{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "terminal" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Diagnostic" 7 9 1 {CSTYLE "" -1 -1 "" 0 1 64 128 64 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica " 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 48 "/home/m262f99/KOEPF/works heetsV.4/gosperdemo.mws" }{MPLTEXT 1 0 0 "" }}{PARA 258 "" 0 "" {TEXT 257 59 "Math 262a, Fall 1999, Glenn Tesler\nGosper Sum demo\n10/17/99 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "This tells Maple to print out extra info \+ as it does the computation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infol evel[sum] := 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%$s umG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sum(k^2,k);" }} {PARA 6 "" 1 "" {TEXT -1 82 "sum/indefnew: indefinite summation\nsum /indefnew: indefinite summation finished" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"kG\"\"$#\"\"\"F&*$F%\"\"##!\"\"F*F%#F(\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(sumtools);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,%)HypersumG%+SumtohyperG%0extended_gosperG%'g osperG%/hyperrecursionG%)hypersumG%*hypertermG%)simpcombG%-sumrecursio nG%+sumtohyperG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Koepf wrote th is routine that comes with Maple V.4, so the notation agrees with his \+ book." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sumtools[gosper](k^2,k);" }}{PARA 6 "" 1 "" {TEXT -1 369 "sumtools[gosper] a( k )/a( k \+ -1):= k^2/(k-1)^2\nsumtools[gosper] Gosper's algorithm applicabl e\nsumtools[gosper] p:= k^2\nsumtools[gosper] q:= 1\nsumtools[ gosper] r:= 1\nsumtools[gosper] degreebound:= 3\nsumtools[gosp er] solving equations to find f\nsumtools[gosper] Gosper's algorit hm successful\nsumtools[gosper] f:= 1/6*k*(k+1)*(2*k+1)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"kG\"\"\"!\"\"F'F'F&F',&F&\"\"#F(F'F '#F'\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sumtools[gospe r](k!,k);" }}{PARA 6 "" 1 "" {TEXT -1 293 "sumtools[gosper] a( k \+ )/a( k -1):= k\nsumtools[gosper] Gosper's algorithm applicab le\nsumtools[gosper] p:= 1\nsumtools[gosper] q:= k\nsumtools[g osper] r:= 1\nsumtools[gosper] degreebound:= -1\nsumtools[gosp er] Gosper's algorithm: no closed form antidifference exists" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sumtools[gosper](k*k!,k);" }}{PARA 6 "" 1 "" {TEXT -1 347 "sumtools[gosper] a( k )/a( k -1):= k^2/(k-1)\nsum tools[gosper] Gosper's algorithm applicable\nsumtools[gosper] p:= \+ k\nsumtools[gosper] q:= k\nsumtools[gosper] r:= 1\nsumtools[ gosper] degreebound:= 0\nsumtools[gosper] solving equations to f ind f\nsumtools[gosper] Gosper's algorithm successful\nsumtools[gosp er] f:= 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*factorialG6#%\"kG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sumtools[gosper](binomi al(n,k),k);" }}{PARA 6 "" 1 "" {TEXT -1 307 "sumtools[gosper] a( k )/a( k -1):= -(-n+k-1)/k\nsumtools[gosper] Gosper's algori thm applicable\nsumtools[gosper] p:= 1\nsumtools[gosper] q:= n -k+1\nsumtools[gosper] r:= k\nsumtools[gosper] degreebound:= - 1\nsumtools[gosper] Gosper's algorithm: no closed form antidifferenc e exists" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sumtools[gosper](binomial(n,k)*(-1)^k,k); " }}{PARA 6 "" 1 "" {TEXT -1 356 "sumtools[gosper] a( k )/a( k -1):= (-n+k-1)/k\nsumtools[gosper] Gosper's algorithm applicab le\nsumtools[gosper] p:= 1\nsumtools[gosper] q:= -n+k-1\nsumto ols[gosper] r:= k\nsumtools[gosper] degreebound:= 0\nsumtools[ gosper] solving equations to find f\nsumtools[gosper] Gosper's alg orithm successful\nsumtools[gosper] f:= -1/n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"kG\"\"\"%\"nG!\"\"-%)binomialG6$F'F%F&)F(F%F&F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sumtools[gosper]((4*k+1 )*k!/(2*k+1)!,k);" }}{PARA 6 "" 1 "" {TEXT -1 374 "sumtools[gosper] \+ a( k )/a( k -1):= 1/2*(4*k+1)/(4*k-3)/(2*k+1)\nsumtools[gos per] Gosper's algorithm applicable\nsumtools[gosper] p:= 4*k+1\n sumtools[gosper] q:= 1\nsumtools[gosper] r:= 4*k+2\nsumtools[g osper] degreebound:= 0\nsumtools[gosper] solving equations to fi nd f\nsumtools[gosper] Gosper's algorithm successful\nsumtools[gospe r] f:= -1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"kG\"\"#\"\" \"F(F(-%*factorialG6#F&F(-F*6#F%!\"\"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Attempt the inverse problem: verify that this function at k minus at k-1 gives the summand." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sumk := k -> -k*binomial(n,k)*(-1)^k/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sumkG:6#%\"kG6\"6$%)operatorG%&arrowGF(,$**9$\"\"\"- %)binomialG6$%\"nGF.F/)!\"\"F.F/F3F5F5F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "dsumk := simplify(sumk(k)-sumk(k-1));" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%&dsumkG,$*()!\"\"%\"kG\"\"\",(*&F)F*-%)binomia lG6$%\"nGF)F*F**&-F.6$F0,&F)F*F(F*F*F)F*F*F2F(F*F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(**%\"kG\"\"\"%\"nG!\"\"-%)binomialG6$F'F%F&)F(F%F&F(* ,F,F&F'F(F%\"\"#,(F'F&F%F(F&F&F(F)F&F(*,F,F&F'F(F%F&F/F(F)F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#**%\"kG\"\"\"-%)binomialG6$%\"nGF$F%)!\"\"F$F%,( F)F+F$F%F+F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"kG\"\"\"-%)binomialG6$%\" nGF$F%)!\"\"F$F%,(F)F+F$F%F+F%F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Maple is stubborn, many commands may have to be used to get it to \+ the form we want." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "convert(\",GAM MA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%\"kG\"\"\"-%&GAMMAG6#,&%\"n GF%F%F%F%-F'6#,&F$F%F%F%!\"\"-F'6#,(F*F%F$F.F%F%F.)F.F$F%,(F*F.F$F%F.F %F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "combine(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*.%\"kG\"\"\"-%&GAMMAG6#,&%\"nGF%F%F%F %-F'6#,&F$F%F%F%!\"\"-F'6#,(F*F%F$F.F%F%F.)F.F$F%,(F*F.F$F%F.F%F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$**)!\"\"%\"kG\"\"\"-%&GAMMAG6#,(%\"nGF(F'F&\" \"#F(F&-F*6#F'F&-F*6#,&F-F(F(F(F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert(\",binomial);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)!\"\"%\"kG\"\"\"-%)binomialG6$%\"nG,&F'F(F&F(F(F&" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 66 "================================= =================================" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sumtool s[gosper](k!/k^2,k);" }}{PARA 6 "" 1 "" {TEXT -1 309 "sumtools[gosper] a( k )/a( k -1):= 1/k*(k-1)^2\nsumtools[gosper] Gosper 's algorithm applicable\nsumtools[gosper] p:= 1\nsumtools[gosper] \+ q:= (k-1)^2\nsumtools[gosper] r:= k\nsumtools[gosper] degree bound:= -2\nsumtools[gosper] Gosper's algorithm: no closed form an tidifference exists" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sumtools[gosper]((k-3)*k!,k) ;" }}{PARA 6 "" 1 "" {TEXT -1 353 "sumtools[gosper] a( k )/a( \+ k -1):= (k-3)/(k-4)*k\nsumtools[gosper] Gosper's algorithm appl icable\nsumtools[gosper] p:= k-3\nsumtools[gosper] q:= k\nsumt ools[gosper] r:= 1\nsumtools[gosper] degreebound:= 0\nsumtools [gosper] solving equations to find f\nsumtools[gosper] Gosper's al gorithm: no closed form antidifference exists" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 "A rand omly chosen summand as above will not always be gosper-summable, but i f we start with a hypergeometric sum s(k) and compute the difference a (k)=s(k+1)-a(k), then s(k) = const + sum up to k-1 of a(k), so we're g uaranteed success." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sk := (k^2+2* k+5)*(2*k+3)!/(3*k+4)!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#skG*(,(* $%\"kG\"\"#\"\"\"F(F)\"\"&F*F*-%*factorialG6#,&F(F)\"\"$F*F*-F-6#,&F(F 0\"\"%F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ak := simp lify(subs(k=k+1,sk)-sk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#akG,$** ,,*$%\"kG\"\"$\"$e\"*$F)\"\"#\"$I%F)\"$y'\"$X%\"\"\"*$F)\"\"%\"#FF1,&F )F*\"\"(F1!\"\"-%&GAMMAG6#,&F)F*\"\"'F1F7-F96#,&F)F-F3F1F1#F7F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sumtools[gosper](ak,k);" }} {PARA 6 "" 1 "" {TEXT -1 493 "sumtools[gosper] a( k )/a( k - 1):= 2/3*(2*k+3)*(158*k^3+430*k^2+678*k+445+27*k^4)/(50*k^3+118*k^2 +184*k+66+27*k^4)/(3*k+5)/(3*k+7)\nsumtools[gosper] Gosper's algorit hm applicable\nsumtools[gosper] p:= 158*k^3+430*k^2+678*k+445+27*k ^4\nsumtools[gosper] q:= 4*k+6\nsumtools[gosper] r:= 3*(3*k+7) *(3*k+5)\nsumtools[gosper] degreebound:= 2\nsumtools[gosper] sol ving equations to find f\nsumtools[gosper] Gosper's algorithm succes sful\nsumtools[gosper] f:= -8-4*k-k^2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"kG\"\"$\"\"&\"\"\"F(,(*$F%\"\"#F(F%F+F'F(F(-%&GA MMAG6#,&F%F&\"\"'F(!\"\"-F-6#,&F%F+\"\"%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sum(ak,k);" }}{PARA 6 "" 1 "" {TEXT -1 693 "sum/ indefnew: indefinite summation\nsum/extgosper: applying Gosper alg orithm to a( k ):= -1/3*(158*k^3+430*k^2+678*k+445+27*k^4)/(3*k+ 7)/GAMMA(3*k+6)*GAMMA(2*k+4)\nsum/gospernew: a( k )/a( k -1) := 2/3*(2*k+3)*(158*k^3+430*k^2+678*k+445+27*k^4)/(50*k^3+118*k^2+1 84*k+66+27*k^4)/(3*k+5)/(3*k+7)\nsum/gospernew: Gosper's algorithm a pplicable\nsum/gospernew: p:= 158*k^3+430*k^2+678*k+445+27*k^4\nsu m/gospernew: q:= 4*k+6\nsum/gospernew: r:= 3*(3*k+7)*(3*k+5)\n sum/gospernew: degreebound:= 2\nsum/gospernew: solving equations to find f\nsum/gospernew: Gosper's algorithm successful\nsum/gosper new: f:= -8-4*k-k^2\nsum/indefnew: indefinite summation finished " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"kG\"\"$\"\"&\"\"\"F(,(*$F% \"\"#F(F%F+F'F(F(-%&GAMMAG6#,&F%F&\"\"'F(!\"\"-F-6#,&F%F+\"\"%F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sumtools[gosper](1/(k*(k+1) ),k);" }}{PARA 6 "" 1 "" {TEXT -1 356 "sumtools[gosper] a( k )/a ( k -1):= 1/(k+1)*(k-1)\nsumtools[gosper] Gosper's algorithm \+ applicable\nsumtools[gosper] p:= 1\nsumtools[gosper] q:= k-1\n sumtools[gosper] r:= k+1\nsumtools[gosper] degreebound:= 1\nsu mtools[gosper] solving equations to find f\nsumtools[gosper] Gospe r's algorithm successful\nsumtools[gosper] f:= -1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"kG!\"\"F&\"\"\"F'F%F&,&F%F'F'F'F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*$%\"kG!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "q version" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "read `qsum .mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OCopyright~1998,~~Harald~Bo eing~&~Wolfram~KoepfG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;Konrad-Zuse -Zentrum~BerlinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "qgosper ((-1)^k * q^binomial(k,2) * qbinomial(n,k,q),q,k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*,,&!\"\"\"\"\")%\"qG%\"kGF'F',&)F)%\"nGF'F&F'F&)F& F*F')F)-%)binomialG6$F*\"\"#F'-%*qbinomialG6%F-F*F)F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "qsk := \":" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "This means:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Sum(( -1)^t * q^binomial(t,2) * qbinomial(n,t,q),t=c..k-1)=qsk;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*()!\"\"%\"tG\"\"\")%\"qG-%)binomialG 6$F*\"\"#F+-%*qbinomialG6%%\"nGF*F-F+/F*;%\"cG,&%\"kGF+F)F+,$*,,&F)F+) F-F:F+F+,&)F-F5F+F)F+F))F)F:F+)F--F/6$F:F1F+-F36%F5F:F-F+F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Check it: make sure the difference in the supposed sum at k+1 & k is the original summand" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " qd:=qsimplify(subs(k=k+1,qsk)-qsk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%#qdG*0,&)%\"qG%\"nG\"\"\"!\"\"F*F*-%,qpochhammerG6%*$F'F+F(%\"kGF*) F(*&F)F*F0F*F*-F-6%)F(F0F(F)F*,&F+F*F5F*F+-F-6%F'F(F0F+-F-6%F(F(F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Fnk := (-1)^k * q^binomia l(k,2)*qbinomial(n,k,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FnkG*() !\"\"%\"kG\"\"\")%\"qG-%)binomialG6$F(\"\"#F)-%*qbinomialG6%%\"nGF(F+F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qsimplify(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%,qpochhammerG6%*$)%\"qG%\"nG!\"\"F )%\"kG\"\"\")F)*&F*F-F,F-F--F%6%F)F)F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "It represented them differently, so we can't tell by insp ection... so try again." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "qsimplif y(Fnk-qd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "What are the \"p,q,r\"? Call them \"P,Q,R\" in stead, due to the double use of Q:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "trace(qgosper);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(qgosperG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "qgosper(Fnk,q,k);" }}{PARA 9 "" 1 "" {TEXT -1 72 "\{--> enter qgosper, args = (-1)^k*q^binomial(k ,2)*qbinomial(n,k,q), q, k" }}{PARA 11 "" 1 "" {TEXT -1 51 "(other tra ce information deleted; K stands for q^k)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PQRG7%\"\"\",&%\"KGF&)%\"qG%\"nG!\"\",&F,F&*&F*F&F(F &F&" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 9 "" 1 "" {TEXT -1 96 "< -- exit qgosper (now at top level) = -(-1+q^k)/(q^n-1)*(-1)^k*q^binomi al(k,2)*qbinomial(n,k,q)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&! \"\"\"\"\")%\"qG%\"kGF'F',&)F)%\"nGF'F&F'F&)F&F*F')F)-%)binomialG6$F* \"\"#F'-%*qbinomialG6%F-F*F)F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "untrace(qgosper);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %(qgosperG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Now try the q-binomial theorem, series fo rm, truncated:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(qpochhammer(a ,q,k1)/qpochhammer(q,q,k1)*x^k1,k1=0..k-1)=`?`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(-%,qpochhammerG6%%\"aG%\"qG%#k1G\"\"\"-F)6% F,F,F-!\"\")%\"xGF-F./F-;\"\"!,&%\"kGF.F1F.%\"?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "qgosper(qpochhammer(a,q,k)/qpochhammer(q,q,k) *x^k,q,k);" }}{PARA 8 "" 1 "" {TEXT -1 62 "Error, (in qgosper) No q-hy pergeometric antidifference exists." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "So we can't t runcate the series. Can we do the infinite sum?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "Sum(qpochhammer(a,q,k)/qpochhammer(q,q,k)*x^k,k=0..i nfinity)=qpochhammer(a*x,q,infinity)/qpochhammer(x,q,infinity) * `?`; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(-%,qpochhammerG6%%\"aG %\"qG%\"kG\"\"\"-F)6%F,F,F-!\"\")%\"xGF-F./F-;\"\"!%)infinityG*(-F)6%* &F+F.F3F.F,F7F.-F)6%F3F,F7F1%\"?GF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "qkfreerec(qpochhammer(a,q,k)/qpochhammer(q,q,k)*x^k,q ,k,n,0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&%\"aG6$\"\"!\"\"\" F*-%\"FG6$%\"nG%\"kGF*!\"\"*&F&F*-F,6$,&F*F*F.F*F/F*F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Oops, there's no n! So try it for the sp ecial case a=q^n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "qgosper(qpochhammer(q^n,q,k) /qpochhammer(q,q,k)*x^k,q,k);" }}{PARA 8 "" 1 "" {TEXT -1 62 "Error, ( in qgosper) No q-hypergeometric antidifference exists." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "So we still can't do the indefinite trun cated sum. Use Sister Celine's algorithm to do the infinite sum inste ad." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "qkfreerec(qpochhammer(q^n,q,k)/qpochhammer(q,q,k),q ,k,n,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(&%\"aG6$\"\"\"F)F)) %\"qG%\"nGF)-%\"FG6$,&F)F)F,F)%\"kGF)!\"\"*&F&F)-F.6$F,,&F1F)F)F)F)F2* &F&F)-F.6$F0F6F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " qfasenmyer(qpochhammer(q^n,q,k)/qpochhammer(q,q,k)*x^k,q,k,s(n),1,1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*&%\"xG\"\"\")%\"qG%\"nGF)F )!\"\"F)F)-%\"sG6#,&F)F)F,F)F)F)-F/6#F,F)\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "rsolve(\",s(n));" }}{PARA 6 "" 1 "" {TEXT -1 331 "sum/indefnew: indefinite summation\nsum/extgosper: applying G osper algorithm to a( _n1 ):= ln(x*q^_n1-1)\nsum/gospernew: a( _n1 )/a( _n1 -1):= -ln(x*q^_n1-1)/(-ln(x*q^_n1-q)+ln(q))\n sum/gospernew: is not rational\nsum/gospernew: Gosper's algorithm \+ not applicable\nsum/indefnew: indefinite summation finished" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*()!\"\"%\"nG\"\"\"-%(productG6$*$,&*& %\"xGF')%\"qG%$_n1GF'F'F%F'F%/F1;\"\"!,&F&F'F%F'F'-%\"sG6#F4F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "(we never turned off involevel[sum ]:=3)" }}{PARA 0 "" 0 "" {TEXT -1 128 "The initial value s(0) has to b e computed separately. Completing this is nearly as hard as the \"hum anoid\" proof we gave before." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 }