{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 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 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 "Courier" 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 "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 "" 3 256 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 }{PSTYLE "R3 Font 0" -1 257 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" -1 258 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 45 "/home/m262f99/KOEPF/works heetsV.4/hw5ansm.mws" }{MPLTEXT 1 0 0 "" }}{PARA 256 "" 0 "" {TEXT 257 45 "Math 262a, Fall 1999, Glenn Tesler\nHomework 5" }{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 12 "Koepf 6 .7(e)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 75 "Firs t, see if any of the earlier ways we learned to do this are applicable ." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Fe \+ := k * binomial(n,k) * binomial(s,k);\nrhse := s*binomial(n+s-1,n-1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FeG*(%\"kG\"\"\"-%)binomialG6$% \"nGF&F'-F)6$%\"sGF&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rhseG*&% \"sG\"\"\"-%)binomialG6$,(%\"nGF'F&F'!\"\"F',&F,F'F-F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gosper(Fe,k);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in gosper) no hypergeometric term antidifference \+ exists" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Creative telescoping" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sumrecu rsion(Fe,k=1..n,f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"nG\" \"\"-%\"fG6#,&F&F'F'F'F'!\"\"*&,&%\"sGF'F&F'F'-F)6#F&F'F'\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sumrecursion(Fe,k,f(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"nG\"\"\"-%\"fG6#,&F&F'F'F'F'! \"\"*&,&%\"sGF'F&F'F'-F)6#F&F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rec_e := rsolve(\{\",f(1)=A\},f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rec_eG**-%&GAMMAG6#,&%\"sG\"\"\"%\"nGF+F+-F'6#F ,!\"\"-F'6#,&F*F+F+F+F/%\"AGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs(A=eval(subs(k=1,n=1,Fe)), rec_e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%&GAMMAG6#,&%\"sG\"\"\"%\"nGF)F)-F%6#F*!\"\"-F%6#,&F (F)F)F)F-F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert(\" ,binomial);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"sG\"\"\"-%)binomia lG6$,(F$F%%\"nGF%!\"\"F%F$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "S ister Celine's algorithm" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fasenmy er(Fe,k,f(n),1);" }}{PARA 8 "" 1 "" {TEXT -1 76 "Error, (in kfreerec) \+ no kfree recurrence equation of order (, 1, 1, ) exists" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fasenmyer(Fe,k,f(n),2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&%\"nG\"\"\"F(F(F(-%\"fG6#,&F'F(\"\"#F (F(F(*&,(F'F(F(F(%\"sGF(F(-F*6#F&F(!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "It's a shift of the same recurrence found by CT, so th e solution will be the same." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 9 "WZ method" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F := Fe / rhse;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG*,%\"kG\"\"\"-%)b inomialG6$%\"nGF&F'-F)6$%\"sGF&F'F.!\"\"-F)6$,(F+F'F.F'F/F',&F+F'F/F'F /" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R := WZcertificate(F,k ,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG**,&%\"kG\"\"\"!\"\"F(F( F'F(,(%\"nGF)F'F(F)F(F),&F+F(%\"sGF(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "G := F*R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG*2% \"kG\"\"#-%)binomialG6$%\"nGF&\"\"\"-F)6$%\"sGF&F,F/!\"\"-F)6$,(F+F,F/ F,F0F,,&F+F,F0F,F0,&F&F,F0F,F,,(F+F0F&F,F0F,F0,&F+F,F/F,F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Verify the " }{TEXT 260 11 "WZ equation" }{TEXT -1 19 " for this identity:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "WZeq := subs(n=n+1,F) - F = subs(k=k+1,G)-G;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%%WZeqG/,&*,%\"kG\"\"\"-%)binomialG6$,&%\"nGF)F)F)F( F)-F+6$%\"sGF(F)F1!\"\"-F+6$,&F1F)F.F)F.F2F)*,F(F)-F+6$F.F(F)F/F)F1F2- F+6$,(F1F)F.F)F2F),&F.F)F2F)F2F2,&*2,&F(F)F)F)\"\"#-F+6$F.F?F)-F+6$F1F ?F)F1F2F9F2F(F),&F(F)F.F2F2F5F2F)*2F(F@F7F)F/F)F1F2F9F2,&F(F)F2F)F),(F .F2F2F)F(F)F2F5F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simp comb(WZeq);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*6-%&GAMMAG6#%\"sG\"\" \"-F&6#,&%\"nGF)F)F)F),**&F-F)F(F)!\"\"F(F0*&%\"kGF)F(F)F)*&F2F)F-F)F) F)-F&6#,&F(F)F)F)F)-F&6#F-F)-F&6#,(F(F)F)F)F-F)F0-F&6#,(F(F)F)F)F2F0F0 -F&6#F2!\"#-F&6#,(F-F)\"\"#F)F2F0F0F2F0F$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "evalb(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Or with purely rational arit hmetic:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "simpcomb(lhs(WZeq)/F) = \+ simpcomb(rhs(WZeq)/F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(,**&%\" nG\"\"\"%\"sGF)!\"\"F*F+*&%\"kGF)F*F)F)*&F-F)F(F)F)F),(F(F+F+F)F-F)F+, &F*F)F(F)F+F+F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalb(\") ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 146 "Now sum up the equation over k=1..n. These are natura l bounds, so we may extend them.\nLet f(n) = sum(F(n,k), k=1..infinity )\nThe left side has sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Sum('F' (n+1,k)-'F'(n,k),k=1..infinity) =\n f(n+1)-f(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&-%\"FG6$,&%\"nG\"\"\"F-F-%\"kGF--F)6$F,F.! \"\"/F.;F-%)infinityG,&-%\"fG6#F+F--F76#F,F1" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 26 "and the right side has sum" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Sum('G'(n,k+1)-'G'(n,k),k=1. .infinity) =\n 'G'(n,infinity) - 'G'(n,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&-%\"GG6$%\"nG,&%\"kG\"\"\"F.F.F.-F)6$F+F-! \"\"/F-;F.%)infinityG,&-F)6$F+F4F.-F)6$F+F.F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(G,k=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$*2%\"kG\"\"#-%)binomialG6$%\"nGF'\"\"\"-F*6$ %\"sGF'F-F0!\"\"-F*6$,(F0F-F,F-F1F-,&F,F-F1F-F1,&F'F-F1F-F-,(F,F1F1F-F 'F-F1,&F0F-F,F-F1/F'%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "On integers, F has finite k-support for each n, so G does too, so this limit should be 0, even though maple doesn't recognize that." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(G ,k=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Thus, for n=1,2,3,...," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(n+1)-f(n)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,& -%\"fG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 12 "which proves" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Sum( F,k=1..infinity)=constant;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG 6$*,%\"kG\"\"\"-%)binomialG6$%\"nGF(F)-F+6$%\"sGF(F)F0!\"\"-F+6$,(F0F) F-F)F1F),&F-F)F1F)F1/F(;F)%)infinityG%)constantG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 42 "Evaluate the constant using f(1) = F(1,1):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(n=1,k=1,F); eval(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#**-%)binomialG6$\"\"\"F'F'-F%6$%\"sGF' F'F*!\"\"-F%6$F*\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "So we have proved" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Sum(F,k=1..infinity)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*,%\"kG\"\"\"-%)binomialG6$%\"nGF(F)-F+6$%\"s GF(F)F0!\"\"-F+6$,(F0F)F-F)F1F),&F-F)F1F)F1/F(;F)%)infinityGF)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Restricting the summation range to the support and rearranging gives, for n=1,2,3,...," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sum(Fe,k=1..n) = rhse;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(%\"kG\"\"\"-%)binomialG6$%\"nGF(F)-F+6$%\"s GF(F)/F(;F)F-*&F0F)-F+6$,(F-F)F0F)!\"\"F),&F-F)F7F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "The companion identity is obtained by summing the WZ equation o ver n instead of k. Let" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g(k) = \+ Sum(G,n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"k G-%$SumG6$*2F'\"\"#-%)binomialG6$%\"nGF'\"\"\"-F.6$%\"sGF'F1F4!\"\"-F. 6$,(F4F1F0F1F5F1,&F0F1F5F1F5,&F'F1F5F1F1,(F0F5F5F1F'F1F5,&F4F1F0F1F5/F 0;F1%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Summing the le ft side of the WZ equation over n=1,2,... gives" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Sum('F'(n+1,k)-'F'(n,k),n=1..infinity) =\n 'F'(infin ity,k)-'F'(1,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&-%\"FG 6$,&%\"nG\"\"\"F-F-%\"kGF--F)6$F,F.!\"\"/F,;F-%)infinityG,&-F)6$F4F.F- -F)6$F-F.F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "When n=1 we have" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "'F'(1,k)=subs(n=1,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6$\"\"\"%\"kG*,F(F'-%)binomialGF&F'-F +6$%\"sGF(F'F.!\"\"-F+6$F.\"\"!F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "'F'(1,k)=piecewise(k=1,1,0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6$\"\"\"%\"kG-%*PIECEWISE G6$7$F'/F(F'7$\"\"!%*otherwiseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "As n -> infinity," }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "'F'(infinity,k)=limit(F,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6$%)infinityG%\"kG-%&limitG6$*,F(\"\"\"-%)b inomialG6$%\"nGF(F--F/6$%\"sGF(F-F4!\"\"-F/6$,(F1F-F4F-F5F-,&F1F-F5F-F 5/F1F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "It won't do it automati cally, let's do it ourselves." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "si mpcomb(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*4%\"kG!\"\"-%&GAMMAG6#% \"nG\"\"\"-F'6#,(F)F*F*F*F$F%F%-F'6#,&%\"sGF*F*F*F*-F'6#,(F1F*F*F*F$F% F%-F'6#,&F)F*F1F*F%-F'6#,&F)F*F*F*F*-F'6#F$!\"#-F'6#F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "F_n := select(has,\",n); F_no_n := \"\"/F_n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$F_nG**-%&GAMMAG6#%\"n G\"\"\"-F'6#,(F)F*F*F*%\"kG!\"\"F/-F'6#,&F)F*%\"sGF*F/-F'6#,&F)F*F*F*F *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'F_no_nG*,%\"kG!\"\"-%&GAMMAG6# ,&%\"sG\"\"\"F-F-F--F)6#,(F,F-F-F-F&F'F'-F)6#F&!\"#-F)6#F,F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "As n goes to infinity, Gamma(n+A) /Gamma(n+B) ~ n^(A-B) so the as n->infinity, F(n,k) is asymptotically " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F_inf := n^(1-(1-k+s)) * F_no_n ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&F_infG*.)%\"nG,&%\"sG!\"\"%\"k G\"\"\"F,F+F*-%&GAMMAG6#,&F)F,F,F,F,-F.6#,(F)F,F,F,F+F*F*-F.6#F+!\"#-F .6#F)F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "which is" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 "'F'(infinity,k)=piecewise(k>s,infinity, k=s,si mpcomb(subs(k=s,F_inf)), k " 0 "" {MPLTEXT 1 0 4 "k,s;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%\"kG%\"sG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Combining all this, summing the left side of the WZ equation over \+ n=1,2,... gives" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "sumWZlhs := 'pi ecewise(s1', 1,\n 'k< s and k<>1', 0,\n 's=1 and k=1',0,\n '17$F@3/F:F-/F.F-7$F132F-F:FF" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The sum is divergent for s \+ " 0 "" {MPLTEXT 1 0 55 "Sum('G'(n,k+1)-'G'(n,k),n=1..infinity) =\n g( k+1)-g(k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&-%\"GG6$%\"n G,&%\"kG\"\"\"F.F.F.-F)6$F+F-!\"\"/F+;F.%)infinityG,&-%\"gG6#F,F.-F76# F-F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Combining the sum of the left side and the sum of the right side gives g(k+1)-g(k) = the 5 cas es listed just above.\nNow for each s, solve the resulting recurrence \+ for g(k)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 "(A) For s>1, s>=k, \+ we have\n g(s+1)-g(s)=1; g(s)-g(s-1)=g(s-1)-g(s-2)=...=g(3)-g( 2)=0; g(2)-g(1)=-1; g(1)-g(0)=g(0)-g(-1)=...=0\nso given g(s+1 ), then g(k)=g(s+1)-1 for k=2,3,...,s; g(k)=g(s+1) for k=s+1 or 1,0,- 1,-2,...\n(B) For s=1: g(k)=g(s+1) for k<=s+1.\n(C) For k<=s<1: g(k)= g(s+1)-1 for k<=s " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " Now find a n initial value. All of these are expressed in terms of g(s+1)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simpcomb(subs(k=s+1,G));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 274 "so its sum is g(s+1)=0. Then the three cases become\n(A ) For s>1: g(k)=-1 for k=2,3,...,s; g(k)=0 for k=s+1 or 1,0,-1,-2,.. .\n(B) For s=1: g(k)=0 for k<=s+1\n(C) For s<1: g(k)=-1 for k<=s, an d g(s+1)=0.\nCase (A) spelled out in full is: For integer s>1 and int eger k<=s+1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sum(G,n=1..infinity ) = piecewise('k=s+1 or k<=1',0,'2<=k and k<=s',-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*2%\"kG\"\"#-%)binomialG6$%\"nGF(\"\"\"-F+ 6$%\"sGF(F.F1!\"\"-F+6$,(F-F.F1F.F2F.,&F-F.F2F.F2,&F(F.F2F.F.,(F-F2F(F .F2F.F2,&F-F.F1F.F2/F-;F.%)infinityG-%*PIECEWISEG6$7$\"\"!5/F(,&F1F.F. F.1F(F.7$F231F)F(1F(F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "or putt ing all n-free factors on the right side," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "rhs2 := s / (k^2 * binomial(s,k) * (k-1)):\nG2 := G* rhs2:\nSum(G2,n=1..infinity) = piecewise('k=s+1 or k<=1',0,'2<=k and k <=s',-rhs2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$**-%)binomia lG6$%\"nG%\"kG\"\"\"-F)6$,(F+F-%\"sGF-!\"\"F-,&F+F-F2F-F2,(F+F2F,F-F2F -F2,&F+F-F1F-F2/F+;F-%)infinityG-%*PIECEWISEG6$7$\"\"!5/F,,&F1F-F-F-1F ,F-7$,$**F1F-F,!\"#-F)6$F1F,F2,&F,F-F2F-F2F231\"\"#F,1F,F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Let's check it." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "gks := proc(k0,s0)\n global G,n;\n local G0;\n G0 := subs(k=k0,s=s0,G);\n sum(G0,n=1..infinity);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "linalg[matrix](6,6,(k,s)->gk s(k,s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7(7(\"\"!F(F(F (F(F(7(F(!\"\"F*F*F*F*7(F(F(F*F*F*F*7(F(F(F(F*F*F*7(F(F(F(F(F*F*7(F(F( F(F(F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 12 "Koepf 8.5(a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gosper((3*k+2)/(k+2) * binomial(k,k/2),k);" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, (in gosper) algorithm not applicable" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "extended_gosper((3*k+2)/(k+ 2) * binomial(k,k/2),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*,,&%\"k G#\"\"\"\"\"#F(F(F(,&F&#\"\"$F)F(F(!\"\",&F&F,F)F(F(,&F&F(F)F(F--%)bin omialG6$F&,$F&F'F(F(*,,&F&F'F+F(F(,&F&F+#\"\"&F)F(F-,&F&F,F8F(F(,&F&F( F,F(F--F16$,&F&F(F(F(,&F&F'F'F(F(F(" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 6 "8.5(b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "extended_gosper((3* k+4)/(k+4)*binomial(k/2,k/4),k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,* *,,&%\"kG#\"\"\"\"\"%F(F(F(,&F&#\"\"$F)F(F(!\"\",&F&F,F)F(F(,&F&F(F)F( F--%)binomialG6$,$F&#F(\"\"#,$F&F'F(F(*,,&F&F'#\"\"&F)F(F(,&F&F+#\"\"( F)F(F-,&F&F,F=F(F(,&F&F(F:F(F--F16$,&F&F4F4F(,&F&F'F'F(F(F(*,,&F&F'#F, F5F(F(,&F&F+#F:F5F(F-,&F&F,\"#5F(F(,&F&F(\"\"'F(F--F16$,&F&F4F(F(,&F&F 'F4F(F(F(*,,&F&F'F " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 15 "Koepf 8.7(5.21)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "F0 := hyperterm([3*a+1/2,3*a+1,-n],[6*a+1,-n /3+2*a+1],4/3,k);\nr := pochhammer(1/3,n/3)*pochhammer(2/3,n/3) / (poc hhammer(1+2*a,n/3)*pochhammer(-2*a,n/3));\nF := F0/r:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F0G*0-%+pochhammerG6$,&%\"aG\"\"$#\"\"\"\"\"#F- %\"kGF--F'6$,&F*F+F-F-F/F--F'6$,$%\"nG!\"\"F/F--F'6$,&F*\"\"'F-F-F/F7- F'6$,(F6#F7F+F*F.F-F-F/F7)#\"\"%F+F/F--%*factorialG6#F/F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG**-%+pochhammerG6$#\"\"\"\"\"$,$%\"nGF)F* -F'6$#\"\"#F+F,F*-F'6$,&F*F*%\"aGF1F,!\"\"-F'6$,$F5!\"#F,F6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 202 "Compute the first few values of s um_k F0(n,k). Note that -n is an upper parameter, so the sum termina tes at k=n unless a is chosen to make one of the denominator parameter s also be a negative integer." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sumF := nn -> sum(subs(n=nn,F),k=0..nn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sumFG:6#%#nnG6\"6$%)operatorG%&arrowGF(-%$sumG6$-%%s ubsG6$/%\"nG9$%\"FG/%\"kG;\"\"!F4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sumF(1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*4-%+poc hhammerG6$,&%\"aG\"\"$#\"\"\"\"\"#F,\"\"!F,-F&6$,&F)F*F,F,F.F,-F&6$,&F )\"\"'F,F,F.!\"\"-F&6$,&#F-F*F,F)F-F.F6-%&GAMMAG6#F:F6%#PiGF,F*F+-F&6$ ,&F,F,F)F-#F,F*F,-F&6$,$F)!\"#FBF,F:*4F(F,F1F,F4F6F9F6F;F6F>F,F*F+F?F, FCF,#!\")\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "'simplify(sumF(nn))'$nn=1..6;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6(\"\"!F#\"\"\"F#F#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Apply WZ meth od." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "W Zcertificate(F,k,n);" }}{PARA 8 "" 1 "" {TEXT -1 50 "Error, (in WZcert ificate) extended WZ method fails" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "I looked at the code, it doesn't attempt to figure out if it shou ld use m-fold hypergeometric functions; you have to specify m if you w ant it done." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "R := WZcertificate( F,k,n,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,$*.,&%\"aG\"\"'%\" kG\"\"\"F+,(%\"nG!\"\"F(F)F*\"\"$F+F*F+,(F-F.F.F+F*F+F.,(F-F.!\"#F+F*F +F.,(F-F.!\"$F+F*F+F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Verify it:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "G := R*F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The WZ equation becomes F(n+3,k) - F(n,k) = \+ G(n,k+1) - G(n,k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "simpcomb( (su bs(n=n+3,F)-F) - (subs(k=k+1,G) - G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "f(n) = Sum('F(n,k)',k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"nG-%$SumG6$-%\"FG6$F'%\"kG/F.;\" \"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "The left side \+ of the WZ equation sums to f(n+3)-f(n), and the right side sums to G(n ,infinity)-G(n,0).\nThe k-support of G is finite for each n, since G(n ,k+1)/G(n,k) is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ratio(G,k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.,(%\"aG\"\"'\"\"\"F(%\"kG\"\"#F(,( F&\"\"$F(F(F)F(F(,(%\"nG!\"\"!\"$F(F)F(F(F)F/,(F.F/F&F'F)F,F/,&F&F'F)F (F/F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "which vanishes at k=n+3 (unless a=n/6 - k/2 for an integer k>n+3, yielding a pole in this).\n So G(n,infinity)=0." }}{PARA 0 "" 0 "" {TEXT -1 15 "Also, G(n,0) is" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(k=0,G);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "T hus, the WZ equation summed over k yields f(n+3)-f(n)=0-0=0 for int egers n>=0. From the initial conditions above, we have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f(n) = piecewise('n mod 3'=0,1,0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"nG-%*PIECEWISEG6$7$\" \"\"/-%$modG6$F'\"\"$\"\"!7$F2%*otherwiseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "so for integers n>=0," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Sum('F[0](n,k)/r(n)',k=0..infinity)=rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&-&%\"FG6#\"\"!6$%\"nG%\"kG\"\"\"-%\"rG6#F.! \"\"/F/;F,%)infinityG-%*PIECEWISEG6$7$F0/F.F,7$F,%*otherwiseG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Sum('F[0](n,k)',k=0..infinity)=piecewise('n mod 3'=0,'r(n)',0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$-&%\"FG6#\"\"!6$%\"nG%\" kG/F.;F+%)infinityG-%*PIECEWISEG6$7$-%\"rG6#F-/-%$modG6$F-\"\"$F+7$F+% *otherwiseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "and plugging every thing in," }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(F0,k=0..infinity)=piecewise('n mod 3'=0,r,0);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/-%$SumG6$*0-%+pochhammerG6$,&%\"aG\"\"$#\"\"\" \"\"#F/%\"kGF/-F)6$,&F,F-F/F/F1F/-F)6$,$%\"nG!\"\"F1F/-F)6$,&F,\"\"'F/ F/F1F9-F)6$,(F8#F9F-F,F0F/F/F1F9)#\"\"%F-F1F/-%*factorialG6#F1F9/F1;\" \"!%)infinityG-%*PIECEWISEG6$7$**-F)6$#F/F-,$F8FSF/-F)6$#F0F-FTF/-F)6$ ,&F/F/F,F0FTF9-F)6$,$F,!\"#FTF9/-%$modG6$F8F-FJ7$FJ%*otherwiseG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "Koepf 11.7 " }}{PARA 0 "" 0 "" {TEXT -1 11 "Recall that" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "erf(x) = 2/sqrt(Pi) * Int(ex p(-t^2), t=0..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$erfG6#%\"xG,$ *&%#PiG#!\"\"\"\"#-%$IntG6$-%$expG6#,$*$%\"tGF-F,/F6;\"\"!F'\"\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Use Taylor series:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 56 "2/sqrt(Pi) * Int(Sum((-t^2)^k/k!,k=0..infini ty),t=0..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG#!\"\"\"\"#-% $IntG6$-%$SumG6$*&),$*$%\"tGF(F'%\"kG\"\"\"-%*factorialG6#F4F'/F4;\"\" !%)infinityG/F3;F;%\"xGF5F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Do ing it term by term gives" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "2/sqrt (Pi)*Sum((-1)^k * x^(2*k+1) / (k! * (2*k+1)),k=0..infinity);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%# PiG#!\"\"\"\"#-%$SumG6$**)F'%\"kG\"\"\")%\"xG,&F.F(F/F/F/-%*factorialG 6#F.F'F2F'/F.;\"\"!%)infinityGF/F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Convert the sum to hypergeometric notation" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "2/sqrt(Pi) * Sumtohyper((-1)^k * x^(2*k+1) / (k!*(2 *k+1)),k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%#PiG#!\"\"\"\"#%\"x G\"\"\"-%*HypergeomG6%7##F*F(7##\"\"$F(,$*$F)F(F'F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "reset its mea ning, it's used from scratch below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "F := 'F';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGF$" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "Problem 3." }}{PARA 0 "" 0 "" {TEXT -1 14 "The summand is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Fxk := x^(2*k +1)/(2*k+1)!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FxkG*&)%\"xG,&%\"k G\"\"#\"\"\"F+F+-%*factorialG6#F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Clearly (Ek Dx^2 - 1) annihilates this. We can try to f ind a mixed recurrence/diffeq whose coefficients are k-free by brute f orce.:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1439 "# celinexk(F,x,k,xmax,k max)\n# celinexk(F,x,k,xmax,kmax,verbose,c)\n# Apply Celine's alg. to function F of a continuous variable x and a discrete variable k.\n# \+ Use derivatives (d/dx)^0,...,(d/dx)^xmax, and shifts (Ek)^0,...,(Ek)^k max.\n# optional:\n# verbose can be true or false, indicating whether to print intermediate results.\n# c can be x or k, to collect with \+ respect to it. Default k.\ncelinexk := proc(F,x,k,xmax,kmax)\n loc al oper, recdiffeq, dF,\n i,j, Dx, Ek,\n eqs,vars,so ls,\n verbose,c,arg;\n verbose := false; c := k;\n for \+ arg in args[6..nargs] do\n if type(arg,boolean) then verbose := \+ arg else c := arg fi;\n od;\n Dx := cat(`D`,x); Ek := cat(`E`,k) ;\n oper := 0;\n recdiffeq := 0;\n for i from 0 to xmax do\n \+ for j from 0 to kmax do\n oper := oper + a[i,j] * Dx^i * Ek^j ;\n if i=0 then dF := F else dF := diff(F,x$i) fi;\n recdi ffeq := recdiffeq + a[i,j]*simpcomb(subs(k=k+j,dF)/F);\n od od;\n\n if verbose then print(`trial equation`,recdiffeq) fi;\n\n recdi ffeq := numer(recdiffeq);\n recdiffeq := collect(recdiffeq,c);\n \+ if verbose then print(`collected in powers of `,c,recdiffeq) fi;\n\n \+ eqs := \{coeffs(recdiffeq,c)\};\n vars := \{'('a[i,j]'$'i'=0..xm ax)'$'j'=0..kmax\};\n sols := solve(eqs,vars);\n if verbose then print(`final recursion/diffeq operator:`) fi;\n if sols=NULL then \+ RETURN(FAIL)\n else oper := subs(sols,oper); fi;\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "celinexk(Fxk,x,k,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "celinexk(Fxk,x,k,2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*&%\" aG6$\"\"!F'\"\"\"*&&F%6$F'F(F(%#EkGF(F(*(F$F(%#DxG\"\"#F,F(!\"\"*(F*F( F.F/F,F/F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Same thing, calcula tions spelled out:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "celinexk(Fxk, x,k,2,2,true);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%/trial~equationG,4& %\"aG6$\"\"!F(\"\"\"**&F&6$F(F)F)%\"xG\"\"#,&%\"kGF)F)F)!\"\",&F0F.\" \"$F)F1#F)F.*.&F&6$F(F.F)F-\"\"%F/F1F2F1,&F0F)F.F)F1,&F0F.\"\"&F)F1#F) F8*(&F&6$F)F(F),&F0F.F)F)F)F-F1F)*(&F&6$F)F)F)F-F)F/F1F4*,&F&6$F)F.F)F -F3F/F1F2F1F9F1F<**&F&6$F.F(F)F@F)F0F)F-!\"#F.&F&6$F.F)F)**&F&6$F.F.F) F-F.F/F1F2F1F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%%8collected~in~power s~of~G%\"kG,>*&&%\"aG6$\"\"#\"\"!\"\"\"F$\"\"'\"#k*&,&F'\"$![*&&F(6$F, F+F,%\"xGF,\"#KF,F$\"\"&F,*&,*F'\"%g8*&&F(6$F*F,F,F5F*\"#;F2\"$S#*&&F( 6$F+F+F,F5F*F>F,F$\"\"%F,*&,,F2\"$!oF@\"$7\"F;FGF'\"%+=*&&F(6$F,F,F,F5 \"\"$\"\")F,F$FLF,*&,0F@\"$%G*&&F(6$F+F,F,F5FCFCF2\"$+*FI\"#[F;FPF'\"% '4\"*&&F(6$F*F*F,F5FCFCF,F$F*F,*&,2FW\"#=FQFfn*&&F(6$F,F*F,F5F7F*F2\"$ [&F@\"$3$FI\"#%*F'F?F;F[oF,F$F,F,*&&F(6$F+F*F,F5F-F,FW\"#?FgnF7F2\"$? \"FQF`oF@FaoF;FaoFI\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afinal~rec ursion/diffeq~operator:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*&%\"aG6$ \"\"!F'\"\"\"*&&F%6$F'F(F(%#EkGF(F(*(F$F(%#DxG\"\"#F,F(!\"\"*(F*F(F.F/ F,F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "collect(\",\{a[0, 0],a[0,1]\},distributed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%#E kG\"\"\"*&%#DxG\"\"#F&F*!\"\"F'&%\"aG6$\"\"!F'F'F'*&,&F'F'*&F)F*F&F'F+ F'&F-6$F/F/F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "The two opera tors are related by Ek-Dx^2 Ek^2 = Ek*(1-Dx^2 Ek), so just use th e second one. Note it was also possible to see this by inspection.\nT he desired mixed recurrence/diffeq is (RDE)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F(x,k) - diff(F(x,k+1),x$2) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"FG6$%\"xG%\"kG\"\"\"--&%\"DG6$F*F*6#F&6$F(,&F)F* F*F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 41 "f(x) = Sum(F(x,k),k=-infinity..infinity);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$SumG6$-%\"FG6$F'%\"k G/F.;,$%)infinityG!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Sum \+ (RDE) for k=-infinity..+infinity to get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f(x) - diff(f(x),x$2)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"fG6#%\"xG\"\"\"-%%diffG6$-F+6$F%F(F(!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "whose solution is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(\",f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&%$_C1G\"\"\"-%$expGF&F+F+*&%$_C2GF+-F -6#,$F'!\"\"F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The initial c onditions f(0)=0, f'(0)=1 are easily checked, so use them:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dsolve(\{\"\",f(0)=0,D(f)(0)=1\},f(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&*$-%$expGF&\"\"##\" \"\"F-#!\"\"F-F/F/F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s implify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%%sinhG F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 " " 0 "" {TEXT -1 10 "Koepf 12.1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "F nt := t^n * exp(-t^2 - x/t);\ncelinexk(Fnt,t,n,1,1,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FntG*&)%\"tG%\"nG\"\"\"-%$expG6#,&*$F'\"\"#!\" \"*&%\"xGF)F'F0F0F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "celinexk(Fnt,t,n,2,2,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "celinexk(Fnt,t,n,3,3,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*&&%\"aG6$\"\"\"\"\"#F(%\"xGF(!\"\"*&,&F%!\"#*&F%F(% \"nGF(F+F(%#EnGF(F(*&F%F(F1\"\"$F)*(&F&6$F)F)F(F*F(%#DtGF(F+*(,&F5F.*& F5F(F0F(F+F(F7F(F1F(F(*(F%F(F7F(F1F)F(*(F5F(F7F(F1F3F)*(&F&6$F3F)F(F*F (F7F)F+*(,&*&F>F(F0F(F+F>F.F(F7F)F1F(F(*(F5F(F7F)F1F)F(*(F>F(F7F)F1F3F )*(F>F(F7F3F1F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "collec t(\",\{a[1,2],a[2,2],a[3,2]\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(* &,**&,&%\"nG!\"\"!\"#\"\"\"F+%#EnGF+F+%\"xGF)*&%#DtGF+F,\"\"#F+*$F,\" \"$F0F+&%\"aG6$F+F0F+F+*&,**(F'F+F/F+F,F+F+*&F/F0F,F0F+*&F/F+F,F2F0*&F -F+F/F+F)F+&F46$F0F0F+F+*&,**(F'F+F/F0F,F+F+*&F/F0F,F2F0*&F-F+F/F0F)*& F/F2F,F0F+F+&F46$F2F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(a[2,2]=0,a[3,2]=0,a[1,2]=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&%\"nG!\"\"!\"#\"\"\"F)%#EnGF)F)%\"xGF'*&%#DtGF)F*\"\"#F)*$ F*\"\"$F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rde := \":" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 372 "So we have (-n-2) F(n+1,t) - x* F(n,t) + (d/dt) F(n+2,t) + 2 F(n+3,t) = 0.\n(The x is suppressed fro m the parameter list of F.)\nIntegrate with respect to t. The terms a re:\nintegral of (-n-2) F(n+1,t) dt: (-n-2) A(n+1,x)\nintegral of -x *F(n,t) dt: -x*A(n,x)\nintegral of (d/dt) F(n+2,t) dt = F(n+2,infi nity)-F(n+2,0) = 0-0 = 0\nintegral of 2 F(n+3,t) dt = 2 A(n+3,x)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "-(n+2) * A(n+1,x) - x*A(n,x) + 0 + 2*A(n+3,x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&%\"nG \"\"\"\"\"#F(F(-%\"AG6$,&F'F(F(F(%\"xGF(!\"\"*&F.F(-F+6$F'F.F(F/-F+6$, &F'F(\"\"$F(F.F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "There is also a command in Koepf's software that would have done this for us: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "intr ecursion(Fnt,t,A(n,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%\"AG6# ,&%\"nG\"\"\"\"\"$F*!\"#*&,&F)F*\"\"#F*F*-F&6#,&F)F*F*F*F*F**&-F&6#F)F *%\"xGF*F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "(It doesn't un derstand the extra parameter x.)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "I couldn't get a diff e q using Sister Celine's algorithm.\nKoepf's hsum package has a routine based on the continuous Gosper algorithm, and it produces" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "intdiffeq(t^n*exp(-t^2-x/t),t,A(x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"\"-%%diffG6$-F)6$-F )6$-%\"AG6#F&F&F&F&F'F'*&,&%\"nGF'!\"\"F'F'F+F'F5F/\"\"#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 41 } {VIEWOPTS 1 1 0 1 1 1803 }