{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 "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 "Map le 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/hw6ansm.mws" }{MPLTEXT 1 0 0 "" }}{PARA 258 "" 0 "" {TEXT 257 45 "Math 262a, Fall 1999, Glenn Tesler\nHomework 6" }{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 11 "Koepf # 9.1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "recpoly(n*s(n+1)-(n +20)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*L&%&deltaG6#\"\"! \"\"\"%\"nGF(,&F)F(\"#>F(F(,&F)F(\"#=F(F(,&F)F(\"# " 0 "" {MPLTEXT 1 0 35 "recpoly(n*s(n+1)-( n+40)*s(n),s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*^p&%&deltaG6#\" \"!\"\"\"%\"nGF(,&F)F(\"#RF(F(,&F)F(\"#QF(F(,&F)F(\"#PF(F(,&F)F(\"#OF( F(,&F)F(\"#NF(F(,&F)F(\"#MF(F(,&F)F(\"#LF(F(,&F)F(\"#KF(F(,&F)F(\"#JF( F(,&F)F(\"#IF(F(,&F)F(\"#HF(F(,&F)F(\"#GF(F(,&F)F(\"#FF(F(,&F)F(\"#EF( F(,&F)F(\"#DF(F(,&F)F(\"#CF(F(,&F)F(\"#BF(F(,&F)F(\"#AF(F(,&F)F(\"#@F( F(,&F)F(\"#?F(F(,&F)F(\"#>F(F(,&F)F(\"#=F(F(,&F)F(\"# " 0 "" {MPLTEXT 1 0 83 "rec:=\nsumrecursion(hyperterm([-n,a ,a+1/2,b],[2*a,(b-n+1)/2,(b-n)/2+1],1,k),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,(*,,(%\"nG\"\"\"\"\"#F*%\"bGF*F*,(F*F*F)F*F,! \"\"F*,&F,F.F)F*F*,(F)F*%\"aGF+F*F*F*-%\"sG6#,&F)F*F+F*F*F**,,(F)F*F*F *F,F*F*F/F*,(F)F*F1F*F*F*F*,*F*F*F1F+F)F*F,F.F*-F36#,&F)F*F*F*F*!\"#*, F " 0 "" {MPLTEXT 1 0 20 "rec2hyper(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$*.,&%\"nG\"\"\"F'F'F',&F&F'%\"bGF'F',(%\"aG\"\"#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 3 "" 0 "" {TEXT -1 12 "Koepf # 9.12" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 82 "rec:=sumrecursion(\n(-1)^k*binomial(r-s-k,k)*binomi al(r-2*k,n-k)/(r-n-k+1),k,s(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% $recG/,(**,&%\"nG\"\"\"\"\"#F*F*,(F)F+%\"rG!\"\"\"\"$F*F*,*F)F*F*F*F-F .%\"sGF*F*-F16#F(F*F**(,(F)F+F*F*F-F.F*,.*$F)F+F+F)F+*&F-F*F)F*!\"#*&F 1F*F-F*F**$F1F+F.F-F9F*-F16#,&F)F*F*F*F*F***,(F)F+F-F.F.F*F*,(F-F.F)F* F.F*F*,&F)F*F1F.F*-F16#F)F*F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rechyper(rec,s(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<$,$**,(%\"nG\"\"#%\"rG!\"\"F*\"\"\"F+,&F'F+%\"sGF*F+,&F'F+F+F+F*,(F 'F(F+F+F)F*F*F*,$**F&F+,(F)F*F'F+F*F+F+F/F*,(F'F+F)F*F-F+F*F*" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 12 "Koepf # 9.13" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "rec:=sumrecursion(binomial(n,k)*pochhammer(c ,k)*pochhammer(m,n-k)*\nhyperterm([-k,a,b],[c,d],1,j),j,s(k));" }} {PARA 12 "" 1 "" {XPPMATH 20 ">%$recG/,(*,,&%\"kG\"\"\"\"\"#F)F),*%\"m GF)%\"nGF)F(!\"\"F.F)F),*F-F)!\"#F)F,F)F(F.F),(F(F)%\"dGF)F)F)F)-%\"sG 6#F'F)F.*,,&F-F)F(F.F),(F-F)F.F)F(F.F),&%\"cGF)F(F)F),,%\"aGF)%\"bGF)F (F.F:F.F2F.F)-F46#F(F)F)**F8F)F+F), " 0 "" {MPLTEXT 1 0 20 "rec2hyper(rec,s(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<\"" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 12 "Koepf # 9.16" }}}{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 0 "" 0 "" {TEXT -1 6 "Rogers" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 99 "F:=(-1)^k*q^(k*(3*k-1)/2)/qpochhammer(q,q,n+k)/qpoc hhammer(q,q,n-k):\nRE:=qsumrecursion(F,q,k,S(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#REG/,**,,&)%\"qG,$%\"nG\"\"#!\"\"F*\"\"\"F/,&)F*F,F/ F.F/F/F*F/,&F/F/F1F/F/-%\"SG6#F,F/F.*&,.)F*,&F,F-F-F/F/*$F*F-F.*$F*\" \"%F.*$F*\"\"$F.)F*,$F,F>F/)F*,&F,F>F/F/F/F/-F46#,&F,F/F.F/F/F/*(,*F:F /F;F/F=F/F)F/F/F*F/-F46#,&F,F/!\"#F/F/F/*&F*\"\"&-F46#,&F,F/!\"$F/F/F. \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "qrecsolve(RE,q,S(n ),return=qhypergeometric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$*$-% ,qpochhammerG6%%\"qGF)%\"nG!\"\"1\"\"!F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 337 "There is one q-hypergeometric solution (up to multiplica tive constant). Call it S1. The recursion has 2 other linearly indep endent solutions, S2 and S3, that we have not discovered. The general solution is A1*S1+A2*S2+A3*S3 for constants (w.r.t. n) A1,A2,A3.\nChe ck 3 initial conditions to confirm that it is just S1 (=1*S1+0*S2+0*S3 )." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "Fn := proc(n0)\n local \+ k0;\n qsimpcomb(sum(subs(k=k0,n=n0,F),k0=-n0..n0))\nend:\n'Fn(n)' $'n'=0..5; 'qsimplify(Fn(n)/(1/qpochhammer(q,q,n)))'$'n'=0..5;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6(\"\"\",$*$,&!\"\"F#%\"qGF#F'F'*&F&!\"# ,&F(F#F#F#F',$*(F&!\"$F+F',(*$F(\"\"#F#F(F#F#F#F'F'**F&!\"%F+F*,&F0F#F #F#F'F/F',$*,F&!\"&F+F*,,*$F(\"\"%F#*$F(\"\"$F#F0F#F(F#F#F#F'F/F'F4F'F '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"\"F#F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "\"Creative Symmetrizing\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rat:=qsimpcomb(subs(k=-k,F)/F);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ratG)%\"qG%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "qsumrecursion((1+rat)/2*F,q,k,S(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&)%\"qG%\"nG!\"\"\"\"\"F+F+-%\"S G6#F)F+F+-F-6#,&F)F+F*F+F*\"\"!" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 92 "reduceorder(rec,f,n,u,w) -- implement the reduction of order formu la given in the answer key" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 236 " f, n,w are unassigned names; u is an expression in n\nReduction of order, using formulas from answer key.\nrecursion is in f(n), one solution i s u(n), output recursion in w(n) s.t. other solutions are f(n)=u(n) v( n) with w(n)=v(n+1)-v(n)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1139 "redu ceorder := proc(rec, f, n, u, w)\n local f_uv,rec_uv,v, ord,i,ws;\n \n # make a function n -> u(n)*v(n)\n f_uv := unapply(u*v(n),n); \n\n # substitute it in to the recursion\n rec_uv := eval(subs(f =f_uv,rec));\n\n # reexpress v(n+k)'s in terms of w(n)=v(n+1)-v(n) \n ord := getrecorder(rec_uv,v,n);\n for i from ord to 1 by -1 d o\n rec_uv := subs(v(n+i)=v(n+i-1)+w(n+i-1),rec_uv);\n od;\n \n # maple needs coersion to simplify this\n ws := \{'w(n+i)'$'i '=0..ord-1\};\n\n rec_uv := lhs(rec_uv);\n rec_uv := collect(rec _uv,ws);\n\n # all coeffs. involve various u(n+k)'s, but if\n # \+ u is hypergeometric, dividing by u(n) would\n # simplify them all t o rational functions.\n # Equivalently, dividing by the coeff. of t he leading\n # order term should reduce them all to rational.\n \+ \n rec_uv := simpcomb(rec_uv / coeff(rec_uv,w(n+ord-1)));\n \n \+ rec_uv := collect(numer(rec_uv),ws,factor) = 0;\nend:\n\ngetrecorder := proc(rec,f,n)\n local fs;\n fs := indets(rec,function);\n \+ fs := select((func,f) -> evalb(op(0,func)=f),\n fs,f);\n \+ map((func,n) -> op(1,func)-n,fs,n);\n max(op(\"))\nend:" }}}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 14 "A=B p. 165 # 4" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "First do the whole problem fo r f(n). Part (a):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "F := (n,k) - > binomial(3*k,k) * binomial(3*n-3*k,n-k);\nsumF := proc(n)\n loca l k;\n sum(F(n,k),k=0..n)\nend;\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&-%)binom ialG6$,$9%\"\"$F2\"\"\"-F/6$,&9$F3F2!\"$,&F8F4F2!\"\"F4F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sumFG:6#%\"nG6#%\"kG6\"F*-%$sumG6$-%\"FG6 $9$8$/F2;\"\"!F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rec f := sumrecursion(F(n,k),k,f(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %%recfG/,(*(,&%\"nG\"\"\"\"\"#F*F*,&F)F+\"\"$F*F*-%\"fG6#F(F*\"\")*&,( *$F)F+\"#OF)\"#**\"#qF*F*-F/6#,&F)F*F*F*F*!\"'*(,&F)F-\"\"%F*F*,&F)F-F +F*F*-F/6#F)F*\"#\")\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "hyper_f := rechyper(recf,f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(hyper_fG<##\"#F\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "So the only hypergeometric solution is" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f1_sol := n -> (27/4)^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'f1_solG:6#%\"nG6\"6$%)operatorG%&arrowGF( )#\"#F\"\"%9$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "recw : = reduceorder(recf,f,n,f1_sol(n),w);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%recwG/,&*(,&%\"nG\"\"\"\"\"#F*F*,&F)F+\"\"$F*F*-%\"wG6#,&F)F*F*F* F*\"\"**(,&F)F-\"\"%F*F*,&F)F-F+F*F*-F/6#F)F*!\"#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "This leads to a solution for w(n):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "w1_sol := unapply(rsolve(recw,w(n)) ,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'w1_solG:6#%\"nG6\"6$%)opera torG%&arrowGF(,$*0\"\"$#\"\"\"\"\"#%#PiG!\"\"-%\"wG6#\"\"!F0-%&GAMMAG6 #,&9$F1F.F0F3-F96#,&#\"\"%F.F0F " 0 "" {MPLTEXT 1 0 59 "w2_sol := n -> binomial(3*(n+1),n+1)/(3*n+2) * (27/4)^(-n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'w2_solG:6#%\"nG6 \"6$%)operatorG%&arrowGF(*(-%)binomialG6$,&9$\"\"$F2\"\"\",&F1F3F3F3F3 ,&F1F2\"\"#F3!\"\")#\"#F\"\"%,$F1F7F3F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "simplify(w1_sol(n)/w2_sol(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%\"wG6#\"\"!#\"\"#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "(We have n+1 for the book's n because they did reduction of order with the backwards antidifference v(n)-v(n-1), while we used the forwards antidifference v(n+1)-v(n).)" }}{PARA 0 "" 0 "" {TEXT -1 208 "Thus, v(n) = v(0) + w(0)+w(1)+...+w(n-1); there is no chance w hatsoever that this will be hypergeometric for any choice of the const ant v(0), because otherwise, it would have been discovered already by \+ Hyper." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "v1_sol := unapply(Sum(w2_ sol(k),k=0..n-1),n):\nf2_sol := unapply(f1_sol(n)*v1_sol(n),n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'f2_solG:6#%\"nG6\"6$%)operatorG%&ar rowGF(*&)#\"#F\"\"%9$\"\"\"-%$SumG6$*(-%)binomialG6$,&%\"kG\"\"$F " 0 "" {MPLTEXT 1 0 45 "eval f2 := n -> eval(subs(Sum=sum,f2_sol(n)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'evalf2G:6#%\"nG6\"6$%)operatorG%&arrowGF(-%%evalG6#- %%subsG6$/%$SumG%$sumG-%'f2_solG6#9$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqs := 'sumF(n)=A*f1_sol(n)+B*evalf2(n)'$'n'=0..1;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG6$/\"\"\"%\"AG/\"\"',&F(#\"#F \"\"%%\"BG#\"#\")\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s olve(\{eqs\},\{A,B\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"AG\"\" \"/%\"BG#!\"#\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "full_f := subs(\",A*f1_sol(n)+B*f2_sol(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'full_fG,&)#\"#F\"\"%%\"nG\"\"\"*&F&F+-%$SumG6$*(-%)binomialG6$,& %\"kG\"\"$F6F+,&F5F+F+F+F+,&F5F6\"\"#F+!\"\")F',$F5F:F+/F5;\"\"!,&F:F+ F*F+F+#!\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now do this all for g(n). Part (a ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "G := (n,k)->binomial (3*k,k) * binomial(3*n-3*k-2,n-k-1);\nsumG := proc(n)\n local k;\n sum(G(n,k),k=0..n)\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"G G:6$%\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&-%)binomialG6$,$9%\"\"$F2\" \"\"-F/6$,(9$F3F2!\"$!\"#F4,(F8F4F2!\"\"F%%sumGG:6#%\"nG6#%\"kG6\"F*-%$sumG6$-%\"GG6$9$8$/F2; \"\"!F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "recg := sumr ecursion(G(n,k),k,g(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%recgG/, ***,&%\"nG\"\"#\"\"&\"\"\"F,,&F)F*\"\"$F,F,,&F)F,F*F,F,-%\"gG6#,&F)F,F .F,F,!#;*(F-F,,(*$F)F*\"#aF)\"$`\"\"$I\"F,F,-F16#F/F,\"#7*&,**$F)F.\"# FF7\"#sF)\"#w\"#IF,F,-F16#,&F)F,F,F,F,!$C$**,&F)F.F*F,F,,&F)F.F,F,F,F) F,-F16#F)F,\"%(=#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "(b)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "hyper_g := rechyper(recg,g(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(hyper_gG<$#\"#F\"\"%,$*&%\"nG\"\"\" ,&F+\"\"#F,F,!\"\"#F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " ratio(27^n/(n*binomial(2*n,n)),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&%\"nG\"\"\",&F%\"\"#F&F&!\"\"#\"#FF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "So there are two hypergeometric solutions." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 149 "g1_sol := n -> (27/4)^n;\n# g2_sol := n -> (2 7/4)^n * GAMMA(n)/GAMMA(n+1/2); # straigtforward\ng2_sol := n->27^n/(n *binomial(2*n,n)); # simpler looking" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'g1_solG:6#%\"nG6\"6$%)operatorG%&arrowGF()#\"#F\"\"%9$F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g2_solG:6#%\"nG6\"6$%)operatorG%&ar rowGF(*()\"#F9$\"\"\"F/!\"\"-%)binomialG6$,$F/\"\"#F/F1F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "but there is another linearly ind ependent solution because the recursion has order 3. Reduce order to \+ find it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "recw1 := reduce order(recg,g,n,g1(n),w1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&recw1G /,(**%\"nG\"\"\",&F(\"\"$\"\"#F)F),&F(F+F)F)F)-%#w1G6#F(F)\"\"%*(,&F(F ,F+F)F),(*$F(F,\"\"*F(\"#=\"#5F)F)-F/6#,&F(F)F)F)F)!\"%**,&F(F,\"\"&F) F)F3F),&F(F)F,F)F)-F/6#F@F)F6\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Now reduce the order again. We could either find a solution o f recg_w from scratch, or we can use the second solution of the origin al recursion to create such a solution:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "dg2 := simpcomb(g2_sol(n+1)/g1_sol(n+1) - g2_sol(n)/g1_sol(n) ):\ndg2 := unapply(dg2,n); # turn it into a function" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dg2G:6#%\"nG6\"6$%)operatorG%&arrowGF(,$*,)\"#F9$ \"\"\"-%&GAMMAG6#,&F0F1F1F1F1)#F/\"\"%F0!\"\"-F36#F0F1-F36#,&F0\"\"#F? F1F9F9F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Check it:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(eval(subs(w1=dg2,recw1)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Cleaner solution:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "d g2b := n -> 4^n / (n*(2*n+1)*binomial(2*n,n));\nsimplify(dg2(n)/dg2b(n ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dg2bG:6#%\"nG6\"6$%)operator G%&arrowGF(**)\"\"%9$\"\"\"F/!\"\",&F/\"\"#F0F0F1-%)binomialG6$,$F/F3F /F1F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 33 "Do the second reduction of order:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "recw2 := reduceorder(recw1,w1,n,dg2 b(n),w2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&recw2G/,&*(,&%\"nG\"\" $\"\"#\"\"\"F,,&F)F*F,F,F,-%#w2G6#F)F,!\"\"*(,&F)F,F,F,F,,&F)F,F+F,F,- F/6#F3F,\"\"*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The automat ic solution for this is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "w2_auto \+ := rsolve(recw2,w2(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(w2_autoG ,$*0-%&GAMMAG6#,&%\"nG\"\"\"#\"\"#\"\"$F,F,-F(6#,&#F,F/F,F+F,F,-F(6#,& F+F,F.F,!\"#,&F+F,F,F,F,%#PiG!\"\"F/#F,F.-%#w2G6#\"\"!F,F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "and a cleaner looking solution is" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "w2_sol := n -> binomial(3*n,n)*bino mial(2*n,n)/(n+1)/27^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'w2_solG: 6#%\"nG6\"6$%)operatorG%&arrowGF(**-%)binomialG6$,$9$\"\"$F1\"\"\"-F.6 $,$F1\"\"#F1F3,&F1F3F3F3!\"\")\"#FF1F9F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Check it:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simpcomb(e val(subs(w2=w2_sol,recw2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"! F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Then v2 is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v2_sol := gosper(w2_sol(n),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'v2_solG,$**%\"nG\"\"\"-%)binomialG6$,$F'\"\"$F'F( -F*6$,$F'\"\"#F'F()\"#FF'!\"\"#\"\"*F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "So w1 is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "w1_sol := u napply(v2_sol * dg2b(n),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'w1_s olG:6#%\"nG6\"6$%)operatorG%&arrowGF(,$**-%)binomialG6$,$9$\"\"$F2\"\" \")\"#FF2!\"\")\"\"%F2F4,&F2\"\"#F4F4F7#\"\"*F;F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sim pcomb(eval(subs(w1=w1_sol,recw1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "We don't expect thi s to work, because the original recursion doesn't have 3 lin. ind. hyp ergeometric solutions:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v1_sol := gosper(w1_sol(n),n);" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, (in gospe r) no hypergeometric term antidifference exists" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "So instead use" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "v1_sol := Sum(w1_sol(k),k=0..n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'v1_solG-%$SumG6$,$**-%)binomialG6$,$%\"kG\"\"$F.\"\"\")\"#FF.!\"\" )\"\"%F.F0,&F.\"\"#F0F0F3#\"\"*F7/F.;\"\"!,&F3F0%\"nGF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g3_sol := unapply(g1_sol(n) * v1_so l,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g3_solG:6#%\"nG6\"6$%)oper atorG%&arrowGF(*&)#\"#F\"\"%9$\"\"\"-%$SumG6$,$**-%)binomialG6$,$%\"kG \"\"$F " 0 "" {MPLTEXT 1 0 62 " eqs := 'sumG(n)=A*g1_sol(n)+B*g2_sol(n)+C*g3_sol(n)'$'n'=1..3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqs := eval(subs(Sum=sum,\{e qs\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eqsG<%/\"\"\",(%\"AG#\"# F\"\"%%\"BG#F+\"\"#%\"CG#\"$V#\"\")/\"#[,(F)#\"&$o>\"#kF-#\"%hl\"#?F0# \"'b]@\"$G\"/\"\"(,(F)#\"$H(\"#;F-#F2F,F0#\"%Lv\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sols := solve(eqs,\{A,B,C\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "g_full := subs(sols,A*g1_sol(n)+B*g 2_sol(n)+C*g3_sol(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g_fullG,& )#\"#F\"\"%%\"nG#\"\"\"\"\"**&F&F,-%$SumG6$,$**-%)binomialG6$,$%\"kG\" \"$F8F,)F(F8!\"\")F)F8F,,&F8\"\"#F,F,F;#F-F>/F8;\"\"!,&F;F,F*F,F,#F>\" $V#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "which is equivalent to the solution given in A=B, p. 167." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "(c) the book has a complete answer." }{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0 " 41 }{VIEWOPTS 1 1 0 1 1 1803 }