{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 "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 45 "/home/m262f99/KOEPF/works heetsV.4/hw3ansm.mws" }{MPLTEXT 1 0 0 "" }}{PARA 258 "" 0 "" {TEXT 257 45 "Math 262a, Fall 1999, Glenn Tesler\nHomework 3" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "read `hsum.mpl`;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%ZCopyright~1998~~Wolfram~Koepf,~Konra d-Zuse-Zentrum~BerlinG" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf # 5.8(a)" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gospe r(1/(k*(k+10)),k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*8,6%\"kG\"(_J 0#\"'!)GO\"\"\"*$F&\"\"#\"(+\"=N*$F&\"\"$\"(?Z*G*$F&\"\"%\"(DmM\"*$F& \"\"&\"'Q'z$*$F&\"\"'\"&]h'*$F&\"\"(\"%gp*$F&\"\")\"$0%*$F&\"\"*\"#5F) ,&F&F)F@F)!\"\",&F&F)F=F)FC,&F&F)F:F)FC,&F&F)F7F)FC,&F&F)F4F)FC,&F&F)F 1F)FC,&F&F)F.F)FC,&F&F)F+F)FC,&F&F)F)F)FCF&FC#FCFA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Check it:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sk := \": simplify(subs(k=k+1,sk)-sk);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&%\"kG!\"\",&F$\"\"\"\"#5F'F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Conclusion: as an indefinite sum," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Sum(1/(k*(k+10)),k)=sk+C;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-% $SumG6$*&%\"kG!\"\",&F(\"\"\"\"#5F+F)F(,&*8,6F(\"(_J0#\"'!)GOF+*$F(\" \"#\"(+\"=N*$F(\"\"$\"(?Z*G*$F(\"\"%\"(DmM\"*$F(\"\"&\"'Q'z$*$F(\"\"' \"&]h'*$F(\"\"(\"%gp*$F(\"\")\"$0%*$F(\"\"*F,F+,&F(F+FHF+F),&F(F+FEF+F ),&F(F+FBF+F),&F(F+F?F+F),&F(F+F " 0 "" {MPLTEXT 1 0 55 "Sum(1 /(k*(k+10)),k=m..n-1) = subs(k=n,sk)-subs(k=m,sk);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/-%$SumG6$*&%\"kG!\"\",&F(\"\"\"\"#5F+F)/F(;%\"mG,&% \"nGF+F)F+,&*8,6F1\"(_J0#\"'!)GOF+*$F1\"\"#\"(+\"=N*$F1\"\"$\"(?Z*G*$F 1\"\"%\"(DmM\"*$F1\"\"&\"'Q'z$*$F1\"\"'\"&]h'*$F1\"\"(\"%gp*$F1\"\")\" $0%*$F1\"\"*F,F+,&F1F+FMF+F),&F1F+FJF+F),&F1F+FGF+F),&F1F+FDF+F),&F1F+ FAF+F),&F1F+F>F+F),&F1F+F;F+F),&F1F+F8F+F),&F1F+F+F+F)F1F)#F)F,*8,6F/F 5F6F+*$F/F8F9*$F/F;F<*$F/F>F?*$F/FAFB*$F/FDFE*$F/FGFH*$F/FJFK*$F/FMF,F +,&F/F+FMF+F),&F/F+FJF+F),&F/F+FGF+F),&F/F+FDF+F),&F/F+FAF+F),&F/F+F>F +F),&F/F+F;F+F),&F/F+F8F+F),&F/F+F+F+F)F/F)#F+F," }}}{EXCHG {PARA 3 " " 0 "" {TEXT -1 13 "Koepf #5.8(d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gosper(2^k*(k^3-3*k^2-3*k-1)/(k^3*(k+1)^3),k);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&)\"\"#%\"kG\"\"\"F&!\"$" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 13 "Koepf #5.8(i)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gospe r(binomial(m,k)/binomial(n,k),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* *,(%\"nG!\"\"%\"kG\"\"\"F&F(F(,(F%F(F(F(%\"mGF&F&-%)binomialG6$F*F'F(- F,6$F%F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 11 "Koepf #5.13" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 643 "# Rk2ak(Rk,k) gives the term ak from the certificate Rk, using initial term a(0)\n# Rk2ak(Rk,k,false) gives a product form ula for it instead of a closed form\n# Rk2ak(Rk,k,i) uses initial term a(i)\n# Rk2ak(Rk,k,true/false,i) does both\nRk2ak := proc(Rk,k)\n lo cal rat,ak,i,cflag,arg;\n i := 0; cflag := true;\n for arg in args[3 ..nargs] do\n if type(arg,boolean) then cflag := arg\n else i := arg\n fi\n od;\n\n # term ratio a(k+1)/a(k) = (1+R(k))/R(k+1)\n \+ rat := simplify((1+Rk) / subs(k=k+1,Rk));\n rat := subs(k=n,rat);\n \n if cflag then\n ak := a(i) * product(rat,n=i..k-1);\n else\n ak := a(i) * Product(rat,n=i..k-1);\n fi;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Rk2ak(3,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"aG6#\"\"!\"\"\")#\"\"%\"\"$%\"kGF(" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.13(a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Rk2ak(alpha/(alpha-1),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"aG6#\"\"!\"\"\")*&,&%&alphaG\"\"#!\"\"F(F(F,F.%\" kGF(" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.13(b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Rk2ak(k,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"aG6#\"\"!" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.1 3(c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Rk2ak(k^2,k);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*.-%\"aG6#\"\"!\"\"\"-%&GAMMAG6#,&%\"kGF(%\"IG! \"\"F(-F*6#,&F-F(F.F(F(-F*6#,&F-F(F(F(!\"#-F*6#,$F.F/F/-F*6#F.F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Rk2ak(k^2,k,false);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"aG6#\"\"!\"\"\"-%(ProductG6$*&,& F(F(*$%\"nG\"\"#F(F(,&F/F(F(F(!\"#/F/;F',&%\"kGF(!\"\"F(F(" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.13(d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Rk2ak(1/k,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-% \"aG6#\"\"!\"\"\"-%(productG6$*&,&%\"nGF(F(F(\"\"#F.!\"\"/F.;F',&%\"kG F(F0F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "It's unhappy about th e division by 0. Instead we should do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Rk2ak(1/k,k,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"aG6# \"\"\"F'-%&GAMMAG6#,&%\"kGF'F'F'\"\"#-F)6#F,!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"aG6#\"\"\"F'-%&GAMMAG6#,&%\"kGF'F'F'F'F,F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert(\",factorial);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%\"aG6#\"\"\"F'-%*factorialG6#,&%\"kGF'F'F'F 'F+!\"\"F,F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "It's very stubbor n... it should be a(1) * k * k!" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koe pf #5.13(e)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Rk2ak((k-1)/k,k);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"aG6#\"\"!\"\"\"-%(productG6$*(, &%\"nG\"\"#!\"\"F(F(,&F.F(F(F(F(F.!\"#/F.;F',&%\"kGF(F0F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Same problem with division by 0, t ry again." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Rk2ak((k-1)/k,k,1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.-%\"aG6#\"\"\"F()\"\"#%\"kGF(-%&G AMMAG6#,&F+F(#!\"\"F*F(F(-F-6#,&F+F(F(F(F(-F-6#F+!\"#%#PiGF0#F(F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*.)\"\"#,&%\"kG\"\"\"!\"\"F(F(-%\"aG6#F(F(-%&GAM MAG6#,&F'F(#F)F%F(F(-F.6#F'F)F'F(%#PiGF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Rk2ak((k-1)/k,k,1,false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"aG6#\"\"\"F'-%(ProductG6$*(,&%\"nG\"\"#!\"\"F'F', &F-F'F'F'F'F-!\"#/F-;F',&%\"kGF'F/F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.13(f )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Rk2ak((k+1)/k,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%\"aG6#\"\"!\"\"\"-%(productG6$**,&%\"nG\"\" #F(F(F(,&F.F(F(F(F(F.!\"\",&F.F(F/F(F1/F.;F',&%\"kGF(F1F(F(" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Rk2ak((k+1)/k,k,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*0-%\"aG6#\"\"\"F()\"\"#%\"kGF(-%&GAMMAG6#,&F+F(#F(F*F(F(-F-6#,&F+ F(F(F(F(-F-6#F+!\"\"-F-6#,&F+F(F*F(F6%#PiG#F6F*F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 11 "Koepf #5.20" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gosper(binomial(n,k),k);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in gosper) no hypergeometric term antidifference \+ exists" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gosper(binomial(- n,k),k);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in gosper) no hyperge ometric term antidifference exists" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gosper(binomial(5,k),k);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in gosper) no hypergeometric term antidifference exists" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolevel[sum]:=3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%$sumG\"\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 418 "Note that binomial(-n,k) = (-n)(- n-1)...(-n-k+1)/k! = (-1)^k * binomial(n+k-1,k).\nIn the following out put from gosper, using binomial(-5,k), say, seems to give incorrect re sults (they are inconsistent with the computations I did by hand in th e other part of the answer key), probably because it's confused about \+ the top number being negative. But using the alternate formula gives \+ results consistent with what I said." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "bin2 := (n,k) -> (-1)^k * binomial(-n+k-1,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%bin2G:6$%\"nG%\"kG6\"6$%)operatorG% &arrowGF)*&)!\"\"9%\"\"\"-%)binomialG6$,(9$F/F0F1F/F1F0F1F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 210 "for nn from -1 to -5 by -1 do\n print(` gosper`(b inomial(nn,k),k) =\n sumtools[gosper](binomial(nn,k),k));\n \+ print(``=sumtools[gosper](bin2(nn,k),k));\n print(`------------------ -----------------`);\nod;" }}{PARA 6 "" 1 "" {TEXT -1 339 "sumtools[go sper] a( k )/a( k -1):= 1\nsumtools[gosper] Gosper's al gorithm applicable\nsumtools[gosper] p:= 1\nsumtools[gosper] q:= 1\nsumtools[gosper] r:= 1\nsumtools[gosper] degreebound:= 1 \nsumtools[gosper] solving equations to find f\nsumtools[gosper] G osper's algorithm successful\nsumtools[gosper] f:= k" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%(~gosperG6$-%)binomialG6$!\"\"%\"kGF+*&,&F+\" \"\"F*F.F.F'F." }}{PARA 6 "" 1 "" {TEXT -1 344 "sumtools[gosper] a( \+ k )/a( k -1):= -1\nsumtools[gosper] Gosper's algorithm ap plicable\nsumtools[gosper] p:= 1\nsumtools[gosper] q:= -1\nsum tools[gosper] r:= 1\nsumtools[gosper] degreebound:= 0\nsumtool s[gosper] solving equations to find f\nsumtools[gosper] Gosper's a lgorithm successful\nsumtools[gosper] f:= -1/2" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,$)!\"\"%\"kG#F'\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D-----------------------------------G" }}{PARA 6 "" 1 "" {TEXT -1 357 "sumtools[gosper] a( k )/a( k -1):= (k+1) /k\nsumtools[gosper] Gosper's algorithm applicable\nsumtools[gosper] p:= k+1\nsumtools[gosper] q:= 1\nsumtools[gosper] r:= 1\n sumtools[gosper] degreebound:= 2\nsumtools[gosper] solving equat ions to find f\nsumtools[gosper] Gosper's algorithm successful\nsumt ools[gosper] f:= 1/2*k*(k+3)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %(~gosperG6$-%)binomialG6$!\"#%\"kGF+,$**,&F+\"\"\"F/F/!\"\",&F+F/F0F/ F/,&F+F/\"\"#F/F/F'F/#F/F3" }}{PARA 6 "" 1 "" {TEXT -1 358 "sumtools[g osper] a( k )/a( k -1):= -(k+1)/k\nsumtools[gosper] Gos per's algorithm applicable\nsumtools[gosper] p:= k+1\nsumtools[gos per] q:= -1\nsumtools[gosper] r:= 1\nsumtools[gosper] degree bound:= 1\nsumtools[gosper] solving equations to find f\nsumtools[ gosper] Gosper's algorithm successful\nsumtools[gosper] f:= -3/4 -1/2*k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&,&\"\"\"F(%\"kG\"\"# F()!\"\"F)F(#F,\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------ -----------------------G" }}{PARA 6 "" 1 "" {TEXT -1 372 "sumtools[gos per] a( k )/a( k -1):= (k+2)/k\nsumtools[gosper] Gosper 's algorithm applicable\nsumtools[gosper] p:= (k+1)*(k+2)\nsumtool s[gosper] q:= 1\nsumtools[gosper] r:= 1\nsumtools[gosper] de greebound:= 3\nsumtools[gosper] solving equations to find f\nsumto ols[gosper] Gosper's algorithm successful\nsumtools[gosper] f:= \+ 1/3*k*(11+6*k+k^2)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(~gosperG6$-% )binomialG6$!\"$%\"kGF+,$*,,&F+\"\"\"!\"\"F/F/,(\"\"'F/F+\"\"%*$F+\"\" #F/F/,&F+F/F/F/F0,&F+F/F5F/F0F'F/#F/\"\"$" }}{PARA 6 "" 1 "" {TEXT -1 372 "sumtools[gosper] a( k )/a( k -1):= -(k+2)/k\nsumtool s[gosper] Gosper's algorithm applicable\nsumtools[gosper] p:= (k +1)*(k+2)\nsumtools[gosper] q:= -1\nsumtools[gosper] r:= 1\nsu mtools[gosper] degreebound:= 2\nsumtools[gosper] solving equatio ns to find f\nsumtools[gosper] Gosper's algorithm successful\nsumtoo ls[gosper] f:= -7/4-2*k-1/2*k^2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G,$*,,(\"\"\"F(%\"kG\"\"%*$F)\"\"#F,F(,&F)F(F(F(!\"\",&F)F(F,F(F.) F.F)F(-%)binomialG6$F/F)F(#F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D- ----------------------------------G" }}{PARA 6 "" 1 "" {TEXT -1 384 "s umtools[gosper] a( k )/a( k -1):= (k+3)/k\nsumtools[gospe r] Gosper's algorithm applicable\nsumtools[gosper] p:= (k+3)*(k+ 2)*(k+1)\nsumtools[gosper] q:= 1\nsumtools[gosper] r:= 1\nsumt ools[gosper] degreebound:= 4\nsumtools[gosper] solving equations to find f\nsumtools[gosper] Gosper's algorithm successful\nsumtools [gosper] f:= 1/4*k*(k+5)*(k^2+5*k+10)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(~gosperG6$-%)binomialG6$!\"%%\"kGF+,$*0,&F+\"\"\"! \"\"F/F/,&F+F/\"\"%F/F/,(*$F+\"\"#F/F+\"\"$\"\"'F/F/,&F+F/F6F/F0,&F+F/ F5F/F0,&F+F/F/F/F0F'F/#F/F2" }}{PARA 6 "" 1 "" {TEXT -1 389 "sumtools[ gosper] a( k )/a( k -1):= -(k+3)/k\nsumtools[gosper] Go sper's algorithm applicable\nsumtools[gosper] p:= (k+3)*(k+2)*(k+1 )\nsumtools[gosper] q:= -1\nsumtools[gosper] r:= 1\nsumtools[g osper] degreebound:= 3\nsumtools[gosper] solving equations to fi nd f\nsumtools[gosper] Gosper's algorithm successful\nsumtools[gospe r] f:= -1/8*(2*k+5)*(2*k^2+10*k+9)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*0,&%\"kG\"\"#\"\"$\"\"\"F+,(*$F(F)F)F(\"\"'F+F+F+,&F(F+F* F+!\"\",&F(F+F)F+F0,&F(F+F+F+F0)F0F(F+-%)binomialG6$F/F(F+#F0\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%D-----------------------------------G " }}{PARA 6 "" 1 "" {TEXT -1 401 "sumtools[gosper] a( k )/a( k -1):= (k+4)/k\nsumtools[gosper] Gosper's algorithm applicable \nsumtools[gosper] p:= (k+4)*(k+3)*(k+2)*(k+1)\nsumtools[gosper] \+ q:= 1\nsumtools[gosper] r:= 1\nsumtools[gosper] degreebound:= 5\nsumtools[gosper] solving equations to find f\nsumtools[gosper] Gosper's algorithm successful\nsumtools[gosper] f:= 1/5*k*(274+ 225*k+85*k^2+15*k^3+k^4)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%(~gospe rG6$-%)binomialG6$!\"&%\"kGF+,$*0,&F+\"\"\"!\"\"F/F/,,\"$?\"F/F+\"#'** $F+\"\"#\"#Y*$F+\"\"$\"#6*$F+\"\"%F/F/,&F+F/F;F/F0,&F+F/F8F/F0,&F+F/F5 F/F0,&F+F/F/F/F0F'F/#F/\"\"&" }}{PARA 6 "" 1 "" {TEXT -1 399 "sumtools [gosper] a( k )/a( k -1):= -(k+4)/k\nsumtools[gosper] G osper's algorithm applicable\nsumtools[gosper] p:= (k+4)*(k+3)*(k+ 2)*(k+1)\nsumtools[gosper] q:= -1\nsumtools[gosper] r:= 1\nsum tools[gosper] degreebound:= 4\nsumtools[gosper] solving equation s to find f\nsumtools[gosper] Gosper's algorithm successful\nsumtool s[gosper] f:= -93/4-42*k-25*k^2-6*k^3-1/2*k^4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*0,,\"\"$\"\"\"%\"kG\"#K*$F*\"\"#\"#S*$F*F(\"#;*$ F*\"\"%F-F),&F*F)F2F)!\"\",&F*F)F(F)F4,&F*F)F-F)F4,&F*F)F)F)F4)F4F*F)- %)binomialG6$F3F*F)#F4F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D-------- ---------------------------G" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "infolevel[sum]:=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%$sumG\"\"!" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 11 "Koepf #5.21" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "gosper(1/k^2,k);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Erro r, (in gosper) no hypergeometric term antidifference exists" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "read `qsum.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%OCopyright~1998,~~Harald~Boeing~&~Wolfram~KoepfG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%;Konrad-Zuse-Zentrum~BerlinG" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 11 "Koepf #5.25" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q5 25j := qgosper(q^(j*k),q,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&q52 5jG*&,&)%\"qG%\"jG\"\"\"!\"\"F*F+)F(*&F)F*%\"kGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "for jj from 1 to 5 do\n print(` qgospe r`(q^(jj*k),q,k) =\n qgosper(q^(jj*k),q,k))\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)~qgosperG6%)%\"qG%\"kGF(F)*&,(*&%$_C1G\"\" \"F(\"\"#F.*&F-F.F(F.!\"\"F'F.F.,&F(F.F1F.F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)~qgosperG6%)%\"qG,$%\"kG\"\"#F(F**(,&F(\"\"\"!\"\"F .F/,&F(F.F.F.F/,(*&%$_C1GF.F(\"\"%F.*&F3F.F(F+F/F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)~qgosperG6%)%\"qG,$%\"kG\"\"$F(F**(,&F(\"\"\"! \"\"F.F/,(*$F(\"\"#F.F(F.F.F.F/,(*&%$_C1GF.F(F+F/*&F5F.F(\"\"'F.F'F.F. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)~qgosperG6%)%\"qG,$%\"kG\"\"%F (F***,&F(\"\"\"!\"\"F.F/,&F(F.F.F.F/,(F'F.*&%$_C1GF.F(\"\")F.*&F3F.F(F +F/F.,&*$F(\"\"#F.F.F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)~qgosp erG6%)%\"qG,$%\"kG\"\"&F(F**(,&F(\"\"\"!\"\"F.F/,,*$F(\"\"%F.*$F(\"\"$ F.*$F(\"\"#F.F(F.F.F.F/,(F'F.*&%$_C1GF.F(\"#5F.*&F9F.F(F+F/F." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 292 "It added a constant to the antidi fference: _C1*(q^10-q^5)/(q^5-1)=_C1*q^5 is constant with respect to k . The original input summand is rational w.r.t. k, so there is not a \+ unique q-hypergeometric antidifference, but rather an additive \"const ant\" rational w.r.t. q is contained in the answer." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simplify(qgosper(q^(5*k),q,k) - subs(j=5,q525j)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%$_C1G\"\"\"%\"qG\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.26(a)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "qgosper(q^(j*k)*qpochhammer(n,q,k),q,k);" }}{PARA 8 "" 1 "" {TEXT -1 62 "Error, (in qgosper) No q-hypergeometric antidif ference exists." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "fna := j -> q^(j*k) * qpochhammer(n,q,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$fnaG:6#%\"jG6\"6$%)operatorG%&arrowGF(*&)%\"qG*&9$\"\"\"%\"kGF1F1-%, qpochhammerG6%%\"nGF.F2F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for jj from 1 to 5 do\n printf(`at j=%d`,jj);\n print(qgospe r(fna(jj),q,k));\nod;" }}{PARA 6 "" 1 "" {TEXT -1 6 "at j=1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"nG!\"\"-%,qpochhammerG6%F%%\"qG%\"kG \"\"\"F&" }}{PARA 6 "" 1 "" {TEXT -1 6 "at j=2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,(%\"qG!\"\"\"\"\"F'*&%\"nGF')F%%\"kGF'F&F'F)!\"#-%,q pochhammerG6%F)F%F+F'F%F&" }}{PARA 6 "" 1 "" {TEXT -1 6 "at j=3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#**,0*$%\"qG\"\"$!\"\"F&\"\"\"*$F&\"\"# F)F(F)*&%\"nGF))F&,&%\"kGF)F+F)F)F(*&F-F))F&F0F)F)*&F-F+)F&,&F)F)F0F+F )F(F)F-!\"$-%,qpochhammerG6%F-F&F0F)F&F6" }}{PARA 6 "" 1 "" {TEXT -1 6 "at j=4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**%\"nG!\"%,<*$%\"qG\" \"#\"\"\"*$F)\"\"'F+*$F)\"\"&!\"\"F)F+F0F+*$F)\"\"%F0*&F%F+)F),&%\"kGF +F*F+F+F0*&F%F+)F),&F6F+F/F+F+F+*&F%F+)F)F6F+F+*&F%F+)F),&F6F+\"\"$F+F +F0*&F%F*)F),&F+F+F6F*F+F0*&F%F*)F),&F2F+F6F*F+F+*&F%F?)F),&F6F?F?F+F+ F+F+-%,qpochhammerG6%F%F)F6F+F)!\"'F0" }}{PARA 6 "" 1 "" {TEXT -1 6 "a t j=5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**%\"nG!\"&-%,qpochhammerG6 %F%%\"qG%\"kG\"\"\"F*!#5,N*$F*\"\"#!\"\"*$F*\"#5F,*$F*\"\"*F1*$F*\"\"& F0F*F1F,F,*$F*\"\")F1*&F%F,)F*,&F+F,\"\"'F,F,F1*&F%F,)F*,&F+F,F0F,F,F, *&F%F,)F*,&F+F,F5F,F,F,*&F%F,)F*,&F+F,F7F,F,F1*&F%F,)F*,&F+F,\"\"%F,F, F,*&F%F,)F*F+F,F1*&F%F,)F*,&F+F,\"\"(F,F,F1*&F%F,)F*,&F+F,\"\"$F,F,F,* &F%F0)F*,&F7F,F+F0F,F1*&F%F0)F*,&F,F,F+F0F,F,*&F%F0)F*,&F9F,F+F0F,F,*& F%F0)F*,&FJF,F+F0F,F1*&F%FT)F*,&FPF,F+FTF,F,*&F%FT)F*,&F+FTFTF,F,F1*&F %FJ)F*,&F=F,F+FJF,F,F,F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.26(c)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "qgosper((-1)^k*q^binomial(k,2)*qbinomial(n,k,q ),q,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,,&)%\"qG%\"kG\"\"\"!\" \"F)F),&)F'%\"nGF)F*F)F*)F*F(F))F'-%)binomialG6$F(\"\"#F)-%*qbinomialG 6%F-F(F'F)F*" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.26(d)" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "qgosper(q^(j*k)*qbrackets(k,q),q,k );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.,*!\"\"\"\"\")%\"qG,&%\"jGF' F'F'F')F),&%\"kGF'F+F'F&)F)F.F'F')F)*&F+F'F.F'F'-%*qbracketsG6$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 36 "qgosper(q^(2*k)*qbrackets(k,q),q,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*.-%*qbracketsG6$%\"kG%\"qG\"\"\",&)F(F'F)!\"\"F)F ,,&F(F)F,F)F,,&F(F)F)F)F,,(*$F(\"\"#F)F(F)F)F)F,,4*&%$_C1GF)F(\"\"'F)* &F4F)F(\"\"&F)*&F4F)F(\"\"$F,)F(,&F'F1F1F)F,*&F4F)F(F1F,)F(,&F)F)F'F9F ))F(,&F)F)F'F1F,)F(,$F'F9F))F(,$F'F1F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.26(e )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "qgosper(q^binomial(k,2)*qbrack ets(k,q),q,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&)%\"qG%\"kG\"\" \"!\"\"F(F))F&-%)binomialG6$F'\"\"#F(-%*qbracketsG6$F'F&F(" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "Koepf #5.26(f)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "qgosper(q^(-k*(k+1)/2)*qbrackets(k,q),q,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&)%\"qG%\"kG\"\"\"!\"\"F)F*)F',&F(F)*&F (F),&F(F)F)F)F)#F*\"\"#F)-%*qbracketsG6$F(F'F)F*" }}}}{MARK "0 0 0" 41 }{VIEWOPTS 1 1 0 1 1 1803 }