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This work is a follow-up to a talk by Donald Lutz at our Algor ithms seminar, and summarized in [Durand]." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 33 "1. Borel-Laplace resummation and " }{TEXT 261 0 "" } {TEXT -1 9 "Euler acc" }{TEXT 260 0 "" }{TEXT -1 9 "eleration" }} {PARA 0 "" 0 "" {TEXT -1 108 "Starting with a linear differential equa tion with polynomial coefficients satisfied by a formal power series" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^~ = Sum(x[n]*z^n,n = 0 .. infinit y);" "6#/)%\"xG%\"|irG-%$SumG6$*&&F%6#%\"nG\"\"\")%\"zGF-F./F-;\"\"!%) infinityG" }{TEXT 257 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 87 "it is possible to compute a diffe rential equation satisfied by the Borel transform of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 17 ". We assume that " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 31 " is Gevrey 1, which means \+ that " }{XPPEDIT 18 0 "x[n] <= A*c^n*n!;" "6#1&%\"xG6#%\"nG*(%\"AG\"\" \")%\"cGF'F*-%*factorialG6#F'F*" }{TEXT -1 21 " for some constants " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 34 ", so that the Borel transform of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 11 " defined by" }}{PARA 257 " " 0 "" {XPPEDIT 18 0 "y(z) = Sum(x[n]*z^n/n!,n = 0 .. infinity);" "6#/ -%\"yG6#%\"zG-%$SumG6$*(&%\"xG6#%\"nG\"\"\")F'F/F0-%*factorialG6#F/!\" \"/F/;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 121 "is convergent on some neighbourhood of the origin. The Borel tran sform has an \"inverse\", the Laplace transform defined by" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Laplace(y) = Int(exp(-t/z)* y(t),t = 0 .. infinity);" "6#/-%(LaplaceG6#%\"yG-%$IntG6$*&-%$expG6#,$ *&%\"tG\"\"\"%\"zG!\"\"F4F2-F'6#F1F2/F1;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "Provided this integral converges, t he function it defines has " }{XPPEDIT 18 0 "z*x^~;" "6#*&%\"zG\"\"\") %\"xG%\"|irGF%" }{TEXT -1 18 " for expansion as " }{XPPEDIT 18 0 "proc (z) options operator, arrow; 0 end;" "6#R6#%\"zG7\"6$%)operatorG%&arr owG6\"\"\"!F*F*F*" }{TEXT -1 42 " +. Then applying the change of vari able " }{XPPEDIT 18 0 "t = phi(alpha);" "6#/%\"tG-%$phiG6#%&alphaG" } {TEXT -1 69 " to the integral computes the acceleration \"a la Euler\" [Lutz et al.]" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z*x ^~ = Int(exp(-t/z)*y(phi(psi(t))),t = 0 .. infinity);" "6#/*&%\"zG\"\" \")%\"xG%\"|irGF&-%$IntG6$*&-%$expG6#,$*&%\"tGF&F%!\"\"F4F&-%\"yG6#-%$ phiG6#-%$psiG6#F3F&/F3;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "We note" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "y(phi(z)) = Sum(d[n]*z^n,n = 0 .. infinity);" "6#/-%\"yG6#-%$phiG6#%\"zG-%$SumG 6$*&&%\"dG6#%\"nG\"\"\")F*F2F3/F2;\"\"!%)infinityG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "q[n](z) = Int(exp(-t/z)*psi(t)^n,t = 0 .. infinity); " "6#/-&%\"qG6#%\"nG6#%\"zG-%$IntG6$*&-%$expG6#,$*&%\"tG\"\"\"F*!\"\"F 6F5)-%$psiG6#F4F(F5/F4;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 53 " is the functional inverse of the rational function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 47 ". In terms of formal power series , the product " }{XPPEDIT 18 0 "z*x^~;" "6#*&%\"zG\"\"\")%\"xG%\"|irGF %" }{TEXT -1 31 " equals the Taylor expansion of" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "Sum(q[n]*d[n],n = 0 .. infinity);" "6#-%$SumG6$*&&%\"qG 6#%\"nG\"\"\"&%\"dG6#F*F+/F*;\"\"!%)infinityG" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 20 "where the integrals " }{XPPEDIT 18 0 "q[n ];" "6#&%\"qG6#%\"nG" }{TEXT -1 21 " are independent of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "This process is illustrated in \+ the present worksheet using a simple mapping function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 153 " on the double confluent Heun equat ion. The Heun equation is the generic differential equation with four \+ regular singular points located at 0, 1, c, and " }{XPPEDIT 18 0 "infi nity;" "6#%)infinityG" }{TEXT -1 134 ", see [DuLoRi92]. The double con fluent Heun equation is obtained by letting the singularity located a t c tend to the one located at " }{XPPEDIT 18 0 "infinity;" "6#%)infi nityG" }{TEXT -1 127 ", and the singularity located at 1 tend to 0. T he equation obtained then has two irregular \nsingular points located at 0 and " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 77 ". The example we study is the double confluent Heun equation in the f orm " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "heun_infp := z^2*di ff(f(z),z,z)+(z+alpha*z^2+alpha)*diff(f(z),z)+(2*alpha*z^2*beta[1]+alp ha*z^2+alpha^2*z-2*gamma*z+2*alpha*beta[-1]-alpha)*f(z)/(2*z):" "6#>%* heun_infpG,(*&%\"zG\"\"#-%%diffG6%-%\"fG6#F'F'F'\"\"\"F/*&,(F'F/*&%&al phaGF/*$F'\"\"#F/F/F3F/F/-F*6$-F-6#F'F'F/F/*(,.**\"\"#F/F3F/F'\"\"#&%% betaG6#\"\"\"F/F/*&F3F/*$F'\"\"#F/F/*&F3\"\"#F'F/F/*(\"\"#F/%&gammaGF/ F'F/!\"\"*(\"\"#F/F3F/&F@6#,$\"\"\"FKF/F/F3FKF/-F-6#F'F/*&\"\"#F/F'F/F KF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "readlib(gfun):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Since the point of interest is inf inity and the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 66 " packag e works at the origin, we first change the variable (using " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 2 "):" }{TEXT 267 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "heunp:=gfun[algebraicsubs](h eun_infp,z*f-1,f(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&heunpG,(*& ,.*()%\"zG\"\"#\"\"\"%&alphaG\"\"\"&%%betaG6#!\"\"F.!\"#*&F-F,F)F,F.*& )F-F+F,F*F.F2*&%&gammaGF.F*F,F+*&F-F,&F06#F.F.F3F-F2F.-%\"fG6#F*F.F.*& ,(*$F)F,F3*&)F*\"\"$F,F-F,F+*&F*F,F-F,F+F.-%%diffG6$F " 0 "" {XPPEDIT 19 1 "paramform:=[alpha=-1,beta[-1]= 1/2,beta[1]=1/2,gamma=1/3]:" "6#>%*paramformG7&/%&alphaG,$\"\"\"!\"\"/ &%%betaG6#,$\"\"\"F**&\"\"\"\"\"\"\"\"#F*/&F-6#\"\"\"*&\"\"\"F3\"\"#F* /%&gammaG*&\"\"\"F3\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Sub stitution of these parameters in the differential equation gives" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "heun:=subs(paramform,heunp); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%heunG,(*&,&%\"zG#!\"\"\"\"$\"\" #\"\"\"F--%\"fG6#F(F-F-*&,(*$)F(F,\"\"\"!\"#*$)F(F+F5F6F(F6F--%%diffG6 $F.F(F-F-*&F8F5-F:6$F.-%\"$G6$F(F,F-F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "From this equation, we obtain a recurrence equation for t he Taylor series coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "recheunseries:=gfun[diffeqtorec](heun,f(z),u(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%.recheunseriesG<$,(*&%\"nG\"\"\"-%\"uG6#F(F)! \"'*&,(!\"(F)F(!#7*$)F(\"\"#\"\"\"F-F)-F+6#,&F(F)F)F)F)F)*&,&F-F)F(F-F )-F+6#,&F(F)F4F)F)F)/-F+6#\"\"!FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This recurrence yields an efficient procedure to evaluate the coef fici" }{TEXT 266 0 "" }{TEXT -1 17 "ents recursively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "heundiv:=gfun[rectoproc](recheunser ies union \{u(1)=1/2\},u(n),remember);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(heundivGR6#%\"nG6\"6#%)rememberGE\\s#\"\"\"#F,\"\"#\"\"!F/,$* &,(-9!6#,&9$F,!\"#F,\"#7-F46#,&F7F,!\"\"F,!\"(*&,(F3!\"'F:F9*&F:F,F7F, FAF,F7\"\"\"F,FC,&\"\"'F,F7FA!\"\"F=F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "From this procedure, the divergence is clear from the gro wth of the first coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(heundiv(i),i=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61# \"\"\"\"\"##!\"(\"#7#\"$R\"\"$W\"#!%Pm\"%#f##\"'`()f\"&3A'#!)\"z)e')\" (Si'=#\",Z**)3D=\")SY=n#!.H7Y$zr_\"+!)[v@G#\"0^45;%\\`G\",?j<\\$>#!3V. Z`t%*41'*\".g*['))RJ(#\"6JV1$)z2E?]r&\"0+w$*=$R)Q%#!9PQ`%>@#[rxB?T\"2+ ;))\\]Rj*G#\" " 0 "" {XPPEDIT 19 1 "calculheundiv:=proc(heundiv,z)\nlocal total,previous ,last,n;\nprevious:=heundiv(1)*z;total:=previous;last:=heundiv(2)*z^2; \nfor n from 3 while abs(previous)>abs(last) do\ntotal:=total+last;\np revious:=last;\nlast:= heundiv(n)*z^n od;\nuserinfo(1,'heundiv',n,last );\nevalf(total)\nend:" "6#>%.calculheundivGR6$%(heundivG%\"zG7&%&tota lG%)previousG%%lastG%\"nG6\"F.C(>F+*&-F'6#\"\"\"\"\"\"F(F5>F*F+>F,*&-F '6#\"\"#F5*$F(\"\"#F5?(F-\"\"$F5F.2-%$absG6#F,-FB6#F+C%>F*,&F*F5F,F5>F +F,>F,*&-F'6#F-F5)F(F-F5-%)userinfoG6&\"\"\".F'F-F,-%&evalfG6#F*F.F.F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot('calculheundiv'(he undiv,z),z=0..0.3);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7]p7$ \"\"!F(7$$\"+]i9Rl!#7$\"+w[*[C$F,7$$\"+WA)GA\"!#6$\"+Pw))GgF,7$$\"+Qeu i=F2$\"+FMF<\"*F,7$$\"+j3&o]#F2$\"+(3&>=7F27$$\"+pX*y9$F2$\"+#*)G*=:F2 7$$\"+WTAUPF2$\"+[t.%z\"F27$$\"+%*zhdVF2$\"+f0Bv?F27$$\"+%>fS*\\F2$\"+ 1d@iBF27$$\"+>$f%GcF2$\"+\"[0Xk#F27$$\"+Dy,\"G'F2$\"+:&R5$HF27$$\"+8LZF27$$\"+/QBE6Fcp$\"+X=k,]F27$$\"+:o?&=\" Fcp$\"+8G!\\B&F27$$\"+a&4*\\7Fcp$\"+sWo)[&F27$$\"+j=_68Fcp$\"+jwt6dF27 $$\"+Wy!eP\"Fcp$\"+[h&=&fF27$$\"+UC%[V\"Fcp$\"+BCPohF27$$\"+J#>&)\\\"F cp$\"+6InWkF27$$\"+>:mk:Fcp$\"+Fn/'p'F27$$\"+w&QAi\"Fcp$\"+t-r9pF27$$ \"+v4L`;Fcp$\"+JT!H.(F27$$\"+uLU%o\"Fcp$\"+&f#G^rF27$$\"+ZPX#p\"Fcp$\" +1w*==(F27$$\"+?T[+?sF27$$\"+2$* \\/luF27$$\"+M aKs=Fcp$\"+Z0Fcp$\"+h[LjyF27$$\"+:K^+?Fcp$\"+(eMI\"y?Fcp$\"+QgCL')F27$$\"+)[k*z?Fcp$\"+* GD-k)F27$$\"+v;I(3#Fcp$\"+as:o')F27$$\"+i)QY4#Fcp$\"+bO6'p)F27$$\"+OKJ 4@Fcp$\"+*o-@v)F27$$\"+4w)R7#Fcp$\"+0#)>3))F27$$\"+WN2c@Fcp$\"+lBDJ*)F 27$$\"+y%f\")=#Fcp$\"+rp!\\0*F27$$\"+/-a[AFcp$\"+:OY*G*F27$$\"+ial6BFc p$\"+vdlP&*F27$$\"+j@OtBFcp$\"+(*[!Qy*F27$$\"+fL'zV#Fcp$\"+t)zX+\"Fcp7 $$\"+!*>=+DFcp$\"+b#*GI5Fcp7$$\"+E&4Qc#Fcp$\"+g(Hr0\"Fcp7$$\"+g)f`f#Fc p$\"+&3m12\"Fcp7$$\"+%>5pi#Fcp$\"+!*[O%3\"Fcp7$$\"+NfSTEFcp$\"+fdr!4\" Fcp7$$\"+v;!fl#Fcp$\"+cP5(4\"Fcp7$$\"+WOrdEFcp$\"+7\\!z4\"Fcp7$$\"+6c_ fEFcp$\"+lmq)4\"Fcp7$$\"+zvLhEFcp$\"+nb45(*F27$$\"+Y&\\Jm#Fcp$\"+jYu8( *F27$$\"+\"[tnm#Fcp$\"+5^-@(*F27$$\"+;uRqEFcp$\"+4=GG(*F27$$\"+'GXwn#F cp$\"++QsU(*F27$$\"+bJ*[o#Fcp$\"+a-2d(*F27$$\"+k17=FFcp$\"+Jig@)*F27$$ \"+r\"[8v#Fcp$\"+Yd3%))*F27$$\"+Ijy5GFcp$\"+ZKf!***F27$$\"+/)fT(GFcp$ \"+(Q\\'45Fcp7$$\"+1j\"[$HFcp$\"+j#=!>5Fcp7$$\"\"$!\"\"$\"+](=#G5Fcp-% 'COLOURG6&%$RGBG$\"#5FddlF(F(-%+AXESLABELSG6$Q\"z6\"%!G-%%VIEWG6$;F(Fb dl%(DEFAULTG" 2 376 376 376 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 -22808 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "z;" " 6#%\"zG" }{TEXT -1 68 " tends to infinity, the imprecision of this sum mation grows quickly." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 57 "2. Diffe rential equation satisfied by the Borel transform" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 116 "The class of solutions of linear differential equ ations enjoys many closure properties which are implemented in the " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 225 " package for the case o f equations with polynomial coefficients. One of them is closure under Borel transform. Here is the differential equation satisfied by the B orel transform of the divergent solution of the Heun equation:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "bdeqp:=op(select(has,gfun[bo rel](heunp,f(z),'diffeq'),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&b deqpG,**&,&*&%&alphaG\"\"\"&%%betaG6#!\"\"F*\"\"#F)F.F*-%\"fG6#%\"zGF* F**&,**&F3F*F)\"\"\"!\"#*$)F)F/F7F*F/F*%&gammaGF8F*-%%diffG6$F0F3F*F** &,(F3\"\"'*&F)F7&F,6#F*F*F/F)!\"$F*-F=6$F0-%\"$G6$F3F/F*F**&,&*$)F3F/F 7F/F6F8F*-F=6$F0-FI6$F3\"\"$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "and the equation specialized at the parameters" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "bdeq:=subs(paramform,bdeqp):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "From this follows the recurrence satisfie d by the Taylor coefficients of the Borel transform:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "collect(gfun[diffeqtorec](bdeqp,f(z),a(n) ),a,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%&alphaG\"\"\",(&% %betaG6#!\"\"!\"#F&F&%\"nG\"\"#F&-%\"aG6#F-F&F+*(,&F-F&F&F&F&,,*$)F-F. \"\"\"F.F-\"\"%*$)F%F.F7F&F.F&%&gammaGF,F&-F06#F3F&F&*,F%F7F3F7,&F-F&F .F&F&,(F-F.&F)6#F&F,\"\"$F&F&-F06#F?F&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "l15:= gfun[rectoproc](\{subs(paramform,%),a(0)=0,a(1)=1/2\},a(n),list)(15); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$l15G72\"\"!#\"\"\"\"\"##!\"(\"# C#\"$R\"\"$k)#!%Pm\"&3A'#\"'`()f\"(g\\Y(#!)\"z)e')\"++GpV8#\",Z**)3D= \"-+ce5'Q$#!.H7Y$zr_\"0+;wcJx8\"#\"0^45;%\\`G\"1+;+.\"H9-(#!3V.Z`t%*41 '*\"5+![gM4?+Tl##\"6JV1$)z2E?]r&\"8++o\"RohKhq^<#!9PQ`%>@#[rxB?T\";++c 5iOD.d7N(Q\"#\"++;%3tFaCew2c)H\"#!?Fb!Htc\\\\$*HrRQ3 d$\"A++s-z?32S!G!o$G!=9#\"B$3lJrLh,5u(H=%[M\"=%\"D++?Fu&>M#*[I`rVdr'y \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "This list corresponds to th e list above, the " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 27 "th ele ment being scaled by " }{XPPEDIT 18 0 "1/k!;" "6#*&\"\"\"\"\"\"-%*fact orialG6#%\"kG!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "This differential equation and this recurrence can be used to \+ compute (but not necessarily fast) the analytic continuation of the Bo rel transform. " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 93 "3. The coeffic ients of the composition with an algebraic function satisfy a linear r ecurrence" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "This is another clo sure property of solutions of linear differential equation that we exp loit here. The coefficients " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG " }{TEXT -1 15 " are defined by" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "y( phi(z)) = Sum(d[n]*z^n,n = 0 .. infinity);" "6#/-%\"yG6#-%$phiG6#%\"zG -%$SumG6$*&&%\"dG6#%\"nG\"\"\")F*F2F3/F2;\"\"!%)infinityG" }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 28 " is a rational function and " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 15 " is defined by" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "y(z) \+ = Sum(x[n]*z^n/n!,n = 0 .. infinity);" "6#/-%\"yG6#%\"zG-%$SumG6$*(&% \"xG6#%\"nG\"\"\")F'F/F0-%*factorialG6#F/!\"\"/F/;\"\"!%)infinityG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 42 "On the example of the H eun equation, with " }{XPPEDIT 18 0 "phi = 1/((1-t)^2)-1;" "6#/%$phiG, &*&\"\"\"\"\"\"*$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }{TEXT -1 10 " we obtain" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eq:=(1-z)^2*(f+ 1)-1:" "6#>%#eqG,&*&,&\"\"\"\"\"\"%\"zG!\"\"\"\"#,&%\"fGF)\"\"\"F)F)F) \"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dneqp:=gfun[alg ebraicsubs](bdeqp,eq,f(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dneq pG,**&,&%&alphaG\"\"%*&F(\"\"\"&%%betaG6#!\"\"F+!\")F+-%\"fG6#%\"zGF+F +*&,hn!\"%F+%&gammaGF)*&F(\"\"\"&F-6#F+F+\"\"'F4\"#[*&)F(\"\"#F:)F4FAF :!\"'*&F(F:FBF:!$n#F(!\"**&F@F:F4F+F=*&F8F+F4F:!#7*$FBF:!$)>*(F(F:FBF: F;F:\"#g*&F8F:FBF:\"#7*&)F4\"\"$F:F(F:\"$p%*&F4F:F(F:\"#$)*$FQF:\"$1%* (F(F:F;F:F4F:!#I*&)F4F)F:F(F:!$&\\*&)F4\"\"&F:F(F:\"$4$*&)F4F=F:F(F:!$ 0\"*&F@F:FQF:FA*&)F4\"\"(F:F(F:\"#:*$FenF:!$l%*$FhnF:\"$.$*$F\\oF:F]o* $F`oF:Fbo*(F(F:FQF:F;F:!#g*&F8F:FQF:F7*(F(F:FhnF:F;F:FC*(FenF:F(F:F;F: \"#I*$F@F:!\"#F+-%%diffG6$F1F4F+F+*&,RF9F`pF4FIFD\"$i\"F(FRFJ\"#!*FLFY FP!$%QFT!#OFV!$w#FXFOFZ\"$S&Fgn!$o%F[o\"$Y#*&)F4\"\")F:F(F:\"\"**$F_qF :FaqF_o!#sFco\"$`%Feo!$K%Fgo\"$S#FhoFcqFio\"#SF\\pFOF]pFY*(F(F:F\\oF:F ;F:F`pF+-Fbp6$F1-%\"$G6$F4FAF+F+*&,DFD!#:FJF7FP\"#\\FTFAFV\"#CFZ!#\"*F gn\"$0\"F[o!#x*$)F4FaqF:F+F^qFFFbqFFF_o\"#NFco!#hFeo\"#&)Fgo!#qFho\"#M *&FgrF:F(F:F+F+-Fbp6$F1-F\\r6$F4FRF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "and the equation when the parameters are specialized:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dneq:=gfun[algebraicsubs](b deq,eq,f(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dneqG,(*&,,*$)%\"z G\"\"%\"\"\"\"\"**$)F*\"\"$F,!#O*$)F*\"\"#F,\"#hF*!#]F+\"\"\"F7-%%diff G6$-%\"fG6#F*F*F7F7*&,.F.\"$T\"F(!#v*$)F*\"\"&F,\"#:F2!$B\"F*\"#[!\"'F 7F7-F96$F;-%\"$G6$F*F4F7F7*&,.F(\"#UF.!#[FB!#=*$)F*\"\"'F,F0F2\"#FF*FH F7-F96$F;-FL6$F*F0F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The lin ear recurrence satisfied by the " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#% \"nG" }{TEXT -1 18 " follows directly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dnRec:=op(select(has,gfun[diffeqtorec](dneq,f(z),a(n) ),n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dnRecG,.*&,&*$)%\"nG\"\"# \"\"\"\"\"'*$)F*\"\"$F,F0\"\"\"-%\"aG6#F*F1F1*&,*F*!#$*!#OF1F(!#vF.!#= F1-F36#,&F*F1F1F1F1F1*&,*F*\"$o&\"$/%F1F(\"$n#F.\"#UF1-F36#,&F*F1F+F1F 1F1*&,*F*!%$>\"!%w6F1F(!$6%F.!#[F1-F36#,&F*F1F0F1F1F1*&,*F*\"%U5\"%S7F 1F(\"$\"HF.\"#FF1-F36#,&F*F1\"\"%F1F1F1*&,*F(!#yF*!$O$!$![F1F.!\"'F1-F 36#,&F*F1\"\"&F1F1F1" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 17 "4. The in tegrals " }{XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 28 " sati sfy a linear recurrence" }}{PARA 0 "" 0 "" {TEXT -1 68 "The property a bove does not depend on the specific divergent series " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 80 " that one is resumming. This \+ allows one to precompute efficiently the integrals " }{XPPEDIT 18 0 "q [n](z);" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 28 " given the mapping fu nction " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 173 ". Indeed the \+ general theory of holonomic function has recently led to symbolic summ ation and integration algorithms that turn out to apply to the integra l representation of " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6#%\" zG" }{TEXT -1 224 ". The goal of these algorithms is to derive (system s of) linear functional equations, differential or difference, satisfi ed by a sum or an integral. We now proceed to use a prototypical imple mentation of them in the package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 31 " to obtain a recurrence on the " }{XPPEDIT 18 0 "q[n](z) ;" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 138 ". Then, we prove a theorem that by-passes the general theory of holonomic functions, and recover the same recurrence in a more direct way." }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 32 "(This section uses a version of " }{HYPERLNK 17 "Mgfun " 2 "Mgfun" "" }{TEXT -1 30 " that is not distributed yet.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "readlib(Mgfun):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "phi:=1/(1-t)^2-1:" "6#>%$phiG,&*&\"\"\"\"\"\" *$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "F:=exp(-phi/z)*t^n*diff(phi,t):" "6#>%\"FG*(-%$expG6#,$ *&%$phiG\"\"\"%\"zG!\"\"F.F,)%\"tG%\"nGF,-%%diffG6$F+F0F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ct:=Mgfun[creative_telescoping](F,n:: shift,t::diff);" "6#>%#ctG-&%&MgfunG6#%5creative_telescopingG6%%\"FG'% \"nG%&shiftG'%\"tG%%diffG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ctG7$, **&,&%\"zG\"\"\"*&%\"nGF*F)F*F*F*-%#_fG6$F,%\"tGF*F**&,&F)\"\"$F+F3F*- F.6$,&F,F*\"\"#F*F0F*F**&,&F+!\"\"F)F:F*-F.6$,&F,F*F3F*F0F*F**&,(F+!\" $F)F@!\"#F*F*-F.6$,&F,F*F*F*F0F*F***F)\"\"\"F0F*,*F:F*F0F3*$)F0F7FFF@* $)F0F3FFF*F*F-FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h[n](t) :=%[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"hG6#%\"nG6#%\"tG,**&, &%\"zG\"\"\"*&F(F/F.F/F/F/-%#_fG6$F(F*F/F/*&,&F.\"\"$F0F6F/-F26$,&F(F/ \"\"#F/F*F/F/*&,&F0!\"\"F.F=F/-F26$,&F(F/F6F/F*F/F/*&,(F0!\"$F.FC!\"#F /F/-F26$,&F(F/F/F/F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "H[n](t):=%%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"HG6#%\"nG6#% \"tG**%\"zG\"\"\"F*F-,*!\"\"F-F*\"\"$*$)F*\"\"#\"\"\"!\"$*$)F*F0F4F-F- -%#_fG6$F(F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The meaning of \+ the previous computation is that the differential equation" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "h[n](t)+diff(H[n](t),t) = 0;" "6#/,&-&%\"hG 6#%\"nG6#%\"tG\"\"\"-%%diffG6$-&%\"HG6#F)6#F+F+F,\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 91 "holds. This can be viewed as a differential-differen ce relation satisfied by the integrand " }{XPPEDIT 18 0 "f[n](z,t);" " 6#-&%\"fG6#%\"nG6$%\"zG%\"tG" }{TEXT -1 27 ". Now, integrating between " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 64 " returns a non-homogeneous differential equation in the integral" }}{PARA 273 "" 0 "" {XPPEDIT 18 0 "q[n](z) = int(f[n](z,t),t = -1 .. 1);" "6#/-&%\"qG6#%\"nG6#%\"zG -%$intG6$-&%\"fG6#F(6$F*%\"tG/F3;,$\"\"\"!\"\"\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "namely" }}{PARA 271 "" 0 "" {XPPEDIT 18 0 "int(h[n](t),t = -1 .. 1)+H[n](1)-H[n](-1) = 0;" "6#/,(-%$intG6$-&% \"hG6#%\"nG6#%\"tG/F.;,$\"\"\"!\"\"\"\"\"\"\"\"-&%\"HG6#F,6#\"\"\"F5-& F86#F,6#,$\"\"\"F3F3\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 40 "where the integral rewrites in terms of " }{XPPEDIT 18 0 "q[n](z); " "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 3 " as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(h[n](t),t=-1..1)=eval(subs(_f=unapply(q[n](z ),n,t),ct[1]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,**&,&%\" zG\"\"\"*&%\"nGF+F*F+F+F+-%#_fG6$F-%\"tGF+F+*&,&F*\"\"$F,F4F+-F/6$,&F- F+\"\"#F+F1F+F+*&,&F,!\"\"F*F;F+-F/6$,&F-F+F4F+F1F+F+*&,(F,!\"$F*FA!\" #F+F+-F/6$,&F-F+F+F+F1F+F+/F1;F;F+,**&F)\"\"\"-&%\"qG6#F-6#F*F+F+*&F3F J-&FM6#F7FOF+F+*&F:FJ-&FM6#F>FOF+F+*&F@FJ-&FM6#FEFOF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "As to the non-homogeneous part " } {XPPEDIT 18 0 "H[n](1)-H[n](-1);" "6#,&-&%\"HG6#%\"nG6#\"\"\"\"\"\"-&F &6#F(6#,$\"\"\"!\"\"F2" }{TEXT -1 59 ", we readily evaluate it, verify ing that it is 0 by chance." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "H[n](t):=factor(eval(subs(_f=unapply(F,n,t),H[n](t))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"HG6#%\"nG6#%\"tG**%\"zG\"\"\"F*F-),&! \"\"F-F*F-\"\"$\"\"\"%\"fGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "assume(z>0); assume(n,integer); factor(simplify(limit(op(2,%),t =1)-limit(op(2,%),t=-1))): non_hom:=subs([z='z',n='n'],%); z:='z': n:= 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(non_homG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Consequently, we have obtained the follow ing recurrence on the integrals " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG 6#%\"nG6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "collect(eval(subs(_f=unapply(q[n](z),n,t),ct[1])),q,factor)=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&,(*&%\"nG\"\"\"%\"zGF)!\"$F*F +!\"#F)F)-&%\"qG6#,&F(F)F)F)6#F*F)F)*(F*\"\"\"F1F)-&F/6#,&F(F)\"\"$F)F 2F)!\"\"*(F*F4F1F4-&F/6#,&F(F)\"\"#F)F2F)F9*(F*F4F1F4-&F/6#F(F2F)F)\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(subs(n=n-4,% ),q,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(%\"zG\"\"\",&%\"n GF'!\"$F'F'-&%\"qG6#,&F)F'!\"\"F'6#F&F'F0*(F&\"\"\"F(F3-&F-6#,&F)F'!\" #F'F1F'\"\"$*(F&F3F(F3-&F-6#,&F)F'!\"%F'F1F'F'*&,(*&F)F'F&F3F*F&\"\"*F 8F'F'-&F-6#F(F1F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "More \+ generally, a differential equation with respect to " }{XPPEDIT 18 0 "z ;" "6#%\"zG" }{TEXT -1 103 ", or even a system of mixed differential-d ifference equations could be obtained by the same algorithms." }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "The following result had not been noticed by Lutz et al., but might prove useful in numerical computati ons." }}{PARA 0 "" 0 "" {TEXT 263 7 "Theorem" }{TEXT -1 32 ". With the above notations, let " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 " " 0 "" {XPPEDIT 18 0 "sum(p[k](n)*a(n+k),k = 0 .. K) = 0;" "6#/-%$sumG 6$*&-&%\"pG6#%\"kG6#%\"nG\"\"\"-%\"aG6#,&F.F/F,F/F//F,;\"\"!%\"KGF6" } }{PARA 0 "" 0 "" {TEXT -1 150 "be the linear recurrence satisfied by t he Taylor coefficients at the origin of a power series solution of the first-order linear differential equation" }}{PARA 267 "" 0 "" {XPPEDIT 18 0 "diff(G(t),t) = (diff(phi(t),t,t)/diff(phi(t),t)-diff(ph i(t),t)/z)*G(t);" "6#/-%%diffG6$-%\"GG6#%\"tGF**&,&*&-F%6%-%$phiG6#F*F *F*\"\"\"-F%6$-F16#F*F*!\"\"F3*&-F%6$-F16#F*F*F3%\"zGF8F8F3-F(6#F*F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then the integrals " } {XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 23 " satisfy the rec urrence" }}{PARA 268 "" 0 "" {XPPEDIT 18 0 "sum(p[k](-n)*q[n-k-1](z),k = 0 .. K) = 0;" "6#/-%$sumG6$*&-&%\"pG6#%\"kG6#,$%\"nG!\"\"\"\"\"-&% \"qG6#,(F/F1F,F0\"\"\"F06#%\"zGF1/F,;\"\"!%\"KGF<" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 264 6 "Proof." }{TEXT -1 78 " The differential e quation in the statement above is satisfied by the function" }}{PARA 269 "" 0 "" {XPPEDIT 18 0 "e^(-phi(u)/z)*diff(phi(u),u);" "6#*&)%\"eG, $*&-%$phiG6#%\"uG\"\"\"%\"zG!\"\"F.F,-%%diffG6$-F)6#F+F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Since the integrals " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 11 " rewrite as" } }{PARA 270 "" 0 "" {XPPEDIT 18 0 "int(e^(-phi(u)/z)*u^n*diff(phi(u),u) ,u = 0 .. 1);" "6#-%$intG6$*()%\"eG,$*&-%$phiG6#%\"uG\"\"\"%\"zG!\"\"F 1F/)F.%\"nGF/-%%diffG6$-F,6#F.F.F//F.;\"\"!\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 114 "by integration by parts and differentiat ion under the integral sign, they satisfy the announced linear recurre nce." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The following one-line pr ocedure computes a recurrence on the integral" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "q[n](z) = Int(exp(-t/z)*psi(t)^n,t = 0 .. infinity);" " 6#/-&%\"qG6#%\"nG6#%\"zG-%$IntG6$*&-%$expG6#,$*&%\"tG\"\"\"F*!\"\"F6F5 )-%$psiG6#F4F(F5/F4;\"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 50 " is the functiona l inverse of a rational function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" } {TEXT -1 20 ". It takes as input " }{XPPEDIT 18 0 "phi(t),t,g,n,z;" "6 '-%$phiG6#%\"tGF&%\"gG%\"nG%\"zG" }{TEXT -1 115 " where all the argume nts except the first one are symbols that appear in the output linear \+ recurrence relating the " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6 #%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rec qnofz:=proc(phi::ratpoly,t::name,q::name,n::name,z::name)\nlocal gf,a; \n op(select(has,eval(subs(n=-n,a=subs(_A=q,proc(x) _A(-x-1) end), \n gfun[diffeqtorec](diff(gf(t),t)+(diff(phi,t)/z-diff(phi,t,t)/d iff(phi,t))*gf(t),gf(t),a(n)))),n))\nend:" "6#>%)recqnofzGR6''%$phiG%( ratpolyG'%\"tG%%nameG'%\"qGF,'%\"nGF,'%\"zGF,7$%#gfG%\"aG6\"F6-%#opG6# -%'selectG6%%$hasG-%%evalG6#-%%subsG6%/F0,$F0!\"\"/F5-FB6$/%#_AGF.R6#% \"xG7\"F6F6-FK6#,&FNFF\"\"\"FFF6F6F6-&%%gfunG6#%,diffeqtorecG6%,&-%%di ffG6$-F46#F+F+\"\"\"*&,&*&-Ffn6$F(F+FjnF2FFFjn*&-Ffn6%F(F+F+Fjn-Ffn6$F (F+FFFFFjn-F46#F+FjnFjn-F46#F+-F56#F0F0F6F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example: " }{XPPEDIT 18 0 "phi = 1/((1-t)^2)-1;" "6#/%$ phiG,&*&\"\"\"\"\"\"*$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "qnRec:=recqnofz(1/(1-t)^2-1, t,q,n,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&qnRecG,**&,&*&%\"nG\" \"\"%\"zGF*!\"\"F+\"\"$F*-%\"qG6#,&F)F*F,F*F*F**&,&F+!\"*F(F-F*-F/6#,& F)F*!\"#F*F*F**&,(F(!\"$F+\"\"*F8F*F*-F/6#,&F)F*F;F*F*F**&,&F(F*F+F;F* -F/6#,&F)F*!\"%F*F*F*" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 51 "We hav e obtained the same recurrence as when using " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "qnRec:=applyop(factor,[2,2],applyop(collect,[2,1],readlib(isolate )(subs(n=n+1,qnRec),q(n)),q,normal));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&qnRecG/-%\"qG6#%\"nG,$*&,(*&,&%\"zG\"\"'*&F)\"\"\"F/F2!\"$F2-F'6 #,&F)F2!\"\"F2F2F2*&,(F/!\"'F1\"\"$\"\"#F2F2-F'6#,&F)F2!\"#F2F2F2*&,&F /F " 0 "" {MPLTEXT 1 0 27 "phi:=subs(z=u,solve(eq,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,$*&*&%\"uG\"\"\",&!\"#F)F(F)F)\"\"\"*$),&F(F)! \"\"F)\"\"#F,!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ass ume(z>0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q0:=int(exp(-p hi/z)*diff(phi,u),u =0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q0G% #z|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q0:=subs(z='z',q0 ); z:='z':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q0G%\"zG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Note that this initial value is a posteri ori obvious." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Thus we now have \+ both recurrence and initial condition. The solution " }{XPPEDIT 18 0 " q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 276 " to the recurrence equation qn Rec is a dominated solution, which means that any numerical error grow s exponentially. To avoid this, we run the recurrence backwards from a ny non trivial initial conditions. The dominating solution disappears \+ quickly, and we obtain the solution " }{XPPEDIT 18 0 "q[n];" "6#&%\"qG 6#%\"nG" }{TEXT -1 109 " because when the recurrence is run backwards \+ it becomes a dominating solution. We therefore add a parameter " } {XPPEDIT 18 0 "NN;" "6#%#NNG" }{TEXT -1 50 " indicating from where we \+ start running backwards." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "eval(collect(op(2,isolate(subs(n=n+3,qnRec),q(n))),q,normal),q=proc(n ) q(n,z,NN) end);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"qG6%,&%\"nG \"\"\"\"\"$F)%\"zG%#NNGF)-F%6%,&F(F)\"\"#F)F+F,!\"$*&*&,(F+F**&F(F)F+F )F*F0F)F)-F%6%,&F(F)F)F)F+F,F)\"\"\"*&F+\"\"\"F8\"\"\"!\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "qnprocrev:=subs(_REC=%,q=qnpro crev,proc(n,z,NN)\noption remember;\nif n=NN then 0 \nelif n=NN-1 then 0\nelif n=NN-2 then 1\nelse _REC fi end);" "6#>%*qnprocrevG-%%subsG6% /%%_RECG%\"%G/%\"qGF$R6%%\"nG%\"zG%#NNG7\"6#%)rememberG6\"@)/F/F1\"\"! /F/,&F1\"\"\"\"\"\"!\"\"F8/F/,&F1F;\"\"#F=\"\"\"F)F5F5F5" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%*qnprocrevGR6%%\"nG%\"zG%#NNG6\"6#%)rememberGF *@)/9$9&\"\"!/F/,&F0\"\"\"!\"\"F4F1/F/,&F0F4!\"#F4F4,(-F$6%,&F/F4\"\"$ F49%F0F4-F$6%,&F/F4\"\"#F4F>F0!\"$*&*&,(F>F=*&F/F4F>F4F=FBF4F4-F$6%,&F /F4F4F4F>F0F4\"\"\"*&F>\"\"\"FJ\"\"\"!\"\"F4F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Here is a procedure to compute " }{XPPEDIT 18 0 " q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 13 " numerically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`evalf/q/digits`:=0:" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "`evalf/q`:=proc (n,z,NN)\noption remember; \ng lobal `evalf/q`,`evalf/q/digits`;\nif Digits>`evalf/q/digits` \nthen ` evalf/q/digits`:=Digits;\n `evalf/q`:=subsop(4=NULL,op(`evalf/q`)) fi; \nif n = NN then 0 \n elif n = NN-1 then 0 \n elif n = NN-2 then \+ 1.0\n else \nprocname(n+3,z,NN)+(2+3*z+3*z*n)*procname(n+1,z,NN)/(z*(n +1))-3*procname(n+2,z,NN) fi end:" "6#>%(evalf/qGR6%%\"nG%\"zG%#NNG7\" 6#%)rememberG6\"C$@$2%/evalf/q/digitsG%'DigitsGC$>F1F2>F$-%'subsopG6$/ \"\"%%%NULLG-%#opG6#F$@)/F'F)\"\"!/F',&F)\"\"\"\"\"\"!\"\"FA/F',&F)FD \"\"#FF$\"#5!\"\",(-%)procnameG6%,&F'FD\"\"$FDF(F)FD*(,(\"\"#FD*&\"\"$ FDF(FDFD*(\"\"$FDF(FDF'FDFDFD-FO6%,&F'FD\"\"\"FDF(F)FD*&F(FD,&F'FD\"\" \"FDFDFFFD*&\"\"$FD-FO6%,&F'FD\"\"#FDF(F)FDFFF-6$F$F1F-" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dnRec:=gfun[diffeqtorec](\{dneq,D(f)( 0)=1,f(0)=0\},f(z),a(n)):" "6#>%&dnRecG-&%%gfunG6#%,diffeqtorecG6%<%%% dneqG/--%\"DG6#%\"fG6#\"\"!\"\"\"/-F26#F4F4-F26#%\"zG-%\"aG6#%\"nG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dnproc:=gfun[rectoproc](dnR ec,a(n),remember):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Here is a p rocedure to compute " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG" }{TEXT -1 13 " numerically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`ev alf/d/digits`:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`evalf/d `:=proc (n) option remember; global `evalf/d`,`evalf/d/digits`; if n>5 then if Digits>`evalf/d/digits` then `evalf/d/digits`:=Digits;`evalf/ d`:=subsop(4=NULL,op(`evalf/d`)) fi; evalf(-(-225*procname(n-5)-70*pro cname(n-1)+804*procname(n-4)-1011*procname(n-3)+514*procname(n-2)+(165 *procname(n-5)-693*procname(n-4)+1048*procname(n-3)-683*procname(n-2)+ 157*procname(n-1)+(-39*procname(n-5)+195*procname(n-4)-363*procname(n- 3)+309*procname(n-2)-114*procname(n-1)+(-18*procname(n-4)+42*procname( n-3)-48*procname(n-2)+3*procname(n-5)+27*procname(n-1))*n)*n)*n)/((-6+ (12-6*n)*n)*n)) elif n=0 then 0 elif n=1 then 1 elif n=2 then evalf(1/ 3) elif n=3 then evalf(-23/108) else evalf(-2749/3888) fi end:" "6#>%( evalf/dGR6#%\"nG7\"6#%)rememberG6\"@-2\"\"&F'C$@$2%/evalf/d/digitsG%'D igitsGC$>F2F3>F$-%'subsopG6$/\"\"%%%NULLG-%#opG6#F$-%&evalfG6#,$*&,.*& \"$D#\"\"\"-%)procnameG6#,&F'FH\"\"&!\"\"FHFN*&\"#qFH-FJ6#,&F'FH\"\"\" FNFHFN*&\"$/)FH-FJ6#,&F'FH\"\"%FNFHFH*&\"%65FH-FJ6#,&F'FH\"\"$FNFHFN*& \"$9&FH-FJ6#,&F'FH\"\"#FNFHFH*&,.*&\"$l\"FH-FJ6#,&F'FH\"\"&FNFHFH*&\"$ $pFH-FJ6#,&F'FH\"\"%FNFHFN*&\"%[5FH-FJ6#,&F'FH\"\"$FNFHFH*&\"$$oFH-FJ6 #,&F'FH\"\"#FNFHFN*&\"$d\"FH-FJ6#,&F'FH\"\"\"FNFHFH*&,.*&\"#RFH-FJ6#,& F'FH\"\"&FNFHFN*&\"$&>FH-FJ6#,&F'FH\"\"%FNFHFH*&\"$j$FH-FJ6#,&F'FH\"\" $FNFHFN*&\"$4$FH-FJ6#,&F'FH\"\"#FNFHFH*&\"$9\"FH-FJ6#,&F'FH\"\"\"FNFHF N*&,,*&\"#=FH-FJ6#,&F'FH\"\"%FNFHFN*&\"#UFH-FJ6#,&F'FH\"\"$FNFHFH*&\"# [FH-FJ6#,&F'FH\"\"#FNFHFN*&\"\"$FH-FJ6#,&F'FH\"\"&FNFHFH*&\"#FFH-FJ6#, &F'FH\"\"\"FNFHFHFHF'FHFHFHF'FHFHFHF'FHFHFH*&,&\"\"'FN*&,&\"#7FH*&\"\" 'FHF'FHFNFHF'FHFHFHF'FHFNFN/F'\"\"!Fju/F'\"\"\"\"\"\"/F'\"\"#-FA6#*&\" \"\"FH\"\"$FN/F'\"\"$-FA6#,$*&\"#BFH\"$3\"FNFN-FA6#,$*&\"%\\FFH\"%))QF NFNF+6$F$F2F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Finally, the fo llowing procedure computes values of the double confluent Heun functio n as follows. First, an upper value of " }{XPPEDIT 18 0 "NN;" "6#%#NNG " }{TEXT -1 17 " is selected and " }{XPPEDIT 18 0 "2*NN;" "6#*&\"\"#\" \"\"%#NNGF%" }{TEXT -1 70 " is used in the backward recurrence to comp ute the scaling to use for " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\" nG6#%\"zG" }{TEXT -1 84 " in view of the actual initial condition. The n the summation is performed up to the " }{XPPEDIT 18 0 "NN;" "6#%#NNG " }{TEXT -1 63 "th term. If the relative error of the last term is lar ger than " }{XPPEDIT 18 0 "10^(-Digits);" "6#)\"#5,$%'DigitsG!\"\"" } {TEXT -1 7 ", then " }{XPPEDIT 18 0 "NN;" "6#%#NNG" }{TEXT -1 89 " is \+ doubled and the computation starts again. Note that option remember ha s been used in " }{XPPEDIT 18 0 "qnprocrev;" "6#%*qnprocrevG" }{TEXT -1 45 " so as to avoid duplicating some of the work." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "time(evalf(q(0,10.2,3000),21));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$*R!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "valheun:=subs(_q0=q0,proc(z)\n local tot,i,N,NN,l ambda,st,D;\nN:=10;\nst:=time();\ndo \n N:=floor(2*N);\n NN:=N+f loor(sqrt(N))+10;\n D:=Digits+3*ilog10(N) +floor(log(N));\n lamb da:=_q0/evalf(q(0,z,NN),D);\n \n tot:=add(evalf(d(i),D)*evalf(q(i ,z,NN),D),i=1..N)*lambda;\n if abs(evalf(d(N),D)*evalf(q(N,z,NN),D) *lambda) " 0 "" {XPPEDIT 19 1 "infolevel[ valheun]:=1:" "6#>&%*infolevelG6#%(valheunG\"\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "plot(valheun,0..50);" }}{PARA 6 "" 1 "" {TEXT -1 75 "valheun: \"N=\" 80 \"z=\" 1.089857709 \"time:\" .141 \"digits:\" 17" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \+ \"N=\" 160 \"z=\" 2.038137074 \"time:\" .270 \"digits:\" \+ 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \"N=\" 320 \"z=\" \+ 3.104576397 \"time:\" .579 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \"N=\" 320 \"z=\" 4.178084772 \"time: \" .571 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \+ \"N=\" 320 \"z=\" 5.246490950 \"time:\" .599 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 6.237040242 \"time:\" 1.221 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 7.262696659 \"tim e:\" 1.080 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheu n: \"N=\" 640 \"z=\" 8.323431992 \"time:\" 1.129 \"digit s:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \" z=\" 9.380765534 \"time:\" 1.240 \"digits:\" 22" }}{PARA 6 " " 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 10.46836305 \+ \"time:\" 1.201 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 78 "v alheun: \"N=\" 1280 \"z=\" 11.42631952 \"time:\" 2.520 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1 280 \"z=\" 12.50475163 \"time:\" 2.369 \"digits:\" 26" }} {PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 13.58 761188 \"time:\" 2.601 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 14.63114838 \"time: \" 2.750 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 15.57877949 \"time:\" 2.800 \"digits :\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \" z=\" 16.70560455 \"time:\" 3.110 \"digits:\" 26" }}{PARA 6 " " 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 17.66017317 \+ \"time:\" 3.219 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "v alheun: \"N=\" 1280 \"z=\" 18.77056341 \"time:\" 3.340 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2 560 \"z=\" 19.75344692 \"time:\" 7.290 \"digits:\" 26" }} {PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 20.83 182591 \"time:\" 8.091 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 21.85869773 \"time: \" 8.440 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 22.93013074 \"time:\" 9.180 \"digits :\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \" z=\" 23.91404071 \"time:\" 9.710 \"digits:\" 26" }}{PARA 6 " " 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 24.97532053 \+ \"time:\" 10.209 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 " valheun: \"N=\" 2560 \"z=\" 26.07769200 \"time:\" 10.941 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" \+ 2560 \"z=\" 27.03730960 \"time:\" 11.229 \"digits:\" 26" } }{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 28.0 7372290 \"time:\" 11.961 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 29.14443926 \"time: \" 12.589 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun : \"N=\" 2560 \"z=\" 30.19192632 \"time:\" 12.810 \"digi ts:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \+ \"z=\" 31.20542392 \"time:\" 13.741 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 32.33074020 \"time:\" 20.060 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 33.34188693 \"time:\" 12.8 80 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N= \" 5120 \"z=\" 34.42150189 \"time:\" 15.039 \"digits:\" \+ 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" \+ 35.39979350 \"time:\" 17.241 \"digits:\" 27" }}{PARA 6 "" 1 " " {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 36.46932464 \"tim e:\" 19.520 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valhe un: \"N=\" 5120 \"z=\" 37.47567008 \"time:\" 21.760 \"di gits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \+ \"z=\" 38.52759104 \"time:\" 24.110 \"digits:\" 27" }} {PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 39.55 603605 \"time:\" 26.450 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 40.63272267 \"time: \" 28.740 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun : \"N=\" 5120 \"z=\" 41.66969985 \"time:\" 31.479 \"digi ts:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \+ \"z=\" 42.73015878 \"time:\" 33.841 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 43.78183659 \"time:\" 36.099 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 44.74821926 \"time:\" 39.0 00 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N= \" 5120 \"z=\" 45.85580287 \"time:\" 40.981 \"digits:\" \+ 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" \+ 46.84643885 \"time:\" 43.800 \"digits:\" 27" }}{PARA 6 "" 1 " " {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 47.90266342 \"tim e:\" 45.850 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valhe un: \"N=\" 5120 \"z=\" 48.91360511 \"time:\" 48.260 \"di gits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 72 "valheun: \"N=\" 5120 \+ \"z=\" 50.0 \"time:\" 50.699 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 160 \"z=\" 1.563997392 \"t ime:\" 1.450 \"digits:\" 21" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'C URVESG6$7S7$$\"+4x&)*3\"!\"*$\"+7+&3\"H!#57$$\"+#R(*Rc\"F*$\"+mL$Q6&F- 7$$\"+uq8Q?F*$\"+&Q!)ps(F-7$$\"+(RwX5$F*$\"+'epd]\"F*7$$\"+sZ3yTF*$\"+ $Q+sV#F*7$$\"+]4\\Y_F*$\"+gn*z`$F*7$$\"+U-/PiF*$\"+w!)*[p%F*7$$\"+fmpi sF*$\"+nI,9gF*7$$\"+#*>VB$)F*$\"+\"=:D\\(F*7$$\"+Mbw!Q*F*$\"+**)\\'o!* F*7$$\"+0j$o/\"!\")$\"+CI\\y5Fhn7$$\"+_>jU6Fhn$\"+N4'oB\"Fhn7$$\"+j^Z] 7Fhn$\"+4#pCU\"Fhn7$$\"+)=h(e8Fhn$\"+FQ(fh\"Fhn7$$\"+Q[6j9Fhn$\"+1_l3= Fhn7$$\"+\\z(yb\"Fhn$\"+ny]))>Fhn7$$\"+b/cq;Fhn$\"+Y8)z?#Fhn7$$\"+Fhn$\"+4W#*GGF hn7$$\"+\"f#=$3#Fhn$\"+-(*>dIFhn7$$\"+t(pe=#Fhn$\"+.OMyKFhn7$$\"+uI,$H #Fhn$\"+`+z7NFhn7$$\"+rSS\"R#Fhn$\"+enCJPFhn7$$\"+`?`(\\#Fhn$\"+\"30,( RFhn7$$\"++#pxg#Fhn$\"+J7e@UFhn7$$\"+g4t.FFhn$\"+&\\cJW%Fhn7$$\"+!Hst! GFhn$\"+RW7&o%Fhn7$$\"+ERW9HFhn$\"+.%yy$\\Fhn7$$\"+KE>>IFhn$\"+wKw(=&F hn7$$\"+#RU07$Fhn$\"+;#**=V&Fhn7$$\"+?S2LKFhn$\"+Q%zbq&Fhn7$$\"+$p)=ML Fhn$\"+(HVP&fFhn7$$\"+*=]@W$Fhn$\"+:3(4A'Fhn7$$\"+]$z*RNFhn$\"+j?1lkFh n7$$\"+kC$pk$Fhn$\"+df&Rt'Fhn7$$\"+3qcZPFhn$\"+uH%)))pFhn7$$\"+/\"fF&Q Fhn$\"+sQ:dsFhn7$$\"+0OgbRFhn$\"+DKF@vFhn7$$\"+nAFjSFhn$\"+Cdi*z(Fhn7$ $\"+&)*pp;%Fhn$\"+>yVp!)Fhn7$$\"+ye,tUFhn$\"+G40Z$)Fhn7$$\"+fO=yVFhn$ \"+Z!3Si)Fhn7$$\"+E>#[Z%Fhn$\"+$Q-*z))Fhn7$$\"+(G!e&e%Fhn$\"+Wnyu\"*Fh n7$$\"+&)Qk%o%Fhn$\"+`)Q*R%*Fhn7$$\"+UjE!z%Fhn$\"+:%ySs*Fhn7$$\"+60O\" *[Fhn$\"+87Q(***Fhn7$\"#]$\"+U%[#H5!\"(-%'COLOURG6&%$RGBG$\"#5!\"\"\" \"!F`[l-%+AXESLABELSG6$%!GFd[l-%%VIEWG6$;F`[l$FezF`[l%(DEFAULTG" 2 762 762 762 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 0 0 0 0 0 0 }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 117 "This curve is to be contrasted with the irregular plot we got from the same series using summation to the least term." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Sections 1 and 2 of this works heet only depend on the Heun differential equation, and can easily be \+ adapted to any linear differential equation. Sections 3 and 4 compute \+ the recurrences satisfied by the coefficients " }{XPPEDIT 18 0 "q[n]; " "6#&%\"qG6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d[n];" "6#&%\" dG6#%\"nG" }{TEXT -1 53 ", which depend on the choice of the mapping f unction " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 10 " only for " } {XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 47 ", and on the dif ferential equation as well for " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\" nG" }{TEXT -1 62 ". Section 5 details the numerical computations. It d epends on " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 158 " and on th e recurrences found in the previous sections. This worksheet can be ad apted to another mapping function and to another linear differential e quation." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 15 "[Lutz et al.] " }{TEXT 268 40 "On the converge nce of Borel approximants" }{TEXT -1 62 ", by W. Balser, D. A. Lutz a nd R. Sch\344fke, (2000). Preprint." }}{PARA 0 "" 0 "" {TEXT -1 12 "[ DuLoRi92] " }{TEXT 269 77 "Kovacic's Algorithm and Its Application to Some Families of Special Functions" }{TEXT -1 43 ", by Anne Duval and Mich\350le Loday-Richaud, " }{TEXT 270 62 "Applicable Algebra in Engi neering, Communication and Computing" }{TEXT -1 29 ", (1992), vol. 3, \+ p. 211-246." }}{PARA 0 "" 0 "" {TEXT -1 10 "[Durand] " }{TEXT 272 75 "On the convergence of Borel approximants [summary of a talk by Donald Lutz]" }{TEXT -1 45 ", by Marianne Durand, (2001). To appear in: " } {TEXT 271 29 "Algorithms Seminar, 2000-2001" }{TEXT -1 24 ", INRIA Res earch Report." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }