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-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 "" 0 260 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 "" 0 261 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 "" 0 262 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 "" 0 263 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 "" 0 264 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 "" 0 265 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 "" 0 266 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 "" 0 267 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 "" 0 268 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 "" 0 269 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 "" 0 270 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 "" 0 271 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 "" 0 272 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 "" 0 273 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 "" 0 274 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 256 "" 0 "" {TEXT -1 62 "Staircase polygons, a si mplified model for self-avoiding walks" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 282 15 "Frederic Chyzak" }}{PARA 259 " " 0 "" {TEXT -1 30 "(Version of January 16, 1997)\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "A combinatorial problem in statistical physics \+ is that of counting " }{TEXT 258 19 "self-avoiding walks" }{TEXT -1 465 " in an integer lattice. In particular, the cases of dimensions 2 and 3 are related to problems of phase transitions in the classical I sing model of statistical physics. Solving the general model for self -avoiding walks is an old open problem so that one focusses on approxi mate models. Exactly solvable models exist for various models obtaine d by constraining the admissible walks. More and more is known about \+ self-avoiding walks by relaxing these constraints." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Here we follow a thread started by A. J. Guttmann and T. Prellberg [" }{TEXT 262 83 "Staircase polygons, elliptic integ rals, Heun functions, and lattice Green functions" }{TEXT -1 67 ", (19 93), Physical Review E, (47) 4, R2233-2236]. We show how the " } {HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 311 " package may be used to derive in a matter of seconds differential equations that were previo usly guessed in the article above, and how to establish rigorously the singular structure of the ODE's. As a by-product, we derive effectiv e asymptotic estimates. Early results on this subject date back to G. Polya [" }{TEXT 263 41 "On the number of certain lattice polygons" } {TEXT -1 67 ", (1969), Journal of Combinatorial Theory, Series A, (6), 102-105]." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "A strong but simple constraint on the walks is to require them to proceed by positive steps. Throughou t the remainder of this session, we only consider such paths with posi tive steps." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{XPPEDIT 278 0 "d" "I\"dG6\"" }{TEXT 279 30 "-dimensional staircase polygon" } {TEXT -1 40 " is a special self-avoiding walk in the " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 162 "-dimensional integer lattice where all p ositive steps precede all negative steps. This can be interpreted as \+ an (unordered) pair of non-crossing paths of length " }{XPPEDIT 18 0 " n" "I\"nG6\"" }{TEXT -1 69 " from the same origin to the same end. He re is an example of length " }{XPPEDIT 18 0 "n=7" "/%\"nG\"\"(" } {TEXT -1 14 " in dimension " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot([[0,0 ],[2,0],[2,1],[4,1],[4,3],[2,3],[2,2],[1,2],[1,1],[0,1],[0,0]]);" }} {PARA 0 "" 0 "" {INLPLOT "6#-%'CURVESG6$7-7$\"\"!F(7$$\"\"#F(F(7$F*$\" \"\"F(7$$\"\"%F(F-7$F0$\"\"$F(7$F*F37$F*F*7$F-F*7$F-F-7$F(F-F'-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(" 2 345 212 212 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 197 251 0 0 0 0 0 0 }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{XPPEDIT 280 0 "d" "I\"dG 6\"" }{TEXT 281 20 "-dimensional festoon" }{TEXT -1 97 " is obtained f rom a sequence of staircase polygons, with the possible introduction o f unit steps." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Here is an exam ple in dimension 2, involving 3 polygons (thick lines) and 5 unit step s (thin lines)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 729 "\014plo ts[display](\n plot([[0,0],[2,0],[2,2],[1,2],[1,1],[0,1],[0,0]],thi ckness=3),\n plots[textplot]([1/2,3/2,`+`],font=[TIMES,BOLD,18]),\n plots[textplot]([5/2,1/2,`-`],font=[TIMES,BOLD,18]),\n plot([[2 ,2],[3,2]]),\n plot([[3,2],[5,2],[5,5],[6,5],[6,6],[4,6],[4,4],[3,4 ],[3,2]],thickness=3),\n plots[textplot]([7/2,9/2,`+`],font=[TIMES, BOLD,18]),\n plots[textplot]([11/2,7/2,`-`],font=[TIMES,BOLD,18]), \n plot([[6,6],[7,6],[7,7]]),\n plot([[7,7],[10,7],[10,8],[11,8] ,[11,9],[8,9],[8,8],[7,8],[7,7]],thickness=3),\n plots[textplot]([2 1/2,15/2,`+`],font=[TIMES,BOLD,18]),\n plots[textplot]([15/2,17/2,` -`],font=[TIMES,BOLD,18]),\n plot([[11,9],[13,9]]),\nscaling=constr ained,title=`A 2-dimensional festoon`);" }}{PARA 13 "" 1 "" {INLPLOT " 60-%'CURVESG6%7)7$\"\"!F(7$$\"\"#F(F(7$F*F*7$$\"\"\"F(F*7$F.F.7$F(F.F' -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#\"\"$-%%TEXTG6%7$$\"+++ ++]!#5$\"+++++:!\"*%\"+G-%%FONTG6%%&TIMESG%%BOLDG\"#=-F>6%7$$\"+++++DF FFA%\"-GFH-F$6$7$F,7$$F6%7$$\"+++++NFF$\"+++++ XFFFGFH-F>6%7$$\"+++++bFFFfoFSFH-F$6$7%F]o7$$\"\"(F(F[o7$FcpFcpF2-F$6% 7+Fep7$$F7F(Fcp7$Fjp$\"\")F(7$$\"#6F(F\\q7$F_q$\"\"*F(7$F\\qFbq7$F\\qF \\q7$FcpF\\qFepF2F9-F>6%7$$\"++++]5!\")$\"+++++vFFFGFH-F>6%7$F]r$\"+++ ++&)FFFSFH-F$6$7$Faq7$$\"#8F(FbqF2-%&TITLEG6#%8A~2-dimensional~festoon G-%(SCALINGG6#%,CONSTRAINEDG" 2 606 606 606 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 333 223 0 0 0 0 0 0 }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 247 "We orient each of the staircase polygon (i.e., we distinguish between its two consisting paths), so that a festoon beco mes a sequence of oriented polygons, unit steps remaining non-oriented . On the previous display, this orientation is marked by " }{XPPEDIT 18 0 "`+`" "I\"+G6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`-`" "I\"-G6 \"" }{TEXT -1 86 " signs. In this way, we interpret a festoon as an ( oriented) pair of paths of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 26 " with same origin and end." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Denote by " }{XPPEDIT 18 0 "q[d,n]" "&%\"qG6$%\"dG%\"nG" }{TEXT -1 41 " the number of staircase polygons in the " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 39 "-dimensional integer lattice, and call \+ " }{XPPEDIT 18 0 "Q[d](z)" "-&%\"QG6#%\"dG6#%\"zG" }{TEXT -1 33 " the \+ ordinary generating function" }}}{EXCHG {PARA 260 "" 0 "" {XPPEDIT 18 0 "Q[d](z)=Sum(q[d,n]*z^n,n=0..infinity)" "/-&%\"QG6#%\"dG6#%\"zG-%$Su mG6$*&&%\"qG6$F'%\"nG\"\"\")F)F1F2/F1;\"\"!%)infinityG" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "This function has a simple re lation to the ordinary generating function" }}}{EXCHG {PARA 261 "" 0 " " {XPPEDIT 18 0 "P[d](z)=Sum(p[d,n]*z^n,n=0..infinity)" "/-&%\"PG6#%\" dG6#%\"zG-%$SumG6$*&&%\"pG6$F'%\"nG\"\"\")F)F1F2/F1;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "of the number of (ordered) pair s of paths of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 43 " wi th same origin and end. More precisely," }}}{EXCHG {PARA 262 "" 0 "" {XPPEDIT 18 0 "P[d](z)=1/(1+d*z-2*Q[d](z))" "/-&%\"PG6#%\"dG6#%\"zG*& \"\"\"\"\"\",(\"\"\"F,*&F'F,F)F,F,*&\"\"#F,-&%\"QG6#F'6#F)F,!\"\"F7" } {TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "since any pair o f paths can be viewed as a sequence of pairs of non-crossing paths. E quivalently," }}}{EXCHG {PARA 263 "" 0 "" {XPPEDIT 18 0 "Q[d](z)=(1+d* z)/2-1/(2*P[d](z)" "/-&%\"QG6#%\"dG6#%\"zG,&*&,&\"\"\"\"\"\"*&F'F.F)F. F.F.\"\"#!\"\"F.*&\"\"\"F.*&\"\"#F.-&%\"PG6#F'6#F)F.F1F1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "In this session, we proceed to compute the singularities and asymptotics of " }{XPPEDIT 18 0 "Q[d ](z)" "-&%\"QG6#%\"dG6#%\"zG" }{TEXT -1 22 " for small dimensions." }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Dimension 2: an explicit case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "The case of dimension 2 is simp le and yields explicit closed forms for generating functions and count ing sequences, already known to Polya. It is usually solved by hand u sing a Vandermonde convolution. It is readily solved by Maple. The n umber " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"#%\"nG" }{TEXT -1 28 " of pairs of path of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 29 " with same origin and end is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p[2,n]=Sum(binomial(n,k)^2,k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"pG6$\"\"#%\"nG-%$SumG6$*$-%)binomialG6$F(%\"kGF'/F 0;\"\"!F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "because a path is de termined by the choice of its " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 60 " steps in one of the directions of the lattice. The number " } {XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"#%\"nG" }{TEXT -1 69 " of pairs of paths is thus given by the central binomial coefficient:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "p[2,n]=normal(convert(sum(binomial( n,k)^2,k=0..n),factorial),expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"pG6$\"\"#%\"nG*&-%*factorialG6#,$F(F'\"\"\"-F+6#F(!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The generating function " } {XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 32 " of thes e numbers is well-known:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P[2](z)=sum(op(2,\")*z^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"PG6#\"\"#6#%\"zG*$,&\"\"\"F-F*!\"%#!\"\"F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Another closed form for the coeffi cients of this series are gotten by Newton's binomial expansion:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p[2,n]=binomial(-1/2,n)*(-4) ^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"pG6$\"\"#%\"nG*&-%)binomia lG6$#!\"\"F'F(\"\"\")!\"%F(F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 " The following asymptotics for " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"# %\"nG" }{TEXT -1 22 " is obtained by Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "asympt_p[2]:=expand(simplify(subs(cos(Pi*n)=(-1)^ n,convert(asympt(op(2,\"),n,2),polynom)),symbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asympt_pG6#\"\"#*(%#PiG#!\"\"F'%\"nGF*)\"\"%F,\" \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "From the value found for \+ " }{XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 60 ", we \+ get the generating function for the non-crossing paths:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Q[2](z)=(1+2*z-1/op(2,\"\"\"))/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"QG6#\"\"#6#%\"zG,(#\"\"\"F(F-F *F-*$,&F-F-F*!\"%F,#!\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "wh ich Maple easily expands into:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(op(2,\"),z,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7% \"zG\"\"#\"\"\"\"\"\"\"\"#F%\"\"$\"\"&\"\"%\"#9\"\"&\"#U\"\"'\"$K\"\" \"(\"$H%\"\")\"%I9\"\"*-%\"OG6#F'\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "A closed form for the coefficients of this series are got ten by Newton's binomial expansion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "q[2,n]=delta[n,0]/2+delta[n,1]-binomial(1/2,n)*(-4)^n /2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"qG6$\"\"#%\"nG,(&%&deltaG6 $F(\"\"!#\"\"\"F'&F+6$F(F/F/*&-%)binomialG6$F.F(F/)!\"%F(F/#!\"\"F'" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "delta[x,y ]" "&%&deltaG6$%\"xG%\"yG" }{TEXT -1 31 " denotes the Kronecker symbol , " }{XPPEDIT 18 0 "delta[x,y]=1" "/&%&deltaG6$%\"xG%\"yG\"\"\"" } {TEXT -1 16 " if and only if " }{XPPEDIT 18 0 "x=y" "/%\"xG%\"yG" } {TEXT -1 69 ", 0 otherwise. The following evaluation corroborates thi s expansion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(-binomi al(1/2,i)*(-4)^i/2,i=2..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"\" \"\"#\"\"&\"#9\"#U\"$K\"\"$H%\"%I9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The following asymptotics for " }{XPPEDIT 18 0 "q[2,n]" "&%\"qG 6$\"\"#%\"nG" }{TEXT -1 22 " is obtained by Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "asympt_q[2]:=expand(simplify(subs(cos(Pi*n )=(-1)^n,convert(asympt(-(-1)^n*4^n*binomial(1/2,n)/2,n,2),polynom)),s ymbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asympt_qG6#\"\"#,$*( %#PiG#!\"\"F'%\"nG#!\"$F')\"\"%F-\"\"\"#F2F1" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 40 "We check this asymptotic estimate using " }{HYPERLNK 17 "Gfun" 2 "gfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7T%(La placeG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%,diffeqtorecG %)guesseqnG%(guessgfG%0hadamardproductG%0holexprtodiffeqG%)invborelG%, listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+listtolistG%.listtoratp olyG%*listtorecG%-listtoseriesG%5listtoseries/LaplaceG%1listtoseries/e gfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1listtoseries/ogfG%4li sttoseries/revegfG%4listtoseries/revogfG%,maxdegcoeffG%*maxdegeqnG%,ma xordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG%*optionsgfG%,poltod iffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+recG%,rectodiffeqG%* rectoprocG%.seriestoalgeqG%/seriestodiffeqG%2seriestohypergeomG%-serie stolistG%0seriestoratpolyG%,seriestorecG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "The following first order differential eq uation is satisfied by the generating function " }{XPPEDIT 18 0 "Q[2]( z)=1/2+z-sqrt(1-4*z)/2" "/-&%\"QG6#\"\"#6#%\"zG,(*&\"\"\"\"\"\"\"\"#! \"\"F-F)F-*&-%%sqrtG6#,&\"\"\"F-*&\"\"%F-F)F-F/F-\"\"#F/F/" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "2*f(z)+(1-4*z)*diff (f(z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"fG6#%\"zG\"\"#*&,& \"\"\"F+F'!\"%F+-%%diffG6$F$F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "normal(eval(subs(f(z)=sqrt(1-4*z),\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "W e compute a recursion on the coefficients " }{XPPEDIT 18 0 "q[2,n]" "& %\"qG6$\"\"#%\"nG" }{TEXT -1 21 " of the series using " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diffeqtorec(\"\",f(z),u(n)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"#\"\"\"%\"nG!\"%F'-%\"uG 6#F(F'F'*&,&F(F'F'F'F'-F+6#F.F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Next, " }{HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rectoproc]" "" } {TEXT -1 72 " returns a procedure that computes the coefficients in an efficient way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "coeff_of _Q:=rectoproc(\{\",u(0)=1,u(1)=1,u(2)=1\},u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+coeff_of_QG:6#%\"nG6%%\"iG%#u0G%#u1G6\"E\\s$\"\"#\" \"\"\"\"!F/F/F/C%>8%F/?(8$\"\"$F/,&9$F/!\"\"F/%%trueGC$>8&,$*(,&\"\"'F /F5!\"%F/F3F/F5F9F9>F3F=,$*(,&FAF/F8FBF/F3F/F8F9F9F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "We finally check our asymptotic estimate: the following ratio goes to 1 when " }{XPPEDIT 18 0 "n" "I\"nG6\"" } {TEXT -1 18 " goes to infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for i from 100 to 1000 by 100 do i=evalf(coeff_of_Q(i)/subs(n= i,asympt_q[2]),10) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+\"$\"+M 'pP+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+#$\"+'*)z=+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+$$\"+u@D,5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+%$\"+A(Q4+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+&$\"+$y]2+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/\"$+'$\"+Vbi+5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+($\"+8h` +5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+)$\"+b!p/+\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+*$\"+3pT+5!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"%+5$\"+&>v.+\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Using " }{HYPERLNK 17 "series" 2 "series" "" }{TEXT -1 11 " to expand " }{XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 30 " and compute the coefficients " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG 6$\"\"#%\"nG" }{TEXT -1 57 " would be less efficient and require more \+ time and space." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "In higher di mensions, the problem no longer has a known explicit solution. A natu ral tool to attack it are " }{TEXT 259 27 "Bessel generating functions " }{TEXT -1 48 ". The Bessel generating function of a sequence " } {XPPEDIT 18 0 "s[n]" "&%\"sG6#%\"nG" }{TEXT -1 33 " of numbers is defi ned as the sum" }}}{EXCHG {PARA 264 "" 0 "" {XPPEDIT 18 0 "Sum(s[n]*z^ n/n!^2,n=0..infinity)" "-%$SumG6$*(&%\"sG6#%\"nG\"\"\")%\"zGF)F**$-%*f actorialG6#F)\"\"#!\"\"/F);\"\"!%)infinityG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "For instance, the Bessel generatin g function of the constant sequence " }{XPPEDIT 18 0 "s[n]=1" "/&%\"sG 6#%\"nG\"\"\"" }{TEXT -1 17 " is given by the " }{HYPERLNK 17 "modifie d Bessel function" 2 "BesselJ" "" }}}{EXCHG {PARA 265 "" 0 "" {XPPEDIT 18 0 "I[0](2*sqrt(z))=Sum(z^n/n!^2,n=0..infinity)" "/-&%\"IG6 #\"\"!6#*&\"\"#\"\"\"-%%sqrtG6#%\"zGF+-%$SumG6$*&)F/%\"nGF+*$-%*factor ialG6#F5\"\"#!\"\"/F5;F'%)infinityG" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "which we denote by j(z). The Bessel generatin g function of the numbers of pairs of paths with same origin and end a nd positive steps in a " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 44 "- dimensional lattice is given by the product" }}}{EXCHG {PARA 266 "" 0 "" {XPPEDIT 18 0 "Product(j(z[i]),i=1..d)" "-%(ProductG6$-%\"jG6#&%\"z G6#%\"iG/F+;\"\"\"%\"dG" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "z[i]" "&%\"zG6#%\"iG" }{TEXT -1 30 " marks the steps in dimension " }{XPPEDIT 18 0 "i" "I\"iG6\"" } {TEXT -1 65 ". It follows that the Bessel generating function of the \+ numbers " }{XPPEDIT 18 0 "p[d,n]" "&%\"pG6$%\"dG%\"nG" }{TEXT -1 28 " \+ with respect to the length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 17 " of the paths is " }{XPPEDIT 18 0 "j(z)^d" ")-%\"jG6#%\"zG%\"dG" } {TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 58 "Higher dimensio nal cases: computing differential equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "In dimension 2, the generating function " }{XPPEDIT 18 0 "P[2](z)" "-&%\"PG6#\"\"#6#%\"zG" }{TEXT -1 38 " is algebraic. In hig her dimensions, " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" } {TEXT -1 32 " belongs to the larger class of " }{TEXT 260 19 "holonomi c functions" }{TEXT -1 14 ". A function " }{XPPEDIT 18 0 "f(z)" "-%\" fG6#%\"zG" }{TEXT -1 232 " is called holonomic when it satisfies a lin ear differential equation with rational function coefficients. This c lass of functions benefits from numerous closure properties for which \+ algorithms have been implemented in the package " }{HYPERLNK 17 "Gfun " 2 "gfun" "" }{TEXT -1 100 ". In particular, this class is closed un der sum, product, Borel and inverse Borel transforms. The " } {HYPERLNK 17 "Borel transform" 2 "gfun[borel]" "" }{TEXT -1 35 " defin ed in Gfun is related to the " }{HYPERLNK 17 "inverse Laplace transfor m" 2 "inttrans[invlaplace]" "" }{TEXT -1 50 ": formally applied on a f ormal power series, it is" }}}{EXCHG {PARA 267 "" 0 "" {XPPEDIT 18 0 " Borel(Sum(s[n]*z^n,n=0..infinity))=Sum(s[n]*z^n/n!,n=0..infinity)" "/- %&BorelG6#-%$SumG6$*&&%\"sG6#%\"nG\"\"\")%\"zGF-F./F-;\"\"!%)infinityG -F'6$*(&F+6#F-F.)F0F-F.-%*factorialG6#F-!\"\"/F-;F3F4" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "This is enough for us to comp ute differential equations satisfied by the " }{XPPEDIT 18 0 "P[d](z) " "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Dimension 2 revisited" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "This case was solved above in an explicit way." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "We proceed to get " }{XPPEDIT 18 0 "P[2](z)" "-&% \"PG6#\"\"#6#%\"zG" }{TEXT -1 21 ", and more generally " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 16 ", by the formula" }} }{EXCHG {PARA 268 "" 0 "" {XPPEDIT 18 0 "P[d](z)=(Borel@@(-2))(j(z)^d) " "/-&%\"PG6#%\"dG6#%\"zG--%#@@G6$%&BorelG,$\"\"#!\"\"6#)-%\"jG6#F)F' " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We first comp ute a differential equation satisfied by " }{XPPEDIT 18 0 "j" "I\"jG6 \"" }{TEXT -1 58 " starting from a recurrence satisfied by its coeffic ients " }{XPPEDIT 18 0 "u[n]=1/n!^2" "/&%\"uG6#%\"nG*&\"\"\"\"\"\"*$-% *factorialG6#F&\"\"#!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "diff_eq_j:=rectodiffeq(\{(n+1)^2*u(n+1)=u(n),u(0)=1 \},u(n),j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*diff_eq_jG<$/-%\" jG6#\"\"!\"\"\",(-F(6#%\"zGF+-%%diffG6$F-F/!\"\"*&F/F+-F16$F0F/F+F3" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We compute equations satisfied b y the powers " }{XPPEDIT 18 0 "j(z)^d" ")-%\"jG6#%\"zG%\"dG" }{TEXT -1 7 " using " }{HYPERLNK 17 "gfun[poltodiffeq]" 2 "gfun[poltodiffeq] " "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "powe r_j:=proc(d) option remember; poltodiffeq(j(z)^d,[diff_eq_j],[j(z)],j( z)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "For instance, in the \+ case " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" }{TEXT -1 39 ", the followin g system is satisfied by " }{XPPEDIT 18 0 "j(z)^2" "*$-%\"jG6#%\"zG\" \"#" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "powe r_j(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/-%\"jG6#\"\"!\"\"\"/---% #@@G6$%\"DG\"\"#6#F&F'\"\"$/--F0F2F'F1,*-F&6#%\"zG!\"#*&,&F)F)F:!\"%F) -%%diffG6$F8F:F)F)*&-F@6$-F@6$F?F:F:F)F:F1F)*&F:F)FEF)F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We now derive a differential system satis fied by " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 42 ", the ordinary generating function of the " }{XPPEDIT 18 0 "p[d,n] " "&%\"pG6$%\"dG%\"nG" }{TEXT -1 49 ", by computing two inverse Borel \+ transforms with " }{HYPERLNK 17 "gfun[invborel]" 2 "gfun[invborel]" " " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "double _inv_borel:=proc(sys,f,z) option remember; invborel(invborel(sys,f(z), diffeq),f(z),diffeq) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "In t he case " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" }{TEXT -1 10 ", we have: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "double_inv_borel(power_ j(2),j,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"jG6#\"\"!\"\"\",& -F&6#%\"zG\"\"#*&,&!\"\"F)F-\"\"%F)-%%diffG6$F+F-F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The equation is solve by " }{HYPERLNK 17 "dsolve" 2 "dsolve" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(\",j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"jG6#%\"zG*&%\"IG\"\"\",&!\"\"F*F'\"\"%#F,\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "The possible singularities of a holonomic equati on are given by its leading coefficient (on the previous equation, " } {XPPEDIT 18 0 "4*z-1" ",&*&\"\"%\"\"\"%\"zGF%F%\"\"\"!\"\"" }{TEXT -1 75 ") and these singularities yield the asymptotic form of coefficient s by the " }{TEXT 261 30 "method of singularity analysis" }{TEXT -1 55 ". The singularity read on the previous equation is in " } {XPPEDIT 18 0 "1/4" "*&\"\"\"\"\"\"\"\"%!\"\"" }{TEXT -1 39 ", so that the exponential behaviour of " }{XPPEDIT 18 0 "p[2,n]" "&%\"pG6$\"\"# %\"nG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" }{TEXT -1 41 ", which we found in the previous section." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "In the case of dimension 2, we read the singularit y by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "solve(coeff(op(rem ove(type,\"\",equation)),diff(j(z),z)),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 " Dimension 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We perform the same calculations for dimensions 3 and higher." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^3" "*$-%\"jG6# %\"zG\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<'/-%\"jG6#\"\"!\" \"\",,*&,&!\"$F)%\"zG\"\"*F)-F&6#F.F)F)*&,&F.!#?F)F)F)-%%diffG6$F0F.F) F)*&,&F.\"\"(*$F.\"\"#!#5F)-F66$F5F.F)F)*&-F66$-F66$F>F.F.F)F.\"\"$F)* &FCF)F.F<\"\"'/--%\"DG6#F&F'FE/---%#@@G6$FKF " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/-%\"jG6#\"\"!\"\"\",(*&,&%\"zG!\"* \"\"$F)F)-F&6#F-F)F)*&,(*$F-\"\"#!#FF-\"#?!\"\"F)F)-%%diffG6$F0F-F)F)* &,(*$F-F/F.F4\"#5F-F8F)-F:6$F9F-F)F)/---%#@@G6$%\"DGF56#F&F'\"#I/---FF 6$FHF/FIF'\"$e&/--FHFIF'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Ord er and possible singularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",equation)),diff(j(z),z, z)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!#\"\"\"\"\"*F%" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 4" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^4" "*$- %\"jG6#%\"zG\"\"%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(/-% \"jG6#\"\"!\"\"\"/--%\"DG6#F&F'\"\"%/---%#@@G6$F-F/F.F'#\"$z'\"\"'/--- F46$F-\"\"$F.F'#\"$G\"F>/---F46$F-\"\"#F.F'\"#9,.*&,&%\"zG\"#k!\"%F)F) -F&6#FKF)F)*&,(FK!#o*$FKFFFLF)F)F)-%%diffG6$FNFKF)F)*&,&FS!#!*FK\"#:F) -FU6$FTFKF)F)*&,&*$FKF>!#?FS\"#DF)-FU6$FenFKF)F)*&-FU6$-FU6$F\\oFKFKF) FKF/F)*&FaoF)FKF>\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System s atisfied by " }{XPPEDIT 18 0 "P[4](z)" "-&%\"PG6#\"\"%6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel (\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(/-%\"jG6#\"\"!\"\"\"/-- -%#@@G6$%\"DG\"\"$6#F&F'\"%O:/---F.6$F0\"\"#F2F'\"#c/---F.6$F0\"\"%F2F '\"&%=l,**&,&%\"zG!#kF@F)F)-F&6#FEF)F)*&,(*$FEF9!$[%FE\"#o!\"\"F)F)-%% diffG6$FGFEF)F)*&,(*$FEF1!$%QFK\"#!*FE!\"$F)-FP6$FOFEF)F)*&,(*$FEF@FFF T\"#?FKFNF)-FP6$FXFEF)F)/--F0F2F'F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Order and possible singularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",equation)),dif f(j(z),z$3)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F##\"\"\" \"#;#F%\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 5" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^5" "*$-%\"jG6#%\"zG\"\"&" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<)/-%\"jG6#\"\"!\"\"\"/--%\"DG6#F&F'\"\"&/---%#@@G6$F-F/F.F'#\"% F&)\"\")/---F46$F-\"\"%F.F'#\"%&)y\"#C/---F46$F-\"\"$F.F'#\"$X&\"\"'/- --F46$F-\"\"#F.F'#\"#XFP,0*&,(%\"zG\"$&G!\"&F)*$FVFP!$D#F)-F&6#FVF)F)* &,(FY\"$x(FV!$'>F)F)F)-%%diffG6$FenFVF)F)*&,(FV\"#J*$FVFG\"$f#FY!$=&F) -F\\o6$F[oFVF)F)*&,&FY\"#!*Fao!$!GF)-F\\o6$FdoFVF)F)*&,&Fao\"#l*$FVF>! #NF)-F\\o6$FjoFVF)F)*&-F\\o6$-F\\o6$FapFVFVF)FVF/F)*&FfpF)FVF>\"#:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "P[5](z)" "-&%\"PG6#\"\"&6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<)/-%\"jG6#\"\"!\"\"\"/---%#@@G6$%\"DG\"\"%6 #F&F'\"'S#*=/--F0F2F'\"\"&/---F.6$F0\"\"$F2F'\"%qK/---F.6$F0\"\"#F2F' \"#!*/---F.6$F0F7F2F'\")+'[`\",,*&,(*$%\"zGFD\"$+*FP!$&GF7F)F)-F&6#FPF )F)*&,**$FPF=\"%+sFO!%jRFP\"$'>!\"\"F)F)-%%diffG6$FSFPF)F)*&,**$FPF1\" %]&)FW!%,lFO\"$=&FP!\"(F)-Fgn6$FfnFPF)F)*&,**$FPF7\"%+FF[o!%!f#FW\"$!G FO!\"'F)-Fgn6$F`oFPF)F)*&,**$FP\"\"'\"$D#Fdo!$f#F[o\"#NFWFenF)-Fgn6$Fi oFPF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Order and possible sin gularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([sol ve(coeff(op(remove(type,\",equation)),diff(j(z),z$4)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"!F#F##\"\"\"\"#D#F%\"\"*F%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^6" "*$-%\"jG6# %\"zG\"\"'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "power_j(6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<*/-%\"jG6#\"\"!\" \"\"/--%\"DG6#F&F'\"\"'/---%#@@G6$F-\"\"$F.F'\"$m\"/---F46$F-\"\"#F.F' \"#L/---F46$F-F/F.F'#\"'`tt\"#g/---F46$F-\"\"%F.F'#\"%^IFL/---F46$F-\" \"&F.F'#\"&8?$\"#5,2*&,(%\"zG\"%?5!\"'F)*$FenF=!%cMF)-F&6#FenF)F)*&,*F en!$;&Fhn\"%CdF)F)*$FenF6!%/BF)-%%diffG6$FjnFenF)F)*&,(F`o\"%/ZFhn!%OC Fen\"#jF)-Fco6$FboFenF)F)*&,(Fhn\"$,$*$FenFL\"$%yF`o!%3CF)-Fco6$FjoFen F)F)*&,&F`o\"$]$F_p!$+(F)-Fco6$FbpFenF)F)*&,&*$FenFT!#cF_p\"$S\"F)-Fco 6$FhpFenF)F)*&-Fco6$-Fco6$F_qFenFenF)FenF/F)*&FdqF)FenFT\"#@" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "P[6](z)" "-&%\"PG6#\"\"'6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<*/-%\"jG6#\"\"!\"\"\"/---%#@@G6$%\"DG\"\"#6 #F&F'\"$K\"/---F.6$F0\"\"$F2F'\"%wf/---F.6$F0\"\"%F2F'\"'W$R%/---F.6$F 0\"\"&F2F'\")?()4Y,.*&,(*$%\"zGF1\"&CQ\"FM!%?5\"\"'F)F)-F&6#FMF)F)*&,* *$FMF9\"'ON>FL!&cf#FM\"$;&!\"\"F)F)-%%diffG6$FQFMF)F)*&,**$FMF@\"'g,QF U!&w>(FL\"%OCFM!#:F)-Fen6$FZFMF)F)*&,*FL!#DFU\"%3C*$FMFG\"'o>@Fin!&'>^ F)-Fen6$F^oFMF)F)*&,*FU!#5Fin\"$+(*$FMFP\"&?.%Fdo!&g<\"F)-Fen6$FgoFMF) F)*&,*FinFY*$FM\"\"(\"%/BF]p!$%yFdo\"#cF)-Fen6$F`pFMF)F)/--F0F2F'FP/-- -F.6$F0FPF2F'\"+?*H2P'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Order a nd possible singularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",equation)),diff(j(z),z$5)), z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"!F#F#F##\"\"\"\"#O#F%\"# ;#F%\"\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 7" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "j(z)^7." ")-%\"jG6#%\"zG$\"\"(\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "power_j(7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/- %\"jG6#\"\"!\"\"\"/---%#@@G6$%\"DG\"\"%6#F&F'#\"&:n$\"#C/---F.6$F0\"\" $F2F'#\"%X;\"\"'/---F.6$F0\"\"#F2F'#\"#\"*FD/--F0F2F'\"\"(/---F.6$F0FJ F2F'#\"*,-n?\"\"$?(/---F.6$F0F>F2F'#\")p,@FFR/---F.6$F0\"\"&F2F'#\"'d \"[*\"$?\",4*&,**$%\"zGF;\"&D5\"!\"(F)Fao\"%8K*$FaoFD!&55$F)-F&6#FaoF) F)*&,*Fao!%%G\"F`o!&k;&Feo\"&uJ$F)F)F)-%%diffG6$FgoFaoF)F)*&,*Fao\"$F \"F`o\"&Q0&*$FaoF1!&;H\"Feo!&=-\"F)-F_p6$F^pFaoF)F)*&,(F`o!&kk\"Feo\"$ m*Fep\"&S(>F)-F_p6$FhpFaoF)F)*&,(Fep!%e$)F`o\"%,<*$FaoFin\"%u>F)-F_p6$ F_qFaoF)F)*&,&Fep\"%]5Feq!%7:F)-F_p6$FgqFaoF)F)*&,&Feq\"$m#*$FaoF>!#%) F)-F_p6$F]rFaoF)F)*&-F_p6$-F_p6$FdrFaoFaoF)FaoFJF)*&FirF)FaoF>\"#G" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "System satisfied by " }{XPPEDIT 18 0 "P[7](z)" "-&%\"PG6#\"\"(6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "double_inv_borel(\",j,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/---%#@@G6$%\"DG\"\"'6#%\"jG6#\"\"!\",!o@8 f>/-F-F.\"\"\",0*&,**$%\"zG\"\"$!'+Qz*$F8\"\"#\"'!3[#F8!%Ek\"#9F3F3-F- 6#F8F3F3*&,,*$F8\"\"%!)+q!>\"F7\"([Wq&F;!'qYGF8\"%oD!\"#F3F3-%%diffG6$ F@F8F3F3*&,,*$F8\"\"&!)+hQFFD\")#*fBFP\")/N+:FD!(gbS\"F7\"&GH$F;!$!=F3-FL6$FWF8F3F3* &,,*$F8\"\"(!(]\"e`Fen\"(sn'[FP!''4b&FD\"&;n\"F7!$I\"F3-FL6$F[oF8F3F3* &,,*$F8\"\")!']`fF_o\"'o*>'Fen!&3H)FP\"%CIFD!#IF3-FL6$FfoF8F3F3*&,,*$F 8\"\"*!&]?#Fjo\"&Ke#F_o!%[RFen\"$o\"FPFJF3-FL6$FapF8F3F3/---F(6$F*FQF, F.\"*S)yP6/---F(6$F*FEF,F.\"'g6))/---F(6$F*F9F,F.\"%q)*/---F(6$F*F " 0 "" {MPLTEXT 1 0 71 "op(sort([solve(coeff(op(remove(type,\",eq uation)),diff(j(z),z$6)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\" \"!F#F#F#F##\"\"\"\"#\\#F%\"#D#F%\"\"*F%" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "These experimental results suggest a pattern for the sing ularities of the equation in dimension " }{XPPEDIT 18 0 "d" "I\"dG6\" " }{TEXT -1 86 ". More specifically, the leading coefficient of the d ifferential equation is given by" }}}{EXCHG {PARA 272 "" 0 "" {XPPEDIT 18 0 "z^(d-2)*Product(1-(2*k+1)*z,k=0..p)" "*&)%\"zG,&%\"dG\" \"\"\"\"#!\"\"F'-%(ProductG6$,&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'\"\"\"F'F 'F$F'F)/F3;\"\"!%\"pGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "d=2*p+1" "/%\"dG,&*&\"\"#\"\"\"%\"pGF'F'\"\"\"F'" } {TEXT -1 8 ", and by" }}}{EXCHG {PARA 273 "" 0 "" {XPPEDIT 18 0 "z^(d- 2)*Product(1-2*k*z,k=1..p)" "*&)%\"zG,&%\"dG\"\"\"\"\"#!\"\"F'-%(Produ ctG6$,&\"\"\"F'*(\"\"#F'%\"kGF'F$F'F)/F1;\"\"\"%\"pGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "d=2*p" "/%\"dG*&\" \"#\"\"\"%\"pGF&" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "The singular structure of the ODE" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The equation used to describe " }{XPPEDIT 18 0 "j(z)" "-%\"jG6# %\"zG" }{TEXT -1 36 " introduces a \"parasitic\" function, " } {XPPEDIT 18 0 "k(z)=K[0](2*sqrt(z))" "/-%\"kG6#%\"zG-&%\"KG6#\"\"!6#*& \"\"#\"\"\"-%%sqrtG6#F&F/" }{TEXT -1 56 ", also expressed in terms of \+ a modified Bessel function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "op(remove(type,diff_eq_j,equation));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"jG6#%\"zG\"\"\"-%%diffG6$F$F'!\"\"*&F'F(-F*6$F)F'F(F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Both " }{XPPEDIT 18 0 "j(z)" "-%\"j G6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k(z)" "-%\"kG6#%\"zG" } {TEXT -1 30 " are solutions found my Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(\",j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"jG6#%\"zG,&*&%$_C1G\"\"\"-%(BesselIG6$\"\"!,$*$F'#F+\"\"#F3F+F+ *&%$_C2GF+-%(BesselKGF.F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Th e equation above is satisfied by " }{TEXT 256 3 "any" }{TEXT -1 23 " l inear combination of " }{XPPEDIT 18 0 "j(z)" "-%\"jG6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k(z)" "-%\"kG6#%\"zG" }{TEXT -1 51 ". In the same way, the equation used to described " }{XPPEDIT 18 0 "j(z)^d " ")-%\"jG6#%\"zG%\"dG" }{TEXT -1 17 " is satisfied by " }{TEXT 257 3 "any" }{TEXT -1 36 " linear combination of the products " }{XPPEDIT 18 0 "j(z)^l*k(z)^(d-l)" "*&)-%\"jG6#%\"zG%\"lG\"\"\")-%\"kG6#F',&%\"d GF)F(!\"\"F)" }{TEXT -1 54 ". For instance, this is easily checked fo r the cases " }{XPPEDIT 18 0 "d=2" "/%\"dG\"\"#" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "d=3" "/%\"dG\"\"$" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 11 "Dimension 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the differential equation satisfied by " }{XPPEDIT 18 0 "j(z) ^2" "*$-%\"jG6#%\"zG\"\"#" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "op(remove(type,power_j(2),equation));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,*-%\"jG6#%\"zG!\"#*&,&\"\"\"F+F'!\"%F+-%%diffG6 $F$F'F+F+*&-F.6$-F.6$F-F'F'F+F'\"\"#F+*&F'F+F3F+\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Maple is no longer able to solve in terms of special functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "d solve(\",j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"jG6#%\"zG-%&DE SolG6$<#,**&F'\"\"#-%%diffG6$-F06$-F06$-%#_YGF&F'F'F'\"\"\"F8*&F'F8F2F 8\"\"$*&,&F8F8F'!\"%F8F4F8F8F6!\"#<#F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "(The development version of Maple is able to solve.)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We successively plug " }{XPPEDIT 18 0 "j(z)^2" "*$-%\"jG6#%\"zG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " k(z)^2" "*$-%\"kG6#%\"zG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "j(z )*k(z)" "*&-%\"jG6#%\"zG\"\"\"-%\"kG6#F&F'" }{TEXT -1 28 " into the pr evious equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "op(remo ve(type,power_j(2),equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt(z))^2,z),\")));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "normal(eval(subs(j=unapply(BesselK(0,2*sqrt(z))^2,z), \"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 79 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt( z))*BesselK(0,2*sqrt(z)),z),\"\"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Each evaluation yi elds 0, proving that the function is a solution." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Dimension 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the differential equation satisfied by " }{XPPEDIT 18 0 "j (z)^3" "*$-%\"jG6#%\"zG\"\"$" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "op(remove(type,power_j(3),equation));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&!\"$\"\"\"%\"zG\"\"*F'-%\"jG6#F(F'F'*& ,&F(!#?F'F'F'-%%diffG6$F*F(F'F'*&,&F(\"\"(*$F(\"\"#!#5F'-F16$F0F(F'F'* &-F16$-F16$F9F(F(F'F(\"\"$F'*&F>F'F(F7\"\"'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 21 "We successively plug " }{XPPEDIT 18 0 "j(z)^3" "*$-%\"j G6#%\"zG\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "j(z)^2*k(z)" "*&-%\"jG 6#%\"zG\"\"#-%\"kG6#F&\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "j(z)*k( z)^2" "*&-%\"jG6#%\"zG\"\"\"*$-%\"kG6#F&\"\"#F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k(z)^3" "*$-%\"kG6#%\"zG\"\"$" }{TEXT -1 28 " into the \+ previous equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "norma l(eval(subs(j=unapply(BesselI(0,2*sqrt(z))^3,z),\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt(z))^2*BesselK(0,2*sqrt (z)),z),\"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "\"\"\":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 79 "normal(eval(subs(j=unapply(BesselI(0,2*sqrt(z))*Bes selK(0,2*sqrt(z))^2,z),\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "normal(eval(subs(j=unap ply(BesselK(0,2*sqrt(z))^3,z),\"\")));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The ordinary gene rating function " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" } {TEXT -1 42 " is the double inverse Borel transform of " }{XPPEDIT 18 0 "j(z)^d" ")-%\"jG6#%\"zG%\"dG" }{TEXT -1 72 ". In the same way as a bove, the differential equation that vanishes at " }{XPPEDIT 18 0 "P[d ](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 62 " also vanishes at each of \+ the double inverse Borel transforms " }{XPPEDIT 18 0 "(Borel@@(-2))(j( z)^l*k(z)^(d-l))" "--%#@@G6$%&BorelG,$\"\"#!\"\"6#*&)-%\"jG6#%\"zG%\"l G\"\"\")-%\"kG6#F0,&%\"dGF2F1F)F2" }{TEXT -1 62 ". Those transform ha ve a nice integral representation, namely" }}}{EXCHG {PARA 269 "" 0 " " {XPPEDIT 18 0 "(Borel@@(-2))(j(z)^l*k(z)^(d-l))=2*Int(j(z*t)^l*k(z*t )^(d-l)*k(t),t=0...infinity)" "/--%#@@G6$%&BorelG,$\"\"#!\"\"6#*&)-%\" jG6#%\"zG%\"lG\"\"\")-%\"kG6#F1,&%\"dGF3F2F*F3*&\"\"#F3-%$IntG6$*()-F/ 6#*&F1F3%\"tGF3F2F3)-F66#*&F1F3FDF3,&F9F3F2F*F3-F66#FDF3/FD;\"\"!%)inf inityGF3" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "This integral is always convergent at its lower bound. At infinity howeve r, the integrand behaves like" }}}{EXCHG {PARA 270 "" 0 "" {XPPEDIT 18 0 "exp(2*((2*l-d)*sqrt(z)-1)*sqrt(t))/t^((d+1)/4)" "*&-%$expG6#*(\" \"#\"\"\",&*&,&*&\"\"#F(%\"lGF(F(%\"dG!\"\"F(-%%sqrtG6#%\"zGF(F(\"\"\" F0F(-F26#%\"tGF(F()F8*&,&F/F(\"\"\"F(F(\"\"%F0F0" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "It follows that either " } {XPPEDIT 18 0 "2l<=d" "1*&\"\"#\"\"\"%\"lGF%%\"dG" }{TEXT -1 50 " and \+ the integral is defined for any non-negative " }{XPPEDIT 18 0 "z" "I\" zG6\"" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "d<2*l" "2%\"dG*&\"\"#\"\"\" %\"lGF&" }{TEXT -1 33 " and the integral is defined for " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "(d- 2*l)^(-2)" "),&%\"dG\"\"\"*&\"\"#F%%\"lGF%!\"\",$\"\"#F)" }{TEXT -1 63 ". We thus obtain all positive singularities of the conjecture." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 8 "Theorem:" }{TEXT -1 66 " Any homogeneous linear ODE sati sfied by the generating function " }{XPPEDIT 265 0 "P[d](z)" "-&%\"PG6 #%\"dG6#%\"zG" }{TEXT 267 20 " of festoons on the " }{XPPEDIT 268 0 "d " "I\"dG6\"" }{TEXT 269 49 "-dimensional integer lattice is singular a t each " }{XPPEDIT 270 0 "(d-2*l)^(-2)" "),&%\"dG\"\"\"*&\"\"#F%%\"lGF %!\"\",$\"\"#F)" }{TEXT 271 6 ", for " }{XPPEDIT 272 0 "d<2*l" "2%\"dG *&\"\"#\"\"\"%\"lGF&" }{TEXT -1 35 ". More specifically, the function " }{XPPEDIT 266 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT 273 16 " i s singular at " }{XPPEDIT 274 0 "d^(-2)" ")%\"dG,$\"\"#!\"\"" }{TEXT 275 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "Computing the asymptotics: a connection problem" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P [d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 269 " is analytic in a neig hbourhoud of 0, where it is given by its Taylor expansion. On the oth er hand, its dominant singularity (i.e., its singularity closest to 0) and its asymptotic behaviour there are a priori not known. It is nat ural to estimate the asymptotics of " }{XPPEDIT 18 0 "P[d](z)" "-&%\"P G6#%\"dG6#%\"zG" }{TEXT -1 39 " at the smallest positive singularity, \+ " }{XPPEDIT 18 0 "s=d^(-2)" "/%\"sG)%\"dG,$\"\"#!\"\"" }{TEXT -1 60 ", of the equation defining this function. To do so, we use " }{TEXT 283 20 "singularity analysis" }{TEXT -1 3 ". " }{HYPERLNK 17 "First" 1 "" "evaluating the Taylor series" }{TEXT -1 39 ", we evaluate the Ta ylor series at 0. " }{HYPERLNK 17 "Next" 1 "" "computing a basis" } {TEXT -1 78 ", we algorithmically compute a basis of all possible asym ptotic behaviours at " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 32 " of solutions of the equation. " }{HYPERLNK 17 "Finally" 1 "" "the conne ction" }{TEXT -1 105 ", we numerically connect the Taylor series at 0 \+ and a generic linear combination of these asymptotics at " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 50 ", i.e., we equate values at a point \+ between 0 and " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "A common case of singularity of ho lonomic function is that of a " }{TEXT 276 22 "regular singular point " }{TEXT -1 134 ". This is a singular point where the function has a p olynomial behaviour: at such points, the theory expects an expansion o f the form " }{XPPEDIT 18 0 "kappa*z^(p/r)*phi(z^(1/r))" "*(%&kappaG\" \"\")%\"zG*&%\"pGF$%\"rG!\"\"F$-%$phiG6#)F&*&\"\"\"F$F)F*F$" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 149 " is a \+ formal power series. It is therefore natural to look for such an asym ptotics by a method of \"undetermined exponent\". The possible values for " }{XPPEDIT 18 0 "eta=p/r" "/%$etaG*&%\"pG\"\"\"%\"rG!\"\"" } {TEXT -1 32 " are given by a polynomial, the " }{TEXT 277 19 "indicial polynomial" }{TEXT -1 9 ": a zero " }{XPPEDIT 18 0 "eta" "I$etaG6\"" }{TEXT -1 17 " of multiplicity " }{XPPEDIT 18 0 "m+1" ",&%\"mG\"\"\"\" \"\"F$" }{TEXT -1 68 " of the indicial polynomial indicates solutions \+ with the behaviours " }{XPPEDIT 18 0 "z^eta" ")%\"zG%$etaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z^eta*ln(z)" "*&)%\"zG%$etaG\"\"\"-%#lnG6#F$F& " }{TEXT -1 11 ", ..., and " }{XPPEDIT 18 0 "z^eta*ln(z)^m" "*&)%\"zG% $etaG\"\"\")-%#lnG6#F$%\"mGF&" }{TEXT -1 88 ". The following procedu re computes the indicial polynomial of a differential equation." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "indicial_poly:=proc(sys,f,z ) local zero; global eta; option remember;\n zero:=collect(normal(e val(subs(f=proc(z) global eta; z^eta end,op(remove(type,sys,equation)) )))/z^eta,z,normal);\n factor(primpart(coeff(zero,z,ldegree(zero,z) ),eta))\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "As a first remar k, note that " }{XPPEDIT 18 0 "P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" } {TEXT -1 66 " is the only solution of the equation analytic at 0 and s uch that " }{XPPEDIT 18 0 "P[d](0)=1" "/-&%\"PG6#%\"dG6#\"\"!\"\"\"" } {TEXT -1 89 ". To obtain this, let us compute the indicial polynomial s at 0 for the first dimensions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for i from 2 to 7 do i=indicial_poly(double_inv_borel (power_j(i),j,z),j,z) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#,$% $etaG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$,$*$%$etaG\"\"#!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%,$*$%$etaG\"\"$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"&,$*$%$etaG\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"',$*$%$etaG\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"(,$*$%$etaG\"\"'!\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "Therefore, the equation satisfied by " }{XPPEDIT 18 0 " P[d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 14 " in dimension " } {XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 45 " has one solution which is \+ analytic at 0 and " }{XPPEDIT 18 0 "d-1" ",&%\"dG\"\"\"\"\"\"!\"\"" } {TEXT -1 56 " that are singular at 0, with logarithmic singularities. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Connection and asymptotics i n dimension 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The smallest sing ularity is " }{XPPEDIT 18 0 "s=1/9" "/%\"sG*&\"\"\"\"\"\"\"\"*!\"\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "op(sort([ solve(coeff(op(remove(type,double_inv_borel(power_j(3),j,z),equation)) ,diff(j(z),z,z)),z)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!#\"\" \"\"\"*F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We first expand the \+ series at 0 to evaluate " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#% \"zG" }{TEXT -1 10 " at, say, " }{XPPEDIT 18 0 "1/18" "*&\"\"\"\"\"\" \"#=!\"\"" }{TEXT -1 41 ". Next, we recenter the equation around " } {XPPEDIT 18 0 "1/9" "*&\"\"\"\"\"\"\"\"*!\"\"" }{TEXT -1 69 " to study the local behaviour of the solutions there more explicitly." }}} {SECT 1 {PARA 5 "" 0 "evaluating the Taylor series" {TEXT -1 28 "Evalu ating the Taylor series" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "To eva luate the Taylor series, we compute a truncation of it. We actually n eed truncations to several orders to illustrate the slow convergence o f the series close to its singularity. We base on the following ident ity to compute such truncations:" }}}{EXCHG {PARA 271 "" 0 "" {XPPEDIT 18 0 "[z^n]*(phi(t*z)/(1-z))=Sum(c[k]*t^k,k=0..n)" "/*&7#)%\" zG%\"nG\"\"\"*&-%$phiG6#*&%\"tGF(F&F(F(,&\"\"\"F(F&!\"\"F1F(-%$SumG6$* &&%\"cG6#%\"kGF()F.F9F(/F9;\"\"!F'" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "[z^n]*f(z)" "*&7#)%\"zG%\" nG\"\"\"-%\"fG6#F%F'" }{TEXT -1 28 " denotes the coefficient of " } {XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 15 " in the series " } {XPPEDIT 18 0 "f(z)" "-%\"fG6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "I$phiG6\"" }{TEXT -1 15 " is any series " }{XPPEDIT 18 0 "phi (z)=Sum(c[n]*z^n,n=0..infinity)" "/-%$phiG6#%\"zG-%$SumG6$*&&%\"cG6#% \"nG\"\"\")F&F.F//F.;\"\"!%)infinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We start from the equation for " } {XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "double_inv_borel(power_j(3), j,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/-%\"jG6#\"\"!\"\"\",(*&,&% \"zG!\"*\"\"$F)F)-F&6#F-F)F)*&,(*$F-\"\"#!#FF-\"#?!\"\"F)F)-%%diffG6$F 0F-F)F)*&,(*$F-F/F.F4\"#5F-F8F)-F:6$F9F-F)F)/---%#@@G6$%\"DGF56#F&F'\" #I/---FF6$FHF/FIF'\"$e&/--FHFIF'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "We compute " }{XPPEDIT 18 0 "P[3](t*z)" "-&%\"PG6#\"\"$6#*&%\"t G\"\"\"%\"zGF*" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "algebraicsubs(\",j=z*t,j(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"jG6#\"\"!\"\"\",(*&,&%\"tG!\"$*&F-\"\"#%\"zGF)\"\"*F)-F&6#F1 F)F)*&,(F)F)*&F1F)F-F)!#?*&F1F0F-F0\"#FF)-%%diffG6$F3F1F)F)*&,(F1F)*&F 1F0F-F)!#5*&F1\"\"$F-F0F2F)-F<6$F;F1F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "We divide by " }{XPPEDIT 18 0 "(1-z)" ",&\"\"\"\"\"\"%\"z G!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p oltodiffeq(j(z)/(1-z),[\"],[j(z)],j(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/-%\"jG6#\"\"!\"\"\"/--%\"DG6#F&F',&%\"tG\"\"$F)F),(*&,,*&%\"z G\"\"#F0F7\"#O*&F0F7F6F)!\"**&F6F)F0F)!#BF)F)F0F1F)-F&6#F6F)F)*&,.*&F6 F1F0F7\"#XF5!#F*&F6F7F0F)!#SF;\"#?F6F1!\"\"F)F)-%%diffG6$F=F6F)F)*&,.* $F6F7F)*&F6F1F0F)!#5*&F6\"\"%F0F7\"\"*F6FGFD\"#5FAF:F)-FI6$FHF6F)F)" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We extract its coefficients:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rec_j:=diffeqtorec(\",j(z), u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rec_jG<&/-%\"uG6#\"\"#,(* $%\"tGF*\"#:F-\"\"$\"\"\"F0,**&,(F,\"#O*&F-F*%\"nGF0F4*&F-F*F6F*\"\"*F 0-F(6#F6F0F0*&,**&,&F-!#5F,!\"*F0F6F*F0*&,&F-!#]F,!#OF0F6F0F0F,FDF-!#j F0-F(6#,&F6F0F0F0F0F0*&,**&,&F0F0F-\"#5F0F6F*F0*&,&\"\"'F0F-\"#]F0F6F0 F0F-\"#jF8F0F0-F(6#,&F6F0F*F0F0F0*&,(F6!\"'F@F0*$F6F*!\"\"F0-F(6#,&F6F 0F/F0F0F0/-F(6#F0,&F-F/F0F0/-F(6#\"\"!F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "And we compute a procedure, which to " }{XPPEDIT 18 0 "`` (n,t)" "-%!G6$%\"nG%\"tG" }{TEXT -1 36 " associates the truncated seri es at " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 14 " to the order " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "series_j:=rectoproc(rec_j,u(n),params=[t]):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Next, the series can be computed f ormally at " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "normal(series_j(4,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$%\"tG\"\"$\"#$**$F%\"\"#\"#:*$F%\"\"%\"$R'F %F&\"\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "or numerically at \+ a rational number:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "serie s_j(4,1/16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&\"z$)\"&Ob'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "(This is more efficient than compu ting the series, and next evaluating it numerically.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "The following plots display the Taylor e xpansion for increasing an number of terms. Note the behaviour of the series close to the singularity " }{XPPEDIT 18 0 "1/9" "*&\"\"\"\"\" \"\"\"*!\"\"" }{TEXT -1 38 " when we increase the number of terms." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "pl[Taylor]:=plots[display] (\n plot('series_j'(10,z),z=0..1/9,y=0..5,style=point,color=red),\n plot('series_j'(20,z),z=0..1/9,y=0..5,style=point,color=green),\n \+ plot('series_j'(100,z),z=0..1/9,y=0..5,style=point,color=blue),\nti tle=`Taylor series (red=10 terms, green=20 terms, blue=100 terms)`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pl[Taylor];" }}{PARA 13 " " 1 "" {INLPLOT "6(-%'CURVESG6%7S7$\"\"!$\"\"\"F(7$$\"3lFjF=g!>U#!#?$ \"3&3`Cg/bt+\"!#<7$$\"3r]u\"\\\\$>HXF.$\"3%[k-J?/R,\"F17$$\"39,vrc'e!* *oF.$\"3VM0L%>V9-\"F17$$\"3q6c9BGj%G*F.$\"31\"\\x o)e;\"!#>$\"3%*3gQ\"FD$\"3lO-GdctW5F17$$\"3W%Q;, fKRh\"FD$\"3(*eIps_w_5F17$$\"3bvd0`:l\\=FD$\"3*pNXGQ'Hh5F17$$\"3CQ:HiX h%3#FD$\"3,HVTyI FD$\"3TC9:cG]26F17$$\"3q=/B0jO^KFD$\"3P61)oRLv6\"F17$$\"3i*)*Qt*4&>Y$F D$\"3?q,8Oq#p7\"F17$$\"3nyNWklN7PFD$\"3q;YW(Gs%Q6F17$$\"3hxM`DH[CRFD$ \"3[_%)Qf<\"F17$$\"36//PXYHHYFD$\"3z8IFYQr%=\"F17$$\"3F&pe9Q)[d[FD$ \"35xTWe;L(>\"F17$$\"3'G#>(fg%e&4&FD$\"31XcC3)R5@\"F17$$\"3_!GscEJUJ&F D$\"371p&)=U:C7F17$$\"3$R%oHE72]bFD$\"3g')oQL,\"*Q7F17$$\"31ZBxjE/&z&F D$\"3LZqsKX'\\D\"F17$$\"3].DB?5H3gFD$\"3:=F)G5&fp7F17$$\"3Y-\"Rt30'QiF D$\"3&=V%>?<9'G\"F17$$\"3'3!z!\\0UlZ'FD$\"3)3z1!p:7/8F17$$\"3m+CmdpJ4n FD$\"3IBEkM[mA8F17$$\"3$)y)R^kQX$pFD$\"3%yKxTq!fT8F17$$\"34%4P)H*3Y=(F D$\"3TW2\"fN`QO\"F17$$\"3:sF'H?3$4uFD$\"3))f;+QD4&Q\"F17$$\"38mR7QEA\\ wFD$\"3a)*R#><)>49F17$$\"30+jnu2imyFD$\"3#*Rc9ftWK9F17$$\"3t\"o+.O%H/ \")FD$\"3b9[^_!f&f9F17$$\"3!edP?oEzK)FD$\"3<:HU:w&o[\"F17$$\"3!\\;x\\* ooh&)FD$\"36sWV@AX<:F17$$\"3%oX-1BI-z)FD$\"3f`A;7ci\\:F17$$\"3\"3;3F#R \\H!*FD$\"3a^BAq)zfe\"F17$$\"3e-.P&HL*f#*FD$\"3K1lC2f(Qi\"F17$$\"3,/M, P3f&\\*FD$\"3:&)Hg9.!fm\"F17$$\"3Bu2B:qHH(*FD$\"3$3X#RiuA6!=5Ty,>5!#=$\"3]nz#ot(*H\"=F17$$\"3 $3(*pR(>.T5Fby$\"3-i`(>AE$o=F17$$\"3i\\N9JO]k5Fby$\"3Y'p!o!obG$>F17$$ \"3UOY(>+pp3\"Fby$\"3^g!p9B#f+?F17$$\"3$******46666\"Fby$\"3RjX)*)fx13 #F1-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%&POINTG-F$6%7UF'7$F ,$\"3jIX-Y]N25F1F27$F8$\"3wN0L%>V9-\"F17$F=$\"38YvT$\\E#H5F17$FB$\"31C _PNe#)=FFlF0\"F17$FR$\"3xEM'HQ'Hh5 F17$FW$\"37kY')y\\YYLv6\"F17$F_p$\"3 yYHgur#p7\"F17$Fdp$\"3d#R\\\\fs%Q6F17$Fip$\"3gc,'3M#f[6F17$F^q$\"3V1x0 GDyg6F17$Fcq$\"3'[>Id'R(><\"F17$Fhq$\"3o=')zkxr%=\"F17$F]r$\"3'=m\"F17$Fbr$\"34M5MG=067F17$Fgr$\"31mB8BR]D\"F17$Ffs$\"3#4')HM!*z'p7F17$F[t$\"3&>@r[2viG\"F17$F`t $\"36JRHW9L/8F17$Fet$\"3/Zra`x)HK\"F17$Fjt$\"3%H5![%Gv?M\"F17$F_u$\"3N #)o&\\.0YO\"F17$Fdu$\"3(*y)>$3r>'Q\"F17$Fiu$\"3#Q9@B&*[3T\"F17$F^v$\"3 .)fU,L0[V\"F17$Fcv$\"3@rb`P\\,j9F17$Fhv$\"3`Wvj\\ox\"\\\"F17$F]w$\"3HB tCFD_C:F17$Fbw$\"3G^:$G$ykf:F17$Fgw$\"37bM6T*R.g\"F17$F\\x$\"3kJ+7?&yS k\"F17$Fax$\"3)y'RG9$3Wp\"F17$Ffx$\"3AK1jma;^F17$F[z$\"3)>i^/[#4x?F17$$ \"3-$4flJOd2\"Fby$\"3O\\@Tj4=M@F17$F`z$\"3[N?#o#*fm>#F17$$\"3<=t[c+/*4 \"Fby$\"3hRlW*4#fqAF17$Fez$\"3Oc!pc#f]_BF1-Fjz6&F\\[lF(F][lF(F`[l-F$6% 7XF'7$F,$\"3jGX-Y]N25F17$F3$\"3iWE5.U!R,\"F17$F8$\"35N0L%>V9-\"F17$F=$ \"3oXvT$\\E#H5F17$FB$\"3tA_PNeOe'R(><\"F17$Fhq$\"3q]L8lxr%=\"F17$F]r$\"3T>$oR_Qt>\"F17$Fb r$\"33XG.J=067F17$Fgr$\"3.TK')HR]D\"F17$Ffs$\"3.'GwI+!op7F17$F[t$\"3%QGsQIviG\"F17$F`t$\"3Pq&*Gq> L/8F17$Fet$\"3B\\0W5*))HK\"F17$Fjt$\"3dx=16x2U8F17$F_u$\"3$eYr3T\"F17$F^v$\"3ixG\"fE[[V\"F1 7$Fcv$\"3G\"*)34p+JY\"F17$Fhv$\"3ibQ6%HR>\\\"F17$F]w$\"3BU%)p\"F17$Ffx$\"32AksLXyeF17$Ffy$\"3`cA=\"f4^-#F17$$\"3Agnb-yw_5Fby$\"3HDZ^2kz(4#F17$F[z$ \"3oBNw@Yz'=#F17$F_dl$\"3OaIUaj![H#F17$F`z$\"3^7$QG*>.QCF17$$\"3Ix4BHX +$4\"Fby$\"3i8n*\\5rq`#F17$Fgdl$\"3`-eC.*=&eEF17$$\"30fOu$ev]5\"Fby$\" 3B/0F<'>2\"GF17$Fez$\"3xMaU'zKe+$F1-Fjz6&F\\[lF(F(F][lF`[l-%&TITLEG6#% gnTaylor~series~(red=10~terms,~green=20~terms,~blue=100~terms)G-%+AXES LABELSG6$%\"zG%\"yG-%%VIEWG6$;F($\"+66666!#5;F($\"\"&F(" 2 574 574 574 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}}{SECT 1 {PARA 5 "" 0 "co mputing a basis" {TEXT -1 43 "Computing a basis of asymptotics expansi ons" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "To compute asymptotic expan sions at the singularity " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 54 ", we find it convenient to center the equation around " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 39 ". We thus make the change of variable \+ " }{XPPEDIT 18 0 "u=1-9*z" "/%\"uG,&\"\"\"\"\"\"*&\"\"*F&%\"zGF&!\"\" " }{TEXT -1 8 ", i.e., " }{XPPEDIT 18 0 "f(u)=j((1-u)/9)" "/-%\"fG6#% \"uG-%\"jG6#*&,&\"\"\"\"\"\"F&!\"\"F-\"\"*F." }{TEXT -1 6 " (and " } {XPPEDIT 18 0 "z=(1-u)/9" "/%\"zG*&,&\"\"\"\"\"\"%\"uG!\"\"F'\"\"*F)" }{TEXT -1 139 "). To perform this rational change of variable, we app eal to the closure of the class of holonomic functions under algebraic substitution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "d_eq:=co llect(subs(\{j=f,z=u\},algebraicsubs(double_inv_borel(power_j(3),j,z), j-(1-z)/9,j(z))),diff,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%d _eqG,(**%\"uG\"\"\",&F'F(\"\")F(F(,&F'F(!\"\"F(F(-%%diffG6$-F.6$-%\"fG 6#F'F'F'F(F,*&,(F*F(F'!#9*$F'\"\"#!\"$F(F0F(F(*&,&F9F(F'F(F(F2F(F," }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Let us compute the indicial polyn omial at the singularity " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 12 " (now in 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "indicial_p oly(\{d_eq\},f,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%$etaG\"\"#" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Hence, we expect one regular sol ution at 0, and one singular with logarithmic behaviour." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "We therefore expect an analytic solution and a logarithmic solution. This is confirmed by the following forma l expansion:" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Order:=3: dsolve(d_eq,f(u),series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"uG,&*&%$_C1G\"\"\"++F'F+\"\"!#F+\"\"%\" \"\"#\"\"&\"#K\"\"#-%\"OG6#F+\"\"$F+F+*&%$_C2GF+,&*&-%#lnGF&F+F,F+F++) F'#\"\"$\"\")\"\"\"#\"#L\"$G\"\"\"#F5\"\"$F+F+F+" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 129 "We compute but do not display the series for a la rger order, so as to increase the precision in the numerical calculati ons below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Order:=15: ds ol_ser_f:=dsolve(d_eq,f(u),series):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Note that there exist algorithms to perform the previous calcul ation much more efficiently." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "T he following plot displays the asymptotic behaviours at " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 18 " of the solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "plots[display](\n plot(subs(u=1-9*z,c onvert(subs(\{_C1=1,_C2=0\},op(2,dsol_ser_f)),polynom)),\n z=0. .1/5,y=0..5,color=blue),\n plot(subs(u=1-9*z,convert(subs(\{_C1=0,_ C2=-1\},op(2,dsol_ser_f)),polynom)),\n z=0..1/9,y=-2..5,color=g reen),\n plot([[1/9,-2],[1/9,5]],color=black),\ntitle=`Asymptotic b ehaviours at 1/9 (blue=analytic, green=logarithmic)`);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7V7$\"\"!$\"3_0,6z!H6,#!#<7$$\"3\\mmm;arz @!#?$\"3:z[!='QcB>F+7$$\"3(HLLL$3VfVF/$\"3(RXPXw!fY=F+7$$\"3q****\\i&* )fD'F/$\"3`Ejm\"oJqy\"F+7$$\"3Hnmm\"H[D:)F/$\"3Jf'[(>dXL$\"3j]`Jh\">%z;F+7$$\"3SLLLe0$=C\"FD$\"3w@'*p,-3J;F+7$$\"3ILLL 3RBr;FD$\"3J$QKv/&)za\"F+7$$\"3hmm;zjf)4#FD$\"3m0s=P#f,[\"F+7$$\"3LLL$ e4;[\\#FD$\"3MQ%QW]dtU\"F+7$$\"3$)****\\i'y]!HFD$\"3!G1(4&\\$e!Q\"F+7$ $\"3:LL$ezs$HLFD$\"3)[7$Q!HN)Q8F+7$$\"3$*****\\7iI_PFD$\"3$*y8(*y7_-8F +7$$\"3?nmm;_M(=%FD$\"3QQEN`\"z&p7F+7$$\"3_LLL3y_qXFD$\"3#[s3d<)eV7F+7 $$\"3X+++]1!>+&FD$\"3#QTc*)pJr@\"F+7$$\"3f******\\Z/NaFD$\"3N!oRW9sI> \"F+7$$\"3&*******\\$fC&eFD$\"3SF'p2R$*=<\"F+7$$\"3ELL$ez6:B'FD$\"3\"4 ]>+pGT:\"F+7$$\"36mmm;=C#o'FD$\"3[,l?r6eM6F+7$$\"37nmmm#pS1(FD$\"3bYWm 4\")>>6F+7$$\"3y****\\i`A3vFD$\"3\"=BgL!o\\-6F+7$$\"3Wlmmm(y8!zFD$\"3X y7FCgm)3\"F+7$$\"3:++]i.tK$)FD$\"3'[g=>14W2\"F+7$$\"39++](3zMu)FD$\"3( 3nfNmN;1\"F+7$$\"3>nmm\"H_?<*FD$\"3q0=&zBg!\\5F+7$$\"3dmm;zihl&*FD$\"3 -k;\"=mE\"Q5F+7$$\"3VLLL3#G,***FD$\"3U\\c0cV$p-\"F+7$$\"3GLLezw5V5!#=$ \"3!)=kZyB\"f,\"F+7$$\"3)****\\PQ#\\\"3\"Fbt$\"3vi.Setx15F+7$$\"3MLL$e \"*[H7\"Fbt$\"3P>b3%)4at**Fbt7$$\"3!*******pvxl6Fbt$\"3.$o**))=d1))*Fb t7$$\"3'*****\\_qn27Fbt$\"3*fp#z*ebQz*Fbt7$$\"3$****\\i&p@[7Fbt$\"3)y; Nh5EMr*Fbt7$$\"3$*****\\2'HKH\"Fbt$\"3>BfgOD$zi*Fbt7$$\"3UmmmwanL8Fbt$ \"3m8C+'\\-Vb*Fbt7$$\"38+++v+'oP\"Fbt$\"3f#fiUD)zy%*Fbt7$$\"3CLLeR<*fT \"Fbt$\"3NF%)ypX)HT*Fbt7$$\"3-+++&)Hxe9Fbt$\"3U\\m8uOpV$*Fbt7$$\"3Ymm \"H!o-*\\\"Fbt$\"3#\\DP6nI$))Fbt7$$\"3? 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ABELSG6$%\"zG%\"yG-%%VIEWG6$;F($\"+++++?!#5;F`_mFc_m" 2 574 574 574 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 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG " }{TEXT -1 44 " is a linear combination of both behaviours." }}}} {SECT 1 {PARA 5 "" 0 "the connection" {TEXT -1 14 "The connection" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "To perform the connection, we dete rmine values for " }{XPPEDIT 18 0 "_C1" "I$_C1G6\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "_C2" "I$_C2G6\"" }{TEXT -1 15 " above so that " } {XPPEDIT 18 0 "dsol_ser_f" "I+dsol_ser_fG6\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "ser_j[precise]" "&%&ser_jG6#%(preciseG" }{TEXT -1 12 " \+ agree when " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 21 " takes values around " }{XPPEDIT 18 0 "1/18" "*&\"\"\"\"\"\"\"#=!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=25:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "We graphically determine " } {XPPEDIT 18 0 "_C1" "I$_C1G6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 153 "plot(subs(_C1=x,map(solve,\{seq(series_j(50,( 1-(.5+.01*i))/9)=eval(subs(u=.5+.01*i,convert(op(2,dsol_ser_f),polynom ))),i=1..9)\},_C2)),x=0.85984.. .85986);" }}{PARA 13 "" 1 "" {INLPLOT "6--%'CURVESG6$7S7$$\"&%)f)!\"&$!96)G^lr@@E[]8%!#C7$$\":LLLLL$3VfVS)f) !#D$!9w\"\\))p(fL%QO]8%F-7$$\":nmmm;H[D:3%)f)F1$!9-O,<:\"Q&\\g-NTF-7$$ \":LLLLLe0$=CT)f)F1$!9*o1*yOq(pU9]8%F-7$$\":LLLLL3RBr;%)f)F1$!9d415zSP FF+NTF-7$$\":nmmm;zjf)4U)f)F1$!9e3#3?HxL3\"*\\8%F-7$$\":LLLL$e4;[\\U)f )F1$!9ceS.N+X)f)F1$!9T)**fD28%z>\"\\8%F- 7$$\":+++++]Z/Na%)f)F1$!9oe?p\\*))y<+\\8%F-7$$\":+++++]$fC&e%)f)F1$!9E Fg1)oL\\!)))[8%F-7$$\":LLLL$ez6:BY)f)F1$!9H]R#>w+sZy[8%F-7$$\":nmmmm;= C#oY)f)F1$!9R6e-$RFl>m[8%F-7$$\":nmmmmm#pS1Z)f)F1$!9P58*4>'=$zb[8%F-7$ $\":++++]i`A3v%)f)F1$!9pW0V'>F;pV[8%F-7$$\":nmmmmm(y8!z%)f)F1$!9O3])f& 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"fsol:=\{_C1=.85985,_C2=convert(\",`+`)/nops(\")\};" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fsolG<$/%$_C1G$\"&&)f)!\"&/%$_C2G$ !:X&3Y.>c--@xMT!#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Plugging th e previous values into " }{XPPEDIT 18 0 "dsol_ser_f" "I+dsol_ser_fG6\" " }{TEXT -1 56 " yields the following truncated asymptotic expansion o f " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 ":" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "subs(fsol,collect(subs(u= 1-9*z,convert(series(op(2,dsol_ser_f),u,3),polynom)),[ln(1-9*z),z],nor mal));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,(*$%\"zG\"\"#$!:Skn!4uh iT42L_!#CF'$\":w0FO'p/lw$GK4#F+$!:m#=uwENZoK_9e!#D\"\"\"F1-%#lnG6#,&F1 F1F'!\"*F1F1F&$\":uQQ]F\"o\")3&4zC#F+F'$!:)erDu*3q>O7(Q5F+$\":!yhY&H\" p=a^5v%*F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Once again, we co mpute a more precise expansion, but do not display it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "j_num:=subs(fsol,collect(subs(u=1-9 *z,convert(op(2,dsol_ser_f),polynom)),[ln(1-9*z),z],normal)):" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Plotting " }{XPPEDIT 18 0 "P[3](z) " "-&%\"PG6#\"\"$6#%\"zG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We hav e three means to evaluate numerical values of " }{XPPEDIT 18 0 "P[3](z )" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 94 ": its Taylor series at 0, the differential equation it satisfies, the asymptotic expansion at " } {XPPEDIT 18 0 "1/9" "*&\"\"\"\"\"\"\"\"*!\"\"" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The following plot is the numerica lly unstable result of following the graph of " }{XPPEDIT 18 0 "P[3]" "&%\"PG6#\"\"$" }{TEXT -1 103 " by using the differential equation. T he method used is the fourth-order classical Runge-Kutta method." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "pl[RungeKutta]:=subs(\{LINE =POINT,COLOR(RGB,.9,.9,.2)=COLOR(RGB,0,0,0)\},DEtools[DEplot](remove(t ype,double_inv_borel(power_j(3),j,z),equation),j(z),z=0.0001.. 1/9,[[j (.0000001)=1,D(j)(.0000001)=3]],j=0..5,stepsize=.0005)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The following plot is the result of the c onnection." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "pl[connection ]:=plot(j_num,z=0..1/9,y=0..5,style=point,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "plots[display](eval(subs(COLOUR=pr oc() COLOUR(RGB,1.,0,0) end,pl[Taylor])),pl[RungeKutta],pl[connection] ,title=`black=Runge-Kutta, red=Taylor, green=connection`);" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6%7S7$\"\"!$\"\"\"F(7$$\"3lFjF=g!>U#! #?$\"3&3`Cg/bt+\"!#<7$$\"3r]u\"\\\\$>HXF.$\"3%[k-J?/R,\"F17$$\"39,vrc' 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More specifically, the best approximation by the Tayl or series is obtained for a truncation of order 100, whereas the expan sion obtained by connection is of order 15 only." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 53 "We are now able to give an equivalent for the numb er " }{XPPEDIT 18 0 "p[3,n]" "&%\"pG6$\"\"$%\"nG" }{TEXT -1 84 " of pa irs of paths with same origin and end. Retaining only the singular pa rt when " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 14 " tends to 0 of" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Order:=3: dsolve(d_eq,f(u ),series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"uG,&*&%$_C1G \"\"\"++F'F+\"\"!#F+\"\"%\"\"\"#\"\"&\"#K\"\"#-%\"OG6#F+\"\"$F+F+*&%$_ C2GF+,&*&-%#lnGF&F+F,F+F++)F'#\"\"$\"\")\"\"\"#\"#L\"$G\"\"\"#F5\"\"$F +F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "we get the equivalent " }{XPPEDIT 18 0 "_C2*ln(1-9*z)" "*&%$_C2G\"\"\"-%#lnG6#,&\"\"\"F$*&\"\" *F$%\"zGF$!\"\"F$" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "P[3](z)" "-&%\"P G6#\"\"$6#%\"zG" }{TEXT -1 22 ". An asymptotics for " }{XPPEDIT 18 0 "p[3,n]" "&%\"pG6$\"\"$%\"nG" }{TEXT -1 12 " follows by " }{TEXT 284 23 "transfer of singularity" }{TEXT -1 24 ": an asymptotic for the " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 63 "-th coefficient a series is computed using the following table:" }}}{EXCHG {PARA 0 "> " 0 "transf er of singularity" {MPLTEXT 1 0 308 "matrix([[Function,`Asymptotic coe fficient`,` `],[(1-z/rho)^(-alpha)*ln(1-z/rho)^beta,rho^n*n^(alpha-1)* ln(n)^beta/GAMMA(alpha),``(beta>=0 and alpha*` non integer`)],[(1-z/rh o)^(-p)*ln(1-z/rho)^beta,rho^n*n^(p-1)*ln(n)^(beta-1)/GAMMA(alpha),``( beta>=0 and p*` integer`)],[1/ln(1-z/rho),rho^n/n/ln(n)^2,` `]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7%%)FunctionG%7Asymptoti c~coefficientG%\"~G7%*&),&\"\"\"F/*&%\"zGF/%$rhoG!\"\"F3,$%&alphaGF3F/ )-%#lnG6#F.%%betaGF/**)F2%\"nGF/)F=,&F5F/F3F/F/)-F86#F=F:F/-%&GAMMAG6# F5F3-%!G6#31,$F:F3\"\"!*&F5F/%-~non~integerGF/7%*&)F.,$%\"pGF3F/F6F/** F " 0 "" {MPLTEXT 1 0 35 "asympt_p[3 ]:=-subs(fsol,_C2)*9^n/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asymp t_pG6#\"\"$,$*&)\"\"*%\"nG\"\"\"F,!\"\"$\":X&3Y.>c--@xMT!#D" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Asymptotics for " }{XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"zG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We now turn to our original goal: evaluating the coefficients of" }}}{EXCHG {PARA 274 "" 0 "" {XPPEDIT 18 0 "Q[3](z)=(1+3*z-1/P[3](z))/2 " "/-&%\"QG6#\"\"$6#%\"zG*&,(\"\"\"\"\"\"*&\"\"$F-F)F-F-*&\"\"\"F--&% \"PG6#\"\"$6#F)!\"\"F8F-\"\"#F8" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "It follows from the value of " }{XPPEDIT 18 0 "P[3]( z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 39 " that the ordinary generatin g function " }{XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"zG" }{TEXT -1 14 " behaves like " }{XPPEDIT 18 0 "2/3-1/(2*alpha*ln(1-9*z))" ",&* &\"\"#\"\"\"\"\"$!\"\"F%*&\"\"\"F%*(\"\"#F%%&alphaGF%-%#lnG6#,&\"\"\"F %*&\"\"*F%%\"zGF%F'F%F'F'" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "z" "I \"zG6\"" }{TEXT -1 13 " is close to " }{XPPEDIT 18 0 "1/9" "*&\"\"\"\" \"\"\"\"*!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha=.413477" "/% &alphaG$\"'xMT!\"'" }{TEXT -1 52 ". Because of the positivity of the \+ coefficients of " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" } {TEXT -1 39 " never vanishes before its singularity " }{XPPEDIT 18 0 " s=1/9" "/%\"sG*&\"\"\"\"\"\"\"\"*!\"\"" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"zG" }{TEXT -1 32 " also ha s no singularity before " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 33 " , as visualized on the next plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot(2/3-1/2/j_num,z=0..1/9,y=-.1..1,style=point,colo r=green);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7gn7$\"\"!$\":_y $zIMdR*Q-_h\"!#D7$$\":>&=&=&=&=&=g!>U#!#F$\":(znU?Ip,)=%[m;F+7$$\":r.P q.Pq`\\$>HXF/$\":J>[O=F+7$$\":I'H'H'H'H@%*3gQ\"FD$\":w(ep9*>ynZ-T (=F+7$$\":yxxxxxx-fKRh\"FD$\":HZnI\")GsAFYE\">F+7$$\":T2uS2uSKb^'\\=FD $\":B&3sUmz,xVJ_>F+7$$\":+++++++Dc9Y3#FD$\":EGq!z^Q,NF&=*>F+7$$\":[\"[ \"[\"[\"[\")*GIEBFD$\":]6/6lD7k_xE.#F+7$$\":T2uS2uSd6#=RDFD$\":Ty1&)RZ 2IAl)o?F+7$$\":XWWWWWWWpL)yFFD$\":(G@^d1X+q!\\*4@F+7$$\":cbbbbbbbIp%>I FD$\":XKd]G,2.+X;:#F+7$$\":cbbbbbbbIm8D$FD$\":v4V/4*))G(e3B>#F+7$$\":& =&=&=&=&o(*4&>Y$FD$\":4&Gm%4Q)[d[FD$\":n' p*p*R2Zx(R2\\#F+7$$\":#[\"[\"[\"[\"[1Ye&4&FD$\":5q#z>T\">KzC!QDF+7$$\" :/Pq.Pq.iEJUJ&FD$\":/f,#=e(*)z=yAe#F+7$$\":_=&=&=&=&oAr+b&FD$\":f)GY1Q f/!>k4j#F+7$$\":_=&=&=&=NkE/&z&FD$\":EovY)exM:rl#o#F+7$$\":MLLLLLL3-\" H3gFD$\":t+quW\"41.`lGFF+7$$\":jH'H'H'H'z30'QiFD$\":h]g\"zz]Df#f%zFF+7 $$\":cbbbbbbb0UlZ'FD$\":*=\"H>'=r*3coK$GF+7$$\":MLLLLLL$epJ4nFD$\":GS! =iolw3SK()GF+7$$\":MLLLLLLekQX$pFD$\":%*Qj8dhc(4a2THF+7$$\":cbbbbbb0$* 3Y=(FD$\":)3rVA2w#fT*e-IF+7$$\":/Pq.Pq.P?3$4uFD$\":#**=C\")HQzpNnfIF+7 $$\":!*)))))))))))))QEA\\wFD$\":J?aqDXh**4CF7$F+7$$\":kH'H'H'HYv2imyFD $\":IC#\\PM#H<^T>=$F+7$$\":76666666O%H/\")FD$\":R]+i=;]<6G#\\KF+7$$\": r.Pq.PqGoEzK)FD$\":d6Gk^*>HeFG:LF+7$$\":MLLLLLLe*ooh&)FD$\":\">O$eqa,R z\"e(Q$F+7$$\":#[\"[\"[\"[\"[J-B!z)FD$\":]R+-@TR&Q@)>Y$F+7$$\":7666666 O#R\\H!*FD$\":.Sj5YknBk,Xa$F+7$$\":(H'H'H'H'H'HL*f#*FD$\":QotU:;$)*oCL HOF+7$$\":kH'H'H'H'z$3f&\\*FD$\":&)\\p2j%*\\g;TFs$F+7$$\":/Pq.Pq.i,(HH (*FD$\":Vwp6(G>Ar7uBQF+7$$\":BAAAAAAAs[S%**FD$\":2NP6$Q&eGjog#RF+7$$\" :/Pq.Pq.7%y,>5F+$\":#e\">z;xgF88(eSF+7$$\":uS2uS2uS(>.T5F+$\":*4vZWj)f s30x>%F+7$$\":++++++]7j.X1\"F+$\":agU_-_.Pz%G\"Q%F+7$$\":nmmmmmmmJOd2 \"F+$\":I[*od+soA8\\\"\\%F+7$$\":LLLLLL$3-!pp3\"F+$\":z,Rl]CfPvt%GYF+7 $$\":yxxxxFS$HX+$4\"F+$\":%yout;!)>#['=@ZF+7$$\":AAAAAA(fc+/*4\"F+$\": UiF-vvP2+R)Q[F+7$$\":XWWWWpD-#y0-6F+$\":x\"p#pg<#po!oU\"\\F+7$$\":nmmm m;aQev]5\"F+$\":A3GpRo,lC53,&F+7$$\":yxxxFSocY%e16F+$\":*GqBFW/8#>8K2& F+7$$\":*))))))))QE[ZL436F+$\":&p/LD=@L[5p`^F+7$$\":XWWWpv*Q)yZ)36F+$ \":#o- " 0 "" {MPLTEXT 1 0 47 "asympt_q[3]:=-subs(fsol,1/_C2)*9^n/n/ln(n)^2/2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)asympt_qG6#\"\"$,$*()\"\"*%\"nG\" \"\"F,!\"\"-%#lnG6#F,!\"#$\":6\\7Pf'3B!\\c#47!#C" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 38 "Numerical check of the asymptotics of " } {XPPEDIT 18 0 "p[3,n]" "&%\"pG6$\"\"$%\"nG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q[3,n]" "&%\"qG6$\"\"$%\"nG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "To get satisfactory results, we have to work for large " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "NMax:=250;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %NMaxG\"$]#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We create another \+ procedure to compute the coefficients of the series " }{XPPEDIT 18 0 " P[3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 89 "proc_p:=rectoproc(diffeqtorec(double_inv_borel (power_j(3),j,z),j(z),u(n)),u(n),remember);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'proc_pG:6#%\"nG6\"6#%)rememberGE\\s%\"\"\"\"\"$\"\"! F,\"\"#\"#:F-\"#$**&,(-9!6#,&9$F,!\"#F,!\"*-F56#,&F8F,!\"\"F,F-*&,(F4 \"#=F;!#5*&,&F4F:F;\"#5F,F8F,F,F,F8F,F,F,F8F9F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 31 "We get a precise expansion for " }{XPPEDIT 18 0 "P [3](z)" "-&%\"PG6#\"\"$6#%\"zG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "series_p:=convert([seq(proc_p(i)*z^i,i=0..NMax )],`+`):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "from which follows a \+ precise expansion for " }{XPPEDIT 18 0 "Q[3](z)" "-&%\"QG6#\"\"$6#%\"z G" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "series _q:=series(1/2+3*z/2-1/series_p/2,z,NMax+1):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "The asymptotic behaviour of " }{XPPEDIT 18 0 "p[3,n]" " &%\"pG6$\"\"$%\"nG" }{TEXT -1 20 " is easily attained." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for i from 50 to NMax by 50 do i=ev alf(subs(n=i,asympt_p[3])/coeff(series_p,z,i)) od;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"#]$\":PfXLK6*f;B&\\+\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+\"$\":QfG`F$>kpFX-5!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]\"$\":>ijOk%p*)3&>;+\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+#$\":P\"H\"Hz,M*pG?,5!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]#$\":%>@tzYqd%)G&4+\"!#C" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 17 "The behaviour of " }{XPPEDIT 18 0 "q[3,n]" "&%\"qG6$\" \"$%\"nG" }{TEXT -1 79 ", however, is not observed so obviously, due t o a slow logarithmic convergence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for i from 50 to NMax by 50 do i=evalf(subs(n=i,asymp t_q[3])/coeff(series_q,z,i)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ \"#]$\":hwdc8R*[g_DOK!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+\"$\" :,eUX!QY%R%=U&y#!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]\"$\":3w8V T.5&oYW#f#!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$+#$\":!)ovJkK*=]T dxC!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"$]#$\"::9SU_=63xL')R#!#C " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Enlarging " }{XPPEDIT 18 0 "n " "I\"nG6\"" }{TEXT -1 49 ", of course, would lead to a limiting value of 1." }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "To conclude, we compute the table indicating the indicial polynomial at the dominant singularity and th e possible asymptotic behaviours of " }{XPPEDIT 18 0 "P[d](z)" "-&%\"P G6#%\"dG6#%\"zG" }{TEXT -1 7 " there." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "behaviours:=proc(f,x,u) local L,i;\n L:=sort([sol ve(f,x)]);\n convert([seq(u^L[i]*ln(u)^nops(select(type,map(`+`,L[1 ..i-1],-L[i]),integer)),i=1..nops(L))],`+`)\nend:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 71 "M[1]:=d,s,`indicial polynomial at s`,`possib le behaviours for P[d](z)`:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "for i from 2 to 7 do\n M[i]:=i,1/i^2,\n indicial_poly( \{algebraicsubs(double_inv_borel(power_j(i),j,z),j=(1-z)/i^2,j(z))\},j ,z);\n B:=behaviours(M[i][3],eta,u);\n M[i]:=M[i],B\nod:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "matrix([seq([M[i]],i=1..7)]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7)7&%\"dG%\"sG%9indic ial~polynomial~at~sG%@possible~behaviours~for~P[d](z)G7&\"\"##\"\"\"\" \"%,&F/F/%$etaGF-*$%\"uG#!\"\"F-7&\"\"$#F/\"\"**$F2F-,&F/F/-%#lnG6#F4F /7&F0#F/\"#;,$*(F2F/,&F2F-F6F/F/,&F2F/F6F/F/F6,(F/F/*$F4#F/F-F/*&F4F/F =F/F/7&\"\"&#F/\"#D,$*(F2F/,&F2F/!\"#F/F/FFF-F6,*F/F/FJF/*&F4F/F=F-F/* &F4F-F=F8F/7&\"\"'#F/\"#O*,F2F/FFF/FQF/,&F2F/!\"$F/F/,&F2F-FfnF/F/,,F/ F/FJF/*$F4#F8F-F/*&F4F-F=F-F/*&F4F8F=F8F/7&\"\"(#F/\"#\\*,F2F/FFF/FenF /,&F2F/!\"%F/F/FQF-,.F/F/FJF/F[oF/FUF/*&F4F8F=F0F/*&F4F0F=FLF/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "P[d ](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 146 " is a linear combination \+ of the possible behaviours. Solving the connection would indicate whi ch of them appear (with a non-zero coefficient) in " }{XPPEDIT 18 0 "P [d](z)" "-&%\"PG6#%\"dG6#%\"zG" }{TEXT -1 9 ". Next, " }{HYPERLNK 17 "transfer of singularity" 1 "" "transfer of singularity" }{TEXT -1 62 " on the dominant behaviour would yield an asymptotic form for " } {XPPEDIT 18 0 "p[d,n]" "&%\"pG6$%\"dG%\"nG" }{TEXT -1 1 "." }}}}} {MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }