfi:\nfu:=0:\n \n \nif k1 < 0 then\nRETURN(0):\n fi:\nif k1 >= 0 then\nfor ia1 from 0 to k1 do\nfu:=fu+apc[ia1]*k^ia1: \nod:\ngugu:=subs(k=k+1,Q)*fu-R*subs(k=k-1,fu)-P:\ngugu:=expand(gugu): \ndegg:=degree(gugu,k):\n \nfor ia1 from 0 to degg do\neq[ia1+1]:=coef f(gugu,k,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:=con vert(eq,set):\nva1:=convert(va1,set):\nva2:=convert(va2,set):\nva1:=va 1 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 from 0 to ORDER do\ngu:=gu union \{bpc[i1 ]=1\}:\nod:\nfu:=subs(gu,fu):\nope:=subs(gu,ope):\n \nope:=pashet(ope, N):\nope1:=ope*denom(ope):\n \nSDZ:=denom(ope)*subs(k=k+1,Q)*fu/P1/den FAC :\nSDZ1:=subs(k=k-1,SDZ)*simplify1(convert(SUMMAND,factorial),k,-1 ):\nSDZ1:=simplify(SDZ1):\n \nope1a:=0:\nfor i from 0 to degree(ope1,N ) do\nope1a:=ope1a+sort(coeff(ope1,N,i)*N^i):\nod:\n \nRETURN(ope1a,SD Z1):\nend:\n \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 1233 "paper:=proc(SUMMAND,k,n,N,o pe1,SDZ1,NAME,REF)\nlocal SHEM,IDENTITY,RECURRENCE:\n \nif degree(ope1 ,N)=1 then\nSHEM:=IDENTITY:\nelse\nSHEM:=RECURRENCE:\nfi:\nprintf(\"\" ):\nlprintf(\"A PROOF OF THE`,NAME,SHEM):\nprintf(\"\"):\nlprintf(\"By Shalosh B. Ekhad, Temple University, ekhad@math.temple.edu\"):\nprint f(\"\"):\nlprintf(\"I will give a short proof of the following result( `,REF,\").\"):\nprintf(\"\"):\nif degree(ope1,N)=1 then\nlprintf(\"(No te that since the recurrence below is first order, this\"):\nlprintf( \"means that the sum`, SUM(n), `has closed form,and it is\"):\nlprintf (\"easily seen to be equivalent.)\"):\nprintf(\"\"):\nfi:\nlprintf(\"T heorem:Let`, F(n,k), `be given by\"):\nprintf(\"\"):\nprint(SUMMAND): \nprintf(\"\"):\nlprintf(\"and let`, SUM(n),`be the sum of`, F(n,k),` \+ with\"):\nlprintf(\"respect to`, k,`.\"):\nprintf(\"\"):\nlprint(SUM(n ),` satisfies the following linear recurrence equation\"):\nprintf(\" \"):\nprint(yafe(ope1,N,n,SUM(n))):\nprintf(\"=0.\"):\nprintf(\"\"):\n lprintf(\"PROOF: We cleverly construct`, G(n,k),`:=\"):\nprintf(\"\"): \nprint(SDZ1*SUMMAND):\nlprintf(\"with the motive that\"):\nprintf(\" \"):\nprint(yafe(ope1,N,n,F(n,k))):\nlprintf(\"=`,G(n,k+1)-G(n,k), `(c heck!)\"):\nprintf(\"\"):\nlprintf(\"and the theorem follows upon summ ing with respect to`, k,`.QED.\"):\nend:\n \n \n \n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 818 "paper3:=proc(SUMMAND,k,n,N,ope1,SDZ1):\n \+ \nprintf(\"\"):\nlprintf(\"A PROOF OF A RECURRENCE\"):\nprintf(\"\"): \nlprintf(\"By Shalosh B. Ekhad, Temple University, ekhad@math.temple. edu\"):\nprintf(\"\"):\n \nlprintf(\"Theorem:Let`, F(n,k), `be given b y\"):\nprintf(\"\"):\nprint(SUMMAND):\nprintf(\"\"):\nlprintf(\"and le t`, SUM(n),`be the sum of`, F(n,k),` with\"):\nlprintf(\"respect to`, \+ k,`.\"):\nprintf(\"\"):\nlprint(SUM(n),` satisfies the following linea r recurrence equation\"):\nprintf(\"\"):\nprint(yafe(ope1,N,n,SUM(n))) :\nprintf(\"=0.\"):\nprintf(\"\"):\nlprintf(\"PROOF: We cleverly const ruct`, G(n,k),`:=\"):\nprintf(\"\"):\nprint(SDZ1*SUMMAND):\nlprintf(\" with the motive that\"):\nprintf(\"\"):\nprint(yafe(ope1,N,n,F(n,k))): \nlprintf(\"=`,G(n,k+1)-G(n,k), `(check!)\"):\nprintf(\"\"):\nlprintf( \"and the theorem follows upon summing with respect to`, k,`.QED.\"): \nend:\n \n \n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 966 "paperc:=p roc(INTEGRAND,y,x,D,ope1,SDZ1):\nprintf(\"\"):\nprintf(\"\"):\nlprintf (\"A PROOF OF A DIFFERENTIAL EQUATION SATISFIED BY AN INTEGRAL\"):\npr intf(\"\"):\nlprintf(\"By Shalosh B. Ekhad, Temple University, ekhad@m ath.temple.edu\"):\nprintf(\"\"):\nlprintf(\"I will give a short proof of the following result.\"):\nprintf(\"\"):\n \nlprintf(\"Theorem:Let `, F(x,y), `be given by\"):\nprintf(\"\"):\nprint(INTEGRAND):\nprintf( \"\"):\nlprintf(\"and let`, INTEGRAL(x),`be the integral of`, F(x,y),` with\"):\nlprintf(\"respect to`, y,`.\"):\nprintf(\"\"):\nlprint(INTE GRAL(x),` satisfies the following linear differential equation\"):\npr intf(\"\"):\nprint(yafec(ope1,D,x,INTEGRAL(x))):\nprintf(\"=0.\"):\npr intf(\"\"):\nlprintf(\"PROOF: We cleverly construct`, G(x,y),`:=\"):\n printf(\"\"):\nprint(SDZ1*INTEGRAND):\nlprintf(\"with the motive that \"):\nprintf(\"\"):\nprint(yafec(ope1,D,x,F(x,y))):\nlprintf(\"=`,diff (G(x,y),y), `(check!)\"):\nprintf(\"\"):\nlprintf(\"and the theorem fo llows upon integrating with respect to`, y,`.\"):\nend:\n \n" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 940 "paperd:=proc(INTEGRAND,x,n,N ,ope1,SDZ1):\nprintf(\"\"):\nlprintf(\"A PROOF OF A LINEAR RECURRENCE \+ SATISFIED BY AN INTEGRAL\"):\nprintf(\"\"):\nlprintf(\"By Shalosh B. E khad, Temple University, ekhad@math.temple.edu\"):\nprintf(\"\"):\nlpr intf(\"I will give a short proof of the following result.\"):\nprintf( \"\"):\n \nlprintf(\"Theorem:Let`, F(n,x), `be given by\"):\nprintf(\" \"):\nprint(INTEGRAND):\nprintf(\"\"):\nlprintf(\"and let`, INTEGRAL(n ),`be the integral of`, F(n,x),` with\"):\nlprintf(\"respect to`, x,`. \"):\nprintf(\"\"):\nlprint(INTEGRAL(n),` satisfies the following line ar recurrence equation\"):\nprintf(\"\"):\nprint(yafe(ope1,N,n,INTEGRA L(n))):\nprintf(\"=0.\"):\nprintf(\"\"):\nlprintf(\"PROOF: We cleverly construct`, G(n,x),`:=\"):\nprintf(\"\"):\nprint(SDZ1*INTEGRAND):\nlp rintf(\"with the motive that\"):\nprintf(\"\"):\nprint(yafe(ope1,N,n,F (n,x))):\nlprintf(\"=`,diff(G(n,x),x)):\nprintf(\"\"):\nlprintf(\"and \+ the theorem follows upon integrating with respect to`, x,`.QED.\"):\ne nd:\n \n \n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 950 "paperc:=proc (INTEGRAND,y,x,D,ope1,SDZ1):\nlprintf(\"A PROOF OF A DIFFERENTIAL EQUA TION SATISFIED BY AN INTEGRAL\"):\nprintf(\"\"):\nlprintf(\"By Shalosh B. Ekhad, Temple University, ekhad@math.temple.edu\"):\nprintf(\"\"): \nlprintf(\"I will give a short proof of the following result.\"):\npr intf(\"\"):\n \nlprintf(\"Theorem:Let`, F(x,y), `be given by\"):\nprin tf(\"\"):\nprint(INTEGRAND):\nprintf(\"\"):\nlprintf(\"and let`, INTEG RAL(x),`be the integral of`, F(x,y),` with\"):\nlprintf(\"respect to`, y,`.\"):\nprintf(\"\"):\nlprint(INTEGRAL(x),` satisfies the following linear differential equation\"):\nprintf(\"\"):\nprint(yafec(ope1,D,x ,INTEGRAL(x))):\nprintf(\"=0.\"):\nprintf(\"\"):\nlprintf(\"PROOF: We \+ cleverly construct`, G(x,y),`:=\"):\nprintf(\"\"):\nprint(SDZ1*INTEGRA ND):\nlprintf(\"with the motive that\"):\nprintf(\"\"):\nprint(yafec(o pe1,D,x,F(x,y))):\nlprintf(\"=`,diff(G(x,y),y), `(check!)\"):\nprintf( \"\"):\nlprintf(\"and the theorem follows upon integrating with respec t to`, y,`.QED.\"):\nend:\n \n \n \n" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 8722 "zeillim:=proc(SUMMAND,k,n,N,alpha,beta)\nlocal ope ,CERT,lu,k1,i,gu,lu1,lu2:\n \ngu:=Zeillim(SUMMAND,k,n,N,alpha+1,beta+1 ):\nope:=gu[1]:\nCERT:=gu[2]:\nlu:=gu[3]: \nlu1:=subs(k=alpha,SUMMAND) +subs(k=n-beta,SUMMAND):\nlu1:=simplify(lu1):\nlu1:=normal(lu1):\nlu2: =0:\nfor i from 0 to degree(ope,N) do\n lu2:=lu2+coeff(ope,N,i)*simp lify(subs(n=n+i,lu1)):\n od:\n lu2:=normal(lu2):\nope,CERT,normal(expa nd(normal(simplify(expand(normal(lu+lu2)))))):\n \nend:\n \nZeillim:=p roc(SUMMAND,k,n,N,alpha,beta)\nlocal ope,CERT,lu,k1,i,gu:\n \ngu:=zeil (SUMMAND,k,n,N):\nope:=gu[1]:\nCERT:=gu[2]:\n \nlu:=simplify(subs(k=n- beta+1,CERT)*subs(k=n-beta+1,SUMMAND))\n-simplify(subs(k=alpha,CERT)*s ubs(k=alpha,SUMMAND)):\n \nfor i from 0 to degree(ope,N) do\n for k1 f rom 1 to i do\n lu:=lu+coeff(ope,N,i)*simplify(subs(\{n=n+i,k=n-beta +k1\},SUMMAND)):\n od:\nod:\n \ngu,normal(expand(lu)):\n \nend:\n \n \+ \n \nzeilpap3:=proc(SUMMAND,k,n)\nlocal SDZ1, gu, ope1,N:\n \ngu:=zei l4(SUMMAND,k,n,N):\nope1:=gu[1]:\nSDZ1:=gu[2]:\npaper3(SUMMAND,k,n,N,o pe1,SDZ1):\nend:\n \n \n \n \nzeilpap5:=proc(SUMMAND,k,n,NAME,REF)\nlo cal SDZ1, gu, ope1,N:\n \ngu:=zeil4(SUMMAND,k,n,N):\nope1:=gu[1]:\nSDZ 1:=gu[2]:\npaper(SUMMAND,k,n,N,ope1,SDZ1,NAME,REF):\nend:\n \n zeilpap :=proc()\n \nif nargs=5 then\nzeilpap5(args):\nelif nargs=3 then\nzeil pap3(args):\nelse\n ERROR(`zeilpap(SUMMAND,k,n,NAME,REF) or zeilpap(S UMMAND,k,n)\"):\nfi:\nend:\n \n \n \n \nAZpapc:=proc(INTEGRAND,y,x)\nl ocal D,SDZ1,gu,ope1:\ngu:=AZc(INTEGRAND,y,x,D):\nope1:=gu[1]:\nSDZ1:=g u[2]:\npaperc(INTEGRAND,y,x,D,ope1,SDZ1):\nend:\n \n \n \n \nAZpapd:=p roc(INTEGRAND,x,n)\nlocal D,SDZ1, gu, ope1:\ngu:=AZd(INTEGRAND,x,n,D): \nope1:=gu[1]:\nSDZ1:=gu[2]:\npaperd(INTEGRAND,x,n,D,ope1,SDZ1):\nend: \n \n \n \n \n \ngoremd:=proc(N,ORDER)\nlocal i, gu:\ngu:=0:\nfor i fr om 0 to ORDER do\ngu:=gu+(bpc[i])*N^i:\nod:\nRETURN(gu):\nend:\n \ngor emc:=proc(D,ORDER)\nlocal i,gu:\ngu:=0:\nfor i from 0 to ORDER do\ngu: =gu+(bpc[i])*D^i:\nod:\nRETURN(gu):\nend:\n \ngzor:=proc(f,x,n)\nlocal i,gu:\ngu:=f:\nfor i from 1 to n do\ngu:=diff(gu,x):\nod:\nRETURN(gu) :\nend:\n \ngzor1:=proc(a,D,x)\nlocal i,gu:\ngu:=0:\nfor i from 0 to d egree(a,D) do\ngu:=gu+diff(coeff(a,D,i),x)*D^i+coeff(a,D,i)*D^(i+1):\n od:\nend:\n \npashetd:=proc(p,N)\nlocal gu,p1,mekh,i:\np1:=normal(p): \nmekh:=denom(p1):\np1:=numer(p1):\np1:=expand(p1):\ngu:=0:\nfor i fro m 0 to degree(p1,N) do\ngu:=gu+factor(coeff(p1,N,i))*N^i:\nod:\nRETURN (gu,mekh):\nend:\n \npashetc:=proc(p,D)\nlocal gu,p1,i,mekh:\np1:=norm al(p):\nmekh:=denom(p1):\np1:=numer(p1):\np1:=expand(p1):\ngu:=0:\nfor i from 0 to degree(p1,D) do\ngu:=gu+factor(coeff(p1,D,i))*D^i:\nod:\n RETURN(gu,mekh):\nend:\n \n \nAZd:= proc(A,y,n,N)\nlocal ORDER,gu,KAMA :\nKAMA:=8:\n \nfor ORDER from 0 to KAMA do\ngu:=duisd(A,ORDER,y,n,N): \nif gu<>0 then\n RETURN(gu):\nfi:\nod:\n0:\nend:\n \n \nAZc:=proc(A,y ,x,D)\nlocal ORDER,gu,KAMA:\nKAMA:=8:\n \nfor ORDER from 0 to KAMA do \ngu:=duisc(A,ORDER,y,x,D):\nif gu<>0 then\n RETURN(gu):\nfi:\nod:\n0: \nend:\n \n \nduisd:= proc(A,ORDER,y,n,N)\nlocal gorem, conj, yakhas,l u1a,LU1,P,Q,R,S,j1,g,Q1,Q2,l1,l2,mumu,\nmekb1,fu,meka1,k1,gugu,eq,ia1, va1,va2,degg,shad,KAMA,i1:\nKAMA:=10:\ngorem:=goremd(N,ORDER):\n \n \n conj:=gorem:\nyakhas:=0:\n \nfor i1 from 0 to degree(conj,N) do\n \nya khas:=yakhas+coeff(conj,N,i1)*simplify(subs(n=n+i1,A)/A):\nod:\n \nlu1 a:=yakhas:\nLU1:=numer(yakhas):\nyakhas:=1/denom(yakhas):\nyakhas:=nor mal(diff(yakhas,y)/yakhas+diff(A,y)/A):\nP:=LU1:\nQ:=numer(yakhas):\nR :=denom(yakhas):\nj1:=0:\nwhile j1 <= KAMA do\ng:=gcd(R,Q-j1*diff(R,y) ):\nif g <> 1 then\nQ2:=(Q-j1*diff(R,y))/g:\nR:=normal(R/g):\nQ:=norma l(Q2+j1*diff(R,y)):\nP:=P*g^j1:\nj1:=-1:\nfi:\nj1:=j1+1:\nod:\nP:=expa nd(P):\nR:=expand(R):\nQ:=expand(Q):\nQ1:=Q+diff(R,y):\nQ1:=expand(Q1) :\nl1:=degree(R,y):\nl2:=degree(Q1,y):\nmeka1:=coeff(R,y,l1):\nmekb1:= coeff(Q1,y,l2):\nif l1-1 <>l2 \nthen\nk1:=degree(P,y)-max(l1-1,l2):\ne lse\n mumu:= -mekb1/meka1:\n if type(mumu,integer) and mumu > 0 \n t hen\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:\ngu gu:=expand(gugu):\ndegg:=degree(gugu,y):\nfor ia1 from 0 to degg do\ne q[ia1+1]:=coeff(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:=convert(eq,set):\nva1:=convert(va1,set):\nva2:=convert(va2, set):\nva1:=va1 union va2 :\nva1:=solve1(eq,va1):\nfu:=subs(va1,fu):\n gorem:=subs(va1,gorem):\nif fu=0 and gorem=0 then\n RETURN(0):\nfi:\nf or ia1 from 0 to k1 do\ngorem:=subs(apc[ia1]=1,gorem):\nod:\nfor ia1 f rom 0 to ORDER do\ngorem:=subs(bpc[ia1]=1,gorem):\nod:\nfu:=normal(fu) :\n \n \nshad:=pashetd(gorem,N):\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(a pc[ia1]=1,S):\nod:\nfor ia1 from 0 to ORDER do\nS:=subs(bpc[ia1]=1,S): \nod:\n \nRETURN(shad[1],S):\nend:\n \n \nduisc:= proc(A,ORDER,y,x,D) \nlocal KAMA,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,i,shad:\n \nKAMA:=10:\ng orem:=goremc(D,ORDER):\n \nconj:=gorem:\nyakhas:=0:\n \nfor i from 0 t o degree(conj,D) do\nyakhas:=yakhas+normal(coeff(conj,D,i)*gzor(A,x,i) /A):\nyakhas:=normal(yakhas):\nod:\n \nlu1a:=yakhas:\nLU1:=numer(yakha s):\nyakhas:=1/denom(yakhas):\nyakhas:=normal(diff(yakhas,y)/yakhas+di ff(A,y)/A):\nP:=LU1:\nQ:=numer(yakhas):\nR:=denom(yakhas):\nj1:=0:\nwh ile j1 <= KAMA do\ng:=gcd(R,Q-j1*diff(R,y)):\nif g <> 1 then\nQ2:=(Q-j 1*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):\nR:=expand(R):\nQ:=ex pand(Q):\nQ1:=Q+diff(R,y):\nQ1:=expand(Q1):\nl1:=degree(R,y):\nl2:=deg ree(Q1,y):\nmeka1:=coeff(R,y,l1):\nmekb1:=coeff(Q1,y,l2):\nif l1-1 <>l 2 \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\nfu:=fu+apc[ia1]*y^ ia1:\nod:\ngugu:=Q1*fu+R*diff(fu,y)-P:\ngugu:=expand(gugu):\ndegg:=deg ree(gugu,y):\nfor ia1 from 0 to degg do\neq[ia1+1]:=coeff(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:=convert(eq,set):\n va1:=convert(va1,set):\nva2:=convert(va2,set):\nva1:=va1 union va2 :\n va1:=solve1(eq,va1):\nfu:=subs(va1,fu):\ngorem:=subs(va1,gorem):\nif f u=0 and gorem=0 then\n RETURN(0):\nfi:\nfor ia1 from 0 to k1 do\ngorem :=subs(apc[ia1]=1,gorem):\nod:\nfor ia1 from 0 to ORDER do\ngorem:=sub s(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 \n \n \n bdokcertc:=proc(A,y,x,D,ope,S)\nlocal gu,i:\ngu:=0:\n \nfor i from 0 t o degree(ope,D) do\ngu:=gu+coeff(ope,D,i)*simplify(gzor(A,x,i)/A):\ngu :=normal(gu):\nod:\n \ngu:=gu/simplify(diff(S*A,y)/A):\nnormal(gu);\ne nd:\n \n \nbdokcertd:=proc(A,y,n,N,ope,S)\nlocal gu,i:\ngu:=0:\n \nfor i from 0 to degree(ope,N) do\ngu:=gu+coeff(ope,N,i)*simplify( subs(n= n+i,A)/A):\ngu:=normal(gu):\nod:\n \ngu:=gu/simplify(diff(S*A,y)/A):\n normal(gu);\nend:\n \n \n \nbdokcto:=proc(SUMMAND1,ORDER,k,n,N)\nlocal mu,gu,i,G1,ope,lu,SUMMAND:\n \nSUMMAND:=convert(SUMMAND1,factorial): \nmu:=ct(SUMMAND,ORDER,k,n,N):\n \nif mu=0 then\n RETURN(0):\nfi:\n \n ope:=mu[1]:\nG1:=mu[2]*SUMMAND:\ngu:=0:\n \nfor i from 0 to degree(ope ,N) do\ngu:=gu+coeff(ope,N,i)*simplify( subs(n=n+i,SUMMAND)/SUMMAND): \ngu:=normal(gu):\nod:\n \nlu:=simplify(subs(k=k+1,G1)/SUMMAND)-mu[2]: \nlu:=normal(lu):\nnormal(gu/lu);\nend:\n \n \n \n \nbdokct:=proc(SUMM AND1,ORDER,k,n,N)\nlocal mu,gu,i,G1,ope,lu,SUMMAND:\n \nSUMMAND:=conve rt(SUMMAND1,factorial):\nmu:=ct(SUMMAND,ORDER,k,n,N):\n \nif mu=0 then \n RETURN(0):\nfi:\n \nope:=mu[1]:\nG1:=mu[2]*SUMMAND:\ngu:=0:\n \nfor i from 0 to degree(ope,N) do\ngu:=gu+coeff(ope,N,i)*simplify1(SUMMAND ,n,i):\ngu:=normal(gu):\nod:\n \nlu:=subs(k=k+1,mu[2])*simplify1(SUMMA ND,k,1)-mu[2]:\nlu:=normal(lu):\nnormal(gu/lu);\nend:\n \n \nfindrec:= proc(f,DEGREE,ORDER,n,N)\nlocal ope,var,eq,a,i,j,n0,kv,var1,eq1,mu:\ni f (1+DEGREE)*(1+ORDER)+2+ORDER>nops(f) then\nERROR(`Insufficient data \+ for a recurrence of order`,ORDER, `degree`,DEGREE):\nfi:\nope:=0:\nvar :=\{\}:\n \nfor i from 0 to ORDER do\n for j from 0 to DEGREE do\n op e:=ope+a[i,j]*n^j*N^i:\n var:=var union \{a[i,j]\}:\n od:\nod:\n \+ \neq:=\{\}:\n \nfor n0 from 1 to (1+DEGREE)*(1+ORDER)+2 do\n eq1:=0: \n \n for i from 0 to ORDER do\n eq1:=eq1+subs(n=n0,coeff(ope,N,i ))*op(n0+i,f):\n od:\n \n eq:= eq union \{eq1\}:\nod:\n \nvar1:=sol ve1(eq,var):\n \nkv:=\{\}:\n \nfor i from 1 to nops(var1) do\n mu:=op( i,var1):\n \nif op(1,mu)=op(2,mu) then\n kv:= kv union \{op(1,mu)\}: \n fi:\n \nod:\nope:=subs(var1,ope):\n \n \nif nops(kv)>1 then\n prin tf(\" either DEGREE or ORDER are too high\"):\n printf(\"The output \+ is not the minimal possible operator\"):\nfi:\n \nfor i from 1 to nops (kv) do\n ope:=subs(op(i,kv)=1,ope):\nod:\n \nope:\n \nend:\n \+ \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " \n \n \n \n \n" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 258 "tmp:='tmp': tmp:= sort(conve rt(\{anames()\} minus \n \{'tmp',celine,'ever_load',ezra,`ezra/celine` ,'`pldx/Initialized`', '`plots/Initialized`',\n 'tmp_anames',fprintf, printf,'lasterror','`type/interfaceargs`','lastexception',try_celine\} minus tmp_anames, list));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "12 0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }