^0   siLuseMivenhypergeometriccallsequencmainvarshvarintnvarsaxvaransatzdenomposhavarintnvarparameteralgebraicexpressintegrandlistvariablthdifferentialsoughtshiftcorrespondeachintegratauxiliaranyexcludingnullmonomialguesspolynomialtakendenominatorrationalcertificatsynopsimainvarstrienonzeroannihilatorwithpolynomialcoeffoutputformolyxmcasesearchusinggivenlookwhosdenominatorsameindrecbutappropriatdenominatorsautomaticalpartpackagcanusedargsonlafterperformcommandalwayaccessedlongfindrwhenevconflictbetweennamanothnameusedsessuseexamplsolvequationunknowncheersuccescputimesecondalsoversion2.psc2Yv6i1n2.texsecZv6i1n2.logonE.signiture~]v6i1n2.dvi.g2texput.logDe_(v6i1r1.texver[P$EthiopianflagOfficial.gifedv6i1r1.dvis Dtext.psE temp.hlpa.sshk).newsrc-news5.keirc 0 (.netscape$8.ntopH.skel%X.wmrc&p.xsession-errors0'.X.err0 }k.Xauthority0 +misc (.fontalias 23*.xmaplev5rc" +.xftprc}{-.xftpcache" }=.bashrc~ointn: converts multisu m to integration " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 " " {TEXT 23 1 " " }{TEXT 260 19 "CALLING SEQUENCES: " }}{PARA 0 "" 0 " " {TEXT 23 49 " msumtointn(binocoeffs, x, sumvar,sumbound) " }} {PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 3 "" 0 "" {TEXT 23 12 " PARAME TERS:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 56 " binocoeff s : finite product of binomial coeffs " }}{PARA |{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 266 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 256 9 "FUNCTION:" } {TEXT 257 46 " sumtointn: converts summation to integration " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 263 1 "C" }{TEXT 258 1 "A" }{TEXT 264 16 "LLING SEQUENCES:" }{TEXT 259 1 " " }}{PARA 0 "" 0 "" {TEXT 23 49 " sumtointn(binocoeffs, x, sumva r, sumbound) " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 262 1 " " }} {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 260 11 "PARAMETERS:" }{TEXT 261 2 " " }}{PARA 0 "" 0 "" {TEXT 23 60 " binocoeffs : finite product of binomial coeffs " }}{PARA 0 "" 0 "" {TEXT 23 47 " \+ x : algebraic expression " }}{PARA 0 "" 0 "" {TEXT 23 45 "  sumvar : index of summation " }}{PARA 0 "" 0 "" {TEXT 23 66 " sumbound : list of lower and upper bound o f sumvar " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 3 "" 0 " " {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 58 " - sumtointn converts a summation expression of the form: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 179 " \+ -----\n \\\n S(n) := ) (product of binocoeffs) x\n /\n \+ -----\n k" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 23 68 " to its equivalent constant term(C T(F)) expression and output F " }}{PARA 0 "" 0 "" {TEXT 23 50 " wit h the integration variables z_1, z_2, ... " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 64 " - A binomial coef, binomial(a, b), should be entered as [a,b] " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 23 76 " - Whenever there is a conflict betwe en the function name sumtointn " }}{PARA 0 "" 0 "" {TEXT 23 65 " \+ and another name used in the same session, use the long form " }} {PARA 0 "" 0 "" {TEXT 23 22 " Mint['sumtointn']. " }}}{PARA 0 "" 0 " " {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 " " 0 "" {TEXT 23 11 " EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 23 21 "To find the CT form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 2 "" 1 "" {TEXT -1 127 " -----\n \\ k \+ 3\n ) (-1) binomial(n, k)\n /\n \+ -----\n k" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w ith(Mint): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sumtointn([n ,k]^3,(-1)^k,k,[0,infinity]); " }}{PARA 2 "" 1 "" {TEXT -1 166 " \+ /z_1 + 1\\n /z_2 + 1\\n n\n \+ |-------| |-------| (1 - z_1 z_2) , [z_1, z_2]\n \+ \\ z_1 / \\ z_2 /" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 1 " " }} {SECT 0 {PARA 3 "" 0 "" {TEXT 23 9 "SEE ALSO:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 " Mint" "" }{TEXT 267 2 ", " }{HYPERLNK 17 "ssum" 2 "Mint/ssum" "" } {TEXT 266 2 ", " }{HYPERLNK 17 "msumtointn" 2 "Mint/msumtointn" "" } {TEXT 265 2 ", " }{HYPERLNK 17 "msum" 2 "Mint/msum" "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}}{MARK "19 2 5" 4 }{VIEWOPTS 1 1 0 1 1 1803 } " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sumtointn([n ,k]^3,(-1)^k,k,[0,infinity]); " }}{PARA 2 "" 1 "" {TEXT -1 166 " \+ /z_1 + 1\\n /z_2 + 1\\n n\n \+ |-------| |-------| (1 - z_1 z_2) , [z_1, z_2]\n \+ \\ z_1 / \\ z_2 /" }}}{PARA 0 "" 0 "" {TEXT -1|Mint,sumtointn|Mint,msumtointn| Mint,msum| Mint,ssum|Mint,findrec1|Mint,findrec2|Mint,findrec3|Mint,checkrec| Mint,esp| Mint,sym|MintMint| Mint,checkrec|Mint,esp| Mint,findrec1| Mint,findrec2| Mint,findrec3| Mint,msum|Mint,msumtointn| Mint,ssum|Mint,sumtointn|Mint,sym||Bibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctmsumfindnonzerorecurrencdifferentialequatconstanttermexpressgivenmultisumcallsequencbinocoeffsumvarmainvarsumboundorderparameterproductfinitbinomialcoeffalgebraicindicsummationmainvariablelistlowerupperboundeachnonnegatintegdesireceqsynopsicoefenterwhenevconflictbetweenfunctionameanothusesamesesslongformmintexamplindctwithinfinittryingrecurrencewithsolvequationunknowncheersuccescputimesecondspialsossumsumtointnmsumtointned|eith|||ele| elementar|en|ence|enompo|enter||||eq ||||eqn|equal ||equat|||equatio|equation|||||equival ||er|ere|erential|es ||esp ||ession|etric|ex|exampl+ ||||||||||exclud ||excludi|express#||||||||fals|ffs|fi ||find#||||||||findr|findrecM|||||finit|||follow|form/|||||||||||DatE certificat2ec form[ hyperlink\integrat'mainNnameormFreUshiftVsynopsiLuseM, [x,y]) ;" }}{PARA 2 "" 1 "" {TEXT -1 38 " \+ true" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }}}{MARK "0 2" 61 }{VIEWOPTS 1 1 0 1 1 1803 } " 0 "" B@B@ 18 "esp(2,[x,y,z]); \+ " }}{PARA 7 "" 1 "" {TEXT -1 31 "Warning, new definition for Chi" }} {PARA 2 "" 1 "" {TEXT -1 58 " \+ x y + x z + y z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 } XT -1HB@B@ 2 2 \+ 2\n rec := (n + 1) (n - 1) F_[n, x, y] - 3 (2 \+ + 3 n) n checkrec,Mint|esp,Mint| findrec1,Mint| findrec2,Mint| findrec3,Mint| msum,Mint|msumtointn,Mint| ssum,Mint|sumtointn,Mint|sym,Mint||M{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Outp ut" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 Mint| Mint,checkrec|Mint,esp| Mint,findrec1| Mint,findrec2| Mint,findrec3| Mint,msum|Mint,msumtointn| Mint,ssum|Mint,sumtointn|Mint,sym||{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 1 " \+ " }{TEXT 258 9 "FUNCTION:" }{TEXT 259 45 " sumtointn: converts multisu m to integration " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 " " {TEXT 23 1 " " }{TEXT 260 19 "CALLING SEQUENCES: " }}{PARA 0 "" 0 " " {TEXT 23 49 " msumtointn(binocoeffs, x, sumvar,sumbound) " }} {PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 3 "" 0 "" {TEXT 23 12 " PARAME TERS:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 56 " binocoeff s : finite product of binomial coeffs " }}{PARA 0 "" 0 "" {TEXT 23 43 " x : algebraic expression " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 70 " sumbound : list of lower and up per bound of each of sumvar " }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 256 48 " sumvar : list of indices of summation" }} {PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }} {PARA 0 "" 0 "" {TEXT 23 11 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 58 "- msumtointn converts a summation expression of the f orm: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 145 " -----\n \\\n ) (produ ct of binocoeffs) x\n /\n -----\n \+ k1,k2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 67 " to its \+ equivalent constant term(CT(F)) expression and output F " }}{PARA 0 " " 0 "" {TEXT 23 50 " with the integration variables z_1, z_2, ... \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 65 " - A \+ binomial coef, binomial(a,b), should be entered as [a,b]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 76 " - Whenever the re is a conflict between the function name sumtointn " }} {PARA 0 "" 0 "" {TEXT 23 65 " and another name used in the same sess ion, use the long form " }}{PARA 0 "" 0 "" {TEXT 23 23 " Mint['msum tointn']. " }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 23 11 " EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 21 "To find the CT form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 " " 1 "" {TEXT -1 134 " -----\n \\ (i + j)\n ) ( -1) binomial(i + j, i) binomial(n, i) binomial(n, j)\n /\n -----\n i.j" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(Mint):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 77 "msumtointn([i+j,i]*[n,i]*[n,j],(-1)^(i+j),[i,j ],[[0,infinity],[0,infinity]]);" }}{PARA 2 "" 1 "" {TEXT -1 134 " \+ n / 1 \\n\n (-z_ 1) |- ---| , [z_1]\n \\ z_1/" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " } {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 263 2 ", " }{HYPERLNK 17 "ssum" 2 "Mint/ssum" "" }{TEXT 262 2 ", " } {HYPERLNK 17 "sumtointn" 2 "Mint/sumtointn" "" }{TEXT 261 2 ", " } {HYPERLNK 17 "msum" 2 "Mint/msum" "" }}}}{MARK "18" 0 }{VIEWOPTS 1 1 0 1 1 1803 } TEXT -1 0 "" }}{PARA 2 " " 1 "" {TEXT -1 134 " -----\n \\ (i + j)\n ) ( -1) binomial(i + j, i) binomial(n, i) binomial(n, j)\n /\n -----\n i.j" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(Mint):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 77 "msumtointn([i+j,i]*[n,i]*[n,j],(-1)^(i+j),[i,j ],[[0,infinity],[0,infinity]]);" }}{PARA 2 "" 1 "" {TEXT -1 134 " \+ n / 1 \\n\n (-z_ 1) |- ---| , [z_1]\n \\ z_1/" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " } {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 263 2 ", " }{HYPERLNK 17 "ssum" ||| |(|4|=|G|O|R|&XCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1-1/y)^n/x/y, [x,y]) ;" }}{PARA 2 "" 1 "" {TEXT -1 38 " \+ true" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }}}{MARK "0 2" 61 }{VIEWOPTS 1 1 0 1 1 1803 } " 0 "" B@B@ 18 "esp(2,[x,y,z]); \+ " }}{PARA 7 "" 1 "" {TEXT -1 31 "Warning, new definition for Chi" }} {PARA 2 "" 1 "" {TEXT -1 58 " \+ x y + x z + y z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 } XT -1HB@B@ 2 2 \+ 2\n rec := (n + 1) (n - 1) F_[n, x, y] - 3 (2 \+ + 3 n) n |{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 266 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0  0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 257 9 "FUNCTION:" } {TEXT 258 90 " msum: finds a non-zero recurrence-differential equation for the constant term expression " }}{PARA 0 "" 0 "" {TEXT 23 40 " \+ of a given multi sum. " }}{PARA 0 "" 0 "" {TEXT 23 2 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 259 17 "CALLING SEQUENCES" }{TEXT 260 2 ": " }}{PARA 0 "" 0 "" {TEXT 23 61 " msum(binocoeffs , x, sumvars, mainvar, sumbound, order) " }}{PARA 0 "" 0 "" {TEXT 23 54 " msum(binocoeffs, x, sumvars, mainvar, sumbound) " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 261 11 "PARAMETERS:" }{TEXT 262 2 " " }}{PARA 0 "" 0 "" {TEXT 23 59 " \+ binocoeffs : product(finite) of binomial coeffs " }}{PARA 0 "" 0 "" {TEXT 23 45 " x : algebraic expression " }} {PARA 0 "" 0 "" {TEXT 23 45 " sumvars : indices of summat ion " }}{PARA 0 "" 0 "" {TEXT 23 38 " mainvar : main vari able " }}{PARA 0 "" 0 "" {TEXT 23 73 " sumbound : list of \+ lower and upper bound of each of sumvars " }}{PARA 0 "" 0 "" {TEXT 23 88 " order : nonnegative integer(order of the desired r ec. eq. in \"mainvar\") " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 10 "SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 65 " - A binomial coef, binomial(a,b), should be entere d as [a,b]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 " - Whenever there is a conflict between the functio n name msum " }}{PARA 0 "" 0 "" {TEXT 23 79 " and another name use d in the same session, use the long form Mint['msum']. " }}}{PARA 0 " " 0 "" {TEXT 23 2 " " }{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 70 "To f ind a non-zero recurrence-differential equation for the CT form of" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 170 " \+ -----\n \\ (i + j)\n ) (-1) \+ binomial(i + j, i) binomial(n, i) binomial(n, j)\n /\n \+ -----\n i,j" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi th(Mint): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "msum([i+j,i]* [n,i]*[n,j],(-1)^(i+j),[i,j],n,[[0,infinity],[0,infinity]]); " }} {PARA 6 "" 1 "" {TEXT -1 58 " \n... trying to find a non-zero recurren ce eq with order 0" }}{PARA 6 "" 1 "" {TEXT -1 54 "\n could not find a non-zero recurrence eq with order 0" }}{PARA 6 "" 1 "" {TEXT -1 58 " \+ \n... trying to find a non-zero recurrence eq with order 1" }}{PARA 6 "" 1 "" {TEXT -1 39 " ... solving 2 equations for 2 unknowns" }}{PARA 6 "" 1 "" {TEXT -1 50 " Cheers! for the success. CPU Time : .04 secon ds." }}{PARA 2 "" 1 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 412 " \+ \+ n / 1 \\n\n \+ I (-z_1) |- ---|\n \+ \\ z_1/\n [F_[n, z_1] - F_[n + 1, z_1] = D_[z_1, 0], F_[n, z_1] = - 1/2 ------------------]\n \+ z_1 Pi" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 266 2 ", " }{HYPERLNK 17 "ssum" 2 "Mint/ssum" "" }{TEXT 265 2 ", " } {HYPERLNK 17 "sumtointn" 2 "Mint/sumtointn" "" }{TEXT 264 2 ", " } {HYPERLNK 17 "msumtointn" 2 "Mint/msumtointn" "" }{TEXT 263 1 " " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}}{MARK "0 2" 24 }{VIEWOPTS 1 1 0 1 1 1803 } 2 "" 1 "" {TEXT -1 412 " \+ \+ n / 1 \\n\n \+ I (-z_1) |- ---|\n \+ \\ z_1/\n [F_[n, z_1] - F_[n + 1, z_1] = D_[z_1, 0], F_[n, z_1] = - 1/2 ------------------]\n \+ z_1 Pi" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 266 2 ", " }{HYPERLNK 17 "ssum" 2 "Mint/ssum" "" }{TEXT 265 2 ", " } {HYPERLNK 17 "sumtointn" 2 "Mint/sumtointn" "" }{TEXT 264 2 ", " } {HYPERLNK 17 "msumtointn" 2 "Mint/nator|nce ||ncould|nd|ndrec|negat|new ||ng|nite| nnihilator|no|non|||||nonnegat||| nontrivial|norder|normal/ |||||||||||nt|||null|||nvar|oly|onal|onl|only|||operator |||order!||| ||orm|||||||fals|ffs|fi ||find#||||||||findr|findrecM|||||finit|||form/||||||||||||gibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctssumfindnonzerorecurrencdifferentialequatioconstanttermexpressgivensumcallsequencbinocoeffsumvarmainvarsumboundorderbinocoffsparameterfiniteproductbinomialcoeffalgebraicindexsummatmainvariabllistlowerupperboundnonnegatintegdesirreceqsynopsieqnexpressoutputdiffwithconstcoefenterwhenevconflictbetweennameanothusedsamesessuselongformintexamplrecurrncectdixonmintinfinitsolvequationunknowncheersuccescputimesecondpialsomsumsumtointnsumtointntointnmsumtointn|nibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctfindrecfindrecurrencoperatorannihilatgivenhypergeometriccallsequenceintnvaraxvarorderdenompoparameteralgebraicexpressintegrandvariablnamerecurrencdifferentialsoughtlistintegratvarsauxiliarparameteranyexcludnullnonnegatintegrderdesirequatenompoguesspolynomialtakendenominatorrationalcertificatsynopsilytriezeroannihilatorwithcoeffformpolyxmcasesearchdifferentialwhosequallookrationalannihilatorappropriatdenominatorcertificatesautomaticallikeeithabovargsbutatmostdefaultmaxhavingdenominatornnihilatorusenouxiliarpartmintpackagcanusedargonlyafterperformcommandalwayaccesslongwhenevconflictbetweennamanothsamesessexampltryingrecurrnceeqsolvequationunknowncheersuccescputimesecondalso|ibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctsumtointnconvertsummatintegratllingsequencbinocoeffsumvasumboundparameterfinitproductbinomialcoeffalgebraicexpresssumvarindexlistlowerupperboundsynopsiformequivalconstanttermwitvariablcoefenterwhenevconflictbetwennameanothusedsamesessuselongmintexamplfindctithinfinitalsossummsumtointnmsum|ibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctsumtointnconvertmultisuintegratcallsequencmsumtointnbinocoeffsumvarsumboundparamtersfinitproductbinomialcoeffalgebraicexpresslistlowerupperboundeachindicsummatsynopsiormproductequivalconstanttermwithvariablcoefenterwhenevreconflictbetweennameanothusedsamesessionuselongformmintmsumtointnexamplfindinfinitalsossum| |||<|1|3|] 31 "Warning, new definition for Chi" }} {PARA 2 "" 1 "" {TEXT -1 58 " \+ x y + x z + y z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 } XT -1HB@B@ 2 2 \+ 2\n rec := (n + 1) (n - 1) F_[n, x, y] - 3 (2 \+ + 3 n) n (1 B@B@+ 1, x, y] = D_[x, - ((1 + x)\n\n \+ 2 3 2 2 2 2 2 3 2 3 2 \+ 2 2\n (x y n + x y n - x y n - x y - 6 n y + 10 n y \+ - 72 n - 60 n - 4 n y - 12 n) F_[n, x, y])\n\n / 2 \+ 2 2 2 2 2 2 2 \+ 2\n / (x y )] + D_[y, n (1 + y) (12 x y n + 22 n y \+ + 6 y + 2 x y + 20 n y + 10 x y n - 39 n y\n 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 256 9 "FUNCTION:" } {TEXT 257 91 " ssum: finds a non-zero recurrence-differential equatio n for the Constant term expression " }}{PARA 0 "" 0 "" {TEXT 23 33 " \+ of a given sum. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }} {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 258 19 "CALLING SEQUENCES: " }} {PARA 0 "" 0 "" {TEXT 23 60 " ssum(binocoeffs, x, sumvar, mainvar , sumbound, order) " }}{PARA 0 "" 0 "" {TEXT 23 53 " ssum(binocoe ffs, x, sumvar, mainvar, sumbound) " }}{PARA 0 "" 0 "" {TEXT 23 2 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 259 11 "PARAMETERS:" }{TEXT 260 2 " " }}{PARA 0 "" 0 "" {TEXT 23 60 " binocoeffs : fi nite product of binomial coeffs " }}{PARA 0 "" 0 "" {TEXT 23 47 " \+ x : algebraic expression " }}{PARA 0 "" 0 "" {TEXT 23 45 " sumvar : index of summation " }}{PARA 0 "" 0 " " {TEXT 23 40 " mainvar : main variab!le " }}{PARA 0 "" 0 "" {TEXT 23 66 " sumbound : list of lower and upper bo und of sumvar " }}{PARA 0 "" 0 "" {TEXT 23 69 " order \+ : a nonnegative integer(order of the desired " }}{PARA 0 "" 0 "" {TEXT 23 49 " rec. eq. in \"mainvar\") " }} {PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 89 "- ssum \+ finds a non-zero recurrence differential eqn. for the constant term ex pression of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 215 " -----\n \+ \\\n S(n) := ) (product of binocoeffs) x\n \+ /\n -----\n \+ k" }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 67 " and output the rec-diff. eqn. with the const. term expression. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 6"4 " - A binomial coef, binomial(a,b), should be entered as [ a,b]. " }}{PARA 0 "" 0 "" {TEXT 261 1 " " }}{PARA 0 "" 0 "" {TEXT 23 71 " - Whenever there is a conflict between the function name \+ ssum " }}{PARA 0 "" 0 "" {TEXT 23 65 " and another name used in the \+ same session, use the long form " }}{PARA 0 "" 0 "" {TEXT 23 17 " M int['ssum']. " }}}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " EXAMPLES: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 89 "To find a non-zero recurre nce-differential eqn. for the CT expression of Dixon's sum " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 173 " \+ -----\n \\ k 3\n \+ ) (-1) binomial(2 n, k)\n /\n \+ -----\n k" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(Mint):" }}}{EXCHG {PARA 0 "># " 0 "" {MPLTEXT 1 0 42 "ssum([2* n,k]^3,(-1)^k,k,n,[0,infinity],1);" }}{PARA 6 "" 1 "" {TEXT -1 41 " .. . solving 77 equations for 73 unknowns" }}{PARA 6 "" 1 "" {TEXT -1 51 " Cheers! for the success. CPU Time : 4.04 seconds." }}{PARA 2 "" 1 " " {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 1986 " \+ 2\n [3 (2 + 3 n) (1 + 3 n) F_[n, z_1, z_ 2] + (n + 1) F_[n + 1, z_1, z_2] =\n\n D_[z_1, R_[z_1, z_2] F_ [n, z_1, z_2]] + D_[z_2, R_[z_2, z_1] F_[n, z_1, z_2]],\n\n \+ 2 2 \+ 2 4\n R_[z_1, z_2] = 1/4 ((z_1 + 1) (-1 - n + 11 z_1 n + 32 z_1 z_2 n - 39 z_1 z_2 n\n\n 2 4 2 2 \+ 4 2 2 2 3 2 3\n - 38 \+ z_1 z_2 n - 13 z_1 z_2 + 6 n z_1 z_2 + 11 z_1 z_2 n - 12 z_1 \+ z_2 n\n\n 3 2 2 \+ 2 3 2 3\n + 8 z_1 z_2 + $2 z_1 z_2 n - 26 z_1 \+ z_2 n - 12 z_1 z_2 n - 6 z_1 z_2\n\n 3 4 \+ 3 2 3 2 3 3 3 3\n + z_1 \+ z_2 n + 2 z_1 z_2 n + 2 z_1 z_2 + 10 z_1 z_2 + 30 z_1 z_2 n \n\n 3 3 3 2 3 2 2 \+ 3 3 3 2\n + 2 z_1 z_2 + 12 z_1 z_2 n - 6 z _1 z_2 n + 12 z_1 z_2 n + 32 z_1 z_2 n\n\n 2 \+ 2 2 2 2 3 4\n \+ - 6 z_2 n - 12 z_2 n + 5 z_1 - 10 z_1 z_2 - 10 n z_1 z_2 + z_1 \+ z_2\n\n 2 4 3 2 2 \+ 2 2\n - 12 z_1 z_2 n - 2 z_2 + 17 n z_1 z_ 2 + 10 z_1 z_2 + 2 n z_1 z_2 - 6 z_2\n\n 2 \+ 2 / 2\n - 2 z_2 n + 6 z_1 n )) / (z_1 z_2 ),\n /\n\n \+ /z_1 + 1\\(2 n) /z_2 + 1\\(2 n) (2 n)\n % \+ |-------| |-------| (1 - z_1 z_2)\n \+ \\ z_1 / \\ z_2 /\n F_[n, z_1, z_ 2] = - 1/4 ------------------------------------------------]\n \+ 2\n \+ z_1 z_2 Pi" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 " " 0 "" {TEXT 23 1 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 9 " SEE ALSO " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "msum" 2 "Mint/msum" "" }{TEXT -1 1 "," }{HYPERLNK 17 "su mtointn" 2 "`Mint/sumtointn`" "" }{TEXT -1 3 " , " }{HYPERLNK 17 "msum tointn" 2 "Mint/msumtointn" "" }}}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}}{MARK "0 2" 67 }{VIEWOPTS 1 1 0 1 1 1803 } _1 z_2 ),\n /\n\n \+ /z_1 + 1\\(2 n) /z_2 + 1\\(2 n) (2 n)\n %|{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 266 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 267 "times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 269 "times" 0 10 0 0 0 0 0 1 0 0Wintel/ |||||||||||intn|intnvar| | |involv ||ion|||ith|known|late|li|like ||ling|linux/ |||||||||||list+||||||||||lling|long' |||||||||look|||lower||||ly|lynomial|mai|main ||abl|ars|at |||denomina| denominato| denominator|||denompo|||denompol|desir||||diff||| differential |||||dixon|ds|each||||ec|| {VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" 23 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 265 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier)" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 1 " \+ " }{TEXT 256 9 "FUNCTION:" }{TEXT 257 57 " findrec1: finds a recurrenc e operator that annihilate a " }}{PARA 0 "" 0 "" {TEXT 23 52 " \+ given hypergeometric function. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 258 18 "CALLING SEQU ENCES:" }{TEXT 259 1 " " }}{PARA 0 "" 0 "" {TEXT 23 58 " findrec 1(f, n, intnvars, axvars, order, denompoly) " }}{PARA 0 "" 0 "" {TEXT 23 47 " findrec1(f, n, intnvars, axvars, order) " }}{PARA 0 "" 0 "" {TEXT 23 51 " findrec1(f, n, intnvars, axvars, denompoly) \+ " }}{PARA 0 "" 0 "" {TEXT 23 40 " findrec1(f, n, intnvars, axvar s) " }}{PARA 0 "" 0* "" {TEXT 23 32 " findrec1(f, n, intnvars) " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 261 13 " PAR AMETERS: " }{TEXT 260 1 " " }}{PARA 0 "" 0 "" {TEXT 23 56 " \+ f : algebraic expression(integrand) " }}{PARA 0 "" 0 "" {TEXT 23 65 " n : variable name w.r.t. which the rec urrence " }}{PARA 0 "" 0 "" {TEXT 23 61 " (diff erential) operator to be sought " }}{PARA 0 "" 0 "" {TEXT 23 72 " \+ intnvars : list of integration vars. " } }{PARA 0 "" 0 "" {TEXT 23 64 " axvars : list of auxiliary vars(parameters) in f, " }}{PARA 0 "" 0 "" {TEXT 23 63 " \+ (if any), excluding \"n\" and \"intnvars\". " }}{PARA 0 "" 0 "" {TEXT 23 40 " input [] if null " }}{PARA 0 "" 0 "" {TEXT 23 70 " order : non-negative integer(the o rder of the desired " }}{PARA 0 "" 0 "" {TEXT 23 41 " \+ rec. equation n) " }}{PARA 0 "" 0 "+" {TEXT 23 71 " d enompoly : is a list of guessed polynomials to be taken as " }}{PARA 0 "" 0 "" {TEXT 23 66 " denominators of the rati onal certificates. " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 71 "- findrec1(f, n, intnvars, axvars, order, denompo ly) tries to find a " }}{PARA 0 "" 0 "" {TEXT 23 64 " non-zero ann ihilator of f with polynomial coeff in \"n\" and " }}{PARA 0 "" 0 "" {TEXT 23 36 " output a recurrence of the form " }}{PARA 0 "" 0 "" {TEXT 23 51 " poly(n)*F_[n+i,x1,x2,...] + ... = " }} {PARA 0 "" 0 "" {TEXT 23 74 " D_[x1,R1[n, x1,...,xm]F_[n,x1,...,xm]] + ... " }}{PARA 0 "" 0 "" {TEXT 23 75 " \+ In this case findrec1 searches for a recurrence(differential) operator " }}{PARA 0 "" 0 "" {TEXT 23 68 " whose order in \"n\" is equal t o \"order\" and looks for rational " }}{PARA 0 "" 0 "" {TEXT 23 55 " \+ ,certificates whose denominators are \"denompoly\". " }}{PARA 0 " " 0 "" {TEXT 265 1 " " }}{PARA 0 "" 0 "" {TEXT 23 69 " - findrec1(f, \+ n, intnvars, axvars, order) tries to find a non-zero " }}{PARA 0 "" 0 "" {TEXT 23 70 " annihilator the by searching for appropriate denom inators of the " }}{PARA 0 "" 0 "" {TEXT 23 41 " rational certific ates automatically. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 72 " - findrec1(f,n,intnvars,axvars,denompoly) is like \+ either of the above " }}{PARA 0 "" 0 "" {TEXT 23 72 " findrec1(args ),but looks for a non-zero annihilator whose order in " }}{PARA 0 "" 0 "" {TEXT 23 71 " \"n\" is at most 6(the default max order) and ha ving \"denompoly\" as " }}{PARA 0 "" 0 "" {TEXT 23 53 " the denomi nators of the rational certificates. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " - findrec1(f,n,intnvars,axvars) is the like the above findrec1(args), " }}{PARA 0 "" 0 "" {TEXT 23 72 " \+ but in this case findr-ec1 searches for the appropriate denominators " }}{PARA 0 "" 0 "" {TEXT 23 72 " and output (if any) a non-zero a nnihilator whose order in \"n\" is at " }}{PARA 0 "" 0 "" {TEXT 23 14 " most 6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 74 " - findrec1(f,n,intnvars) use this form if there is no a uxiliary vars. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 78 " - This function is part of the Mint package, and so can be used in the form " }}{PARA 0 "" 0 "" {TEXT 23 67 " findrec1(arg s) only after performing the command with(Mint) or " }}{PARA 0 "" 0 " " {TEXT 23 79 " with(Mint,findrec1). The function can always be ac cessed in the long form " }}{PARA 0 "" 0 "" {TEXT 23 28 " Mint[find rec1](args). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " - Whenever there is a conflict between the function nam e findrec1 and " }}{PARA 0 "" 0 "" {TEXT 23 71 " another name used \+ in the same session, use the form Mint['findrec1'" }}}{PA.RA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 10 "EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(Mint):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "findrec1((1+x)^n*(1+y)^n/(1-x*y),n,[x,y]); " } }{PARA 6 "" 1 "" {TEXT -1 58 " \n... trying to find a non-zero recurre nce eq with order 0" }}{PARA 6 "" 1 "" {TEXT -1 54 "\n could not find \+ a non-zero recurrence eq with order 0" }}{PARA 6 "" 1 "" {TEXT -1 58 " \n... trying to find a non-zero recurrence eq with order 1" }}{PARA 6 "" 1 "" {TEXT -1 41 " ... solving 24 equations for 19 unknowns" }} {PARA 6 "" 1 "" {TEXT -1 50 " Cheers! for the success. CPU Time : .38 seconds." }}{PARA 2 "" 1 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 172 " [(2 + 4 n) F_[n, x, y] + (-n - 1) F_[n + 1, x, y] =\n\n \+ D_[x, R_[x, y] F_[n, x, y]] + D_[y, R_[y, x] F_[n, x, y]],\n\n \+ R_[x, y] = - 1/2 (1 + x) (2 x + x y - 3)]" }}}}{PARA 0 "" 0 "" {TEXT 23 4 " /" }}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 10 "SEE ALSO: " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 264 1 ", " }{HYPERLNK 17 "findrec2" 2 "Mint/findrec2" "" }{TEXT 263 2 ", " } {HYPERLNK 17 "findrec3 " 2 "Mint/findrec3" "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 " " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } with order 1" }}{PARA 6 "" 1 "" {TEXT -1 41 " ... solving 24 equations for 19 unknowns" }} {PARA 6 "" 1 "" {TEXT -1 50 " Cheers! for the success. CPU Time : .38 seconds." }}{PARA 2 "" 1 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 172 " [(2 + 4 n) F_[n, x, y] + (-n - 1) F_[n + 1, x, y] =\n\n \+ D_[x, R_[x, y] F_[n, x, y]] + D_[y, R_[y, x] F_[n, x, y]],\n\n \+ R_[x, y] = - 1/2 (1 + x) (2 x + x y - 3)]" }}}}{PARA 0 "" 0 "" {TEXT 23 4 " /|Nibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctmintcheckrectakerecurrencequatrecinvolvessymbolreturntruesatisfcallsequencparametersynopsiusefulparticularverifoutputprocedurfindrecindrecwhenevconflictbetweenfunctionnameanothameusedsamesessuseformexamplwithalsominfind|0ibmintellinuxmaplinputcourinormaltextoutputheadwarnfunctespgeneratelementarsymmetricpolynomialcallsequencparameterintegorderelementarsymunctlistvariablsynopsiionvarswhenevconflictbetweennamendanothusedsameessionuselongformmintexamplwithmintnewdefinitchi|ibmintellinuxmaplinputcourinormaltextoutputheadfunctmintsymcheckwhethgivensymmetricariablcallsequencvarsparameteralgebraicexpresslistvariablsynopsireturntruefalsotherwiswhenevconflictbetweennameanothusedsamesessuseformexamplwith|!ibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctmintfindrecfindrecurrencoperatorannihilategivenhypergeometriccallsequencmainvarshvarintnvarsaxvaransatzdenomposhavarintnvarparameteralgebraicexpressintegrandlistvariablthdifferentialsoughtshiftcorrespondeachintegratauxiliaranyexcludingnullmonomialguesspolynomialtakendenominatorrationalcertificatsynopsimainvarstrienonzeroannihilatorwithpolynomialcoeffoutputformolyxmcasesearchusinggivenlookwhosdenominatorsameindrecbutappropriatdenominatorsautomaticalpartpackagcanusedargsonlafterperformcommandalwayaccessedlongfindrwhenevconflictbetweennamanothnameusedsessuseexamplsolvequationunknowncheersuccescputimesecondalsocess|check|checkrec ||cheer|||||chi ||coef||||coeff|||||||command||||conflict/ |||||||||||const|constant||||contain|convert || correspond ||couri/y|| || | || | |||cpu|||||ct||||default ||definit ||denom ||denomi|denomin|denomina| denominato| denominator|||denompo|||denompol|desir||||diff||| differential |||||dixon|ds|each||||ec||Nibmintellinuxmaplinputcourihyperlinknormaltextoutputheadfunctmintcheckrectakerecurrencequatrecinvolvessymbolreturntruesatisfcallsequencparametersynopsiusefulparticularverifoutputprocedurfindrecindrecwhenevconflictbetweenfunctionnameanothameusedsamesessuseformexamplwithalsominfind|0ibmintellinuxmaplinputcourinormaltextoutputheadwarnfunctespgeneratelementarsymmetricpolynomialcallsequencparameterintegorderelementarsymunctlistvariablsynopsiionvarswhenevconflictbetweennamendanothusedsameessionuselongformmintexamplwithmintnewdefinitchishiftVsynopsiLuseM 0 "" 0 "" {TEXT 23 39 " v : list of \+ variables " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 4 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT 23 106 " - the funct ion esp(order,vars) outputs the elementary symmetric polynomial in the variables v of order n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" |{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 264 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 265 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 266 " Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 5 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 256 9 "FUNCTION:" } {TEXT 257 63 " findrec2: finds a recurrence operator that annihilate a given " }}{PARA 0 "" 0 "" {TEXT 23 46 " hypergeom etric function. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 258 19 "CALLING SEQUENCES: " }}{PARA 0 "" 0 "" {TEXT 23 70 " findrec2(f, mainvars, intnvars, axvars, mainorders , denompoly) " }}{PARA 0 "" 0 "" {TEXT 23 59 " findrec2(f, mainv ars, intnvars, axvars, mainorders) " }}{PARA 0 "" 0 "" {TEXT 23 658 " \+ findrec2(f, mainvars, intnvars, axvars, denompoly) " }}{PARA 0 " " 0 "" {TEXT 23 47 " findrec2(f, mainvars, intnvars, axvars) " } }{PARA 0 "" 0 "" {TEXT 23 39 " findrec2(f, mainvars, intnvars) \+ " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " } {TEXT 259 11 "PARAMETERS:" }{TEXT 260 2 " " }}{PARA 0 "" 0 "" {TEXT 23 57 " f : algebraic expression(integrand) " }} {PARA 0 "" 0 "" {TEXT 23 70 " mainvars : list of variables w.r.t. which the recurrence " }}{PARA 0 "" 0 "" {TEXT 23 61 " \+ (differential) operator to be sought " }}{PARA 0 "" 0 "" {TEXT 23 73 " intnvars : list of integration vars. \+ " }}{PARA 0 "" 0 "" {TEXT 23 65 " axvars \+ : list of auxiliary vars(parameters) in f, " }}{PARA 0 "" 0 "" {TEXT 23 71 " (if any), excluding \"mainvars\" \+ and \"intnvars\". " }}{PARA 0 "" 0 "" {TEXT 23 42 " \+ input [7] if null " }}{PARA 0 "" 0 "" {TEXT 23 75 " mai norders : list of nonnegative integers(orders of the desired " }} {PARA 0 "" 0 "" {TEXT 23 45 " rec. eq. \"mainva rs\") " }}{PARA 0 "" 0 "" {TEXT 23 72 " denompoly : is a li st of guessed polynomials to be taken as " }}{PARA 0 "" 0 "" {TEXT 23 67 " denominators of the rational certificates. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 73 "- f indrec2(f, mainvars, intnvars, axvars, mainorders, denompoly) tries \+ " }}{PARA 0 "" 0 "" {TEXT 23 77 " to find a non-zero annihilator of f with polynomial coeff in \"mainvars\" " }}{PARA 0 "" 0 "" {TEXT 23 40 " and output a recurrence of the form " }}{PARA 0 "" 0 "" {TEXT 23 66 " poly(n)*F_[n1+i1,n2 + i2,..., x1,x2,...] + ... = \+ " }}{PARA 0 "" 0 "" {TEXT 23 74 " D_[x1,R1[n1,n2 ,...,x1,...,xm]F_[n1,n2,...,x1,...,xm]] + ... "8 }}{PARA 0 "" 0 "" {TEXT 23 75 " In this case findrec2 searches for a recurrence(diffe rential) operator " }}{PARA 0 "" 0 "" {TEXT 23 78 " whose order in \+ each \"mainvars\" is equal to the corresponding \"mainorders\" " }} {PARA 0 "" 0 "" {TEXT 23 77 " and looks for rational certificates w hose denominators are \"denompoly\". " }}{PARA 0 "" 0 "" {TEXT 264 2 " " }}{PARA 0 "" 0 "" {TEXT 23 69 " - findrec2(f, mainvars, intnvars , axvars, mainorders) the same as " }}{PARA 0 "" 0 "" {TEXT 23 69 " \+ findrec2(f, mainvars, intnvars, axvars, mainorders, denompoly), " } }{PARA 0 "" 0 "" {TEXT 23 66 " but searches for appropriate denomin ators of the certificates " }}{PARA 0 "" 0 "" {TEXT 23 20 " automat ically. " }}{PARA 0 "" 0 "" {TEXT 265 1 " " }}{PARA 0 "" 0 "" {TEXT 23 74 " - findrec2(f, mainvars, intnvars, axvars, denompoly) is like \+ either of " }}{PARA 0 "" 0 "" {TEXT 23 80 " the above findrec2(arg s), but looks for a non-zero annihilator whose order " }}{PARA 0 "" 0 "" 9{TEXT 23 73 " in each of \"mainvars\" is at most 6(the default max order) and having " }}{PARA 0 "" 0 "" {TEXT 23 67 " \"denompol y\" as the denominators of the rational certificates. " }}{PARA 0 "" 0 "" {TEXT 266 1 " " }}{PARA 0 "" 0 "" {TEXT 23 73 " - findrec2(f,mai nvars,intnvars,axvars) is the like either of the above " }}{PARA 0 "" 0 "" {TEXT 23 75 " findrec2(args),but in this case findrec2 search es for the appropriate " }}{PARA 0 "" 0 "" {TEXT 23 78 " denominato rs and output (if any) a nontrivial annihilator whose order in " }} {PARA 0 "" 0 "" {TEXT 23 31 " each mainvars at most 6. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 78 " - This functi on is part of the Mint package, and so can be used in the form " }} {PARA 0 "" 0 "" {TEXT 23 67 " findrec2(args) only after performing \+ the command with(Mint) or " }}{PARA 0 "" 0 "" {TEXT 23 79 " with(Mi nt,findrec2). The function can always be accessed in the long form " }}{PARA 0 "" 0 "" {TEXT 23 28 " : Mint[findrec2](args). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " - Whenever th ere is a conflict between the function name findrec2 and " }}{PARA 0 " " 0 "" {TEXT 23 74 " another name used in the same session, use the form Mint['findrec2']. " }}}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " EXAMPL ES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mint): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "findrec2((1+x)^n*(1+y)^m/(1-x*y)^k,[n,m],[x,y],[k]); " }}{PARA 7 "" 1 "" {TEXT -1 31 "Warning, new definition for Chi" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... trying to get a recurrence eq with order [0, 0 ] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 68 "\ncould not find a non-zer o recurrence eq with order [0, 0] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... trying to get a recurrence eq with order [1, 0] in [n, m] " }}{PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 19 equations for 13 u;n knowns" }}{PARA 6 "" 1 "" {TEXT -1 49 " Cheers! for the success. CPU \+ Time: .51 seconds." }}{PARA 2 "" 1 "" {TEXT -1 142 " (n + m + 2 - k) \+ F_[n, m, x, y] + (-n - 2 + k) F_[n + 1, m, x, y] =\n\n D_[x, -x (1 + x) F_[n, m, x, y]] + D_[y, (1 + y) F_[n, m, x, y]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 23 3 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT 23 10 " SEE ALSO:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 263 2 ", " } {HYPERLNK 17 "findrec1" 2 "Mint/findrec1" "" }{TEXT 262 2 ", " } {HYPERLNK 17 "findrec3 " 2 "Mint/findrec3" "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}}{MARK "26 4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } order [0, 0] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... trying to get a recurrence eq with order [1, 0] in [n, m] " }}{PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 19 equations for 13 u;|ibmintellinuxmaplinputcourihyperlinknormaltextoutputheadwarnfunctfindrecfindrecurrencoperatorannihilatgivenhypergeometriccallsequencmainvarintnvaraxvarmainorderdenompomainvarsparameteralgebraicexpressintegrandlistvariabldifferentialsoughtintegratvarsauxiliaranyexcludnullmainordernonnegatintegerorderdesirreceqmainvarslistguesspolynomialtakendenominatorrationalcertificatsynopsiindrectrienonzeroannihilatorwithcoeffformpolyxmcasesearchdiffrentialwhoseachequalcorrespondlookhosesamebutappropriatdenominatorautomaticallikeeithabovargatmostdefaultmaxhavingdenompolnvarargsesdenominatonontrivialfunctipartmintpackagcanusedonlyafterperformcommandmintalwayaccesslongwhenevthereconflictbetweennameanothsessuseexamplnewdefinitchitryinggetncouldzersolvequationunknowncheersuccescputimesecondalso HVtC@ the continuous version of the multi-WZ methodx|;{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 ->1 0 } {PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 256 9 "FUNCTION:" } {TEXT 257 62 " Mint[findrec3]: finds a recurrence operator that annihi late " }}{PARA 0 "" 0 "" {TEXT 23 60 " a gi ven hypergeometric function. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }} {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 258 19 "CALLING SEQUENCES: " }} {PARA 0 "" 0 "" {TEXT 23 74 " findrec3(f, mainvars, shvars, intn vars, axvars, ansatz, denompoly) " }}{PARA 0 "" 0 "" {TEXT 23 67 " \+ findrec3(f, mainvars, shavars, intnvars, axvars, ansatz) " }} {PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " } {TEXT 259 13 "PARAMETERS: " }}{PARA 0 "" 0 "" {TEXT 23 57 " \+ f : algebraic expression(integrand) " }}{PARA 0 "" 0 "" {TEXT 23 70 " mainvars : list of variables w.r.t. which th e recurrence " }}{PARA 0 "" 0 "" {TEXT 23 61 " \+ ?(differential) operator to be sought " }}{PARA 0 "" 0 "" {TEXT 23 71 " shvars : shift variables corresponding to each mainvars " }}{PARA 0 "" 0 "" {TEXT 23 73 " intnvars : list of inte gration vars. " }}{PARA 0 "" 0 "" {TEXT 23 65 " axvars : list of auxiliary vars(parameters) in f, " }} {PARA 0 "" 0 "" {TEXT 23 71 " (if any), excludi ng \"mainvars\" and \"intnvars\". " }}{PARA 0 "" 0 "" {TEXT 23 42 " \+ input [] if null " }}{PARA 0 "" 0 "" {TEXT 23 56 " ansatz : list of monomials in the shvars " }}{PARA 0 "" 0 "" {TEXT 23 72 " denompoly : is a list of guessed po lynomials to be taken as " }}{PARA 0 "" 0 "" {TEXT 23 67 " \+ denominators of the rational certificates. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 76 "- findrec3(f, mainva rs, shvars, intnvars, axv@ars, ansatz, denompoly) tries " }}{PARA 0 "" 0 "" {TEXT 23 76 " to find a non-zero annihilator of f with polynom ial coeff in \"mainvars\" " }}{PARA 0 "" 0 "" {TEXT 23 40 " and out put a recurrence of the form " }}{PARA 0 "" 0 "" {TEXT 23 66 " p oly(n)*F_[n1+i1,n2 + i2,..., x1,x2,...] + ... = " }}{PARA 0 "" 0 "" {TEXT 23 74 " D_[x1,R1[n1,n2,...,x1,...,xm]F_[n1, n2,...,x1,...,xm]] + ... " }}{PARA 0 "" 0 "" {TEXT 23 75 " In this \+ case findrec3 searches for a recurrence(differential) operator " }} {PARA 0 "" 0 "" {TEXT 23 69 " using the given ansatz and looks for \+ rational certificates whose " }}{PARA 0 "" 0 "" {TEXT 23 37 " denom inators are \"denompoly\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 23 73 " - findrec3(f, mainvars, shvars, intnvar s, axvars, ansatz) the same as " }}{PARA 0 "" 0 "" {TEXT 23 71 " f indrec3(f, mainvars, shvars, intnvars, axvars, ansatz, denompoy), " }} {PARA 0 "" 0 "" {TEXT 23 67 " but searches for apprAopriate denomina tors of the certificates " }}{PARA 0 "" 0 "" {TEXT 23 21 " automat ically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 78 " - This function is part of the Mint package, and so can be us ed in the form " }}{PARA 0 "" 0 "" {TEXT 23 67 " findrec3(args) onl y after performing the command with(Mint) or " }}{PARA 0 "" 0 "" {TEXT 23 79 " with(Mint,findrec3). The function can always be acce ssed in the long form " }}{PARA 0 "" 0 "" {TEXT 23 28 " Mint[findre c3](args). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " - Whenever there is a conflict between the function nam e findrec3 and " }}{PARA 0 "" 0 "" {TEXT 23 74 " another name used \+ in the same session, use the form Mint['findrec3']. " }}}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mint): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXBT 1 0 80 "findrec3(1/(1-x-x^2-y-y^2)/x^(m+1)/ y^(n+1),[n,m],[N,M],[x,y],[],[N,M,N*M,M^2]); " }}{PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 29 equations for 22 unknowns" }}{PARA 6 " " 1 "" {TEXT -1 50 " Cheers! for the success. CPU Time : .51 seconds. " }}{PARA 2 "" 1 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 477 " (8 + 4 m + 4 n) F_[n, m, x, y] + (8 + 2 n + 4 m) F_[n, m + 1, x, y] + (- 5 m - 10) F_[n, 2 + m, x, y]\n\n + (n + 1) F_[n + 1, m + 1, x, y] + (2 n + 2) F_[n + 1, m, x, y] =\n\n \+ 2\n (4 x - 5 + 4 x ) F_[n, m, x, y] (2 y + 1 ) (2 x + 1) F_[n, m, x, y]\n D_[x, - -------------------------- -----] + D_[y, - ----------------------------------]\n \+ x x" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 4 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: "C }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Mint" 2 "Mint" "" }{TEXT 262 2 ", " }{HYPERLNK 17 "findrec1" 2 "Mint/findrec1" "" }{TEXT 261 2 ", " } {HYPERLNK 17 "findrec2" 2 "Mint/findrec2" "" }{TEXT 260 2 ", " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}}{MARK "23 4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } y] + (8 + 2 n + 4 m) F_[n, m + 1, x, y] + (- 5 m - 10) F_[n, 2 + m, x, y]\n\n + (n + 1) F_[n + 1, m + 1, x, y] + (2 n + 2) F_[n + 1, m, x, y] =\n\n \+ 2\n (4 x - 5 + 4 x ) F_[n, m, x, y] (2 y + 1 ) (2 x + 1) F_[n, m, x, y]\n D_[x, - -------------------------- -----] + D_[y, - ----------------------------------]\n \+ x x" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 4 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: "Cable|abov|||ac|acce|access ||after|||| algebraic#||||||||also#||||||||alway||||ame|ameter|ann|annihi| annihilat || annihilator|||anoth/ |||||||||||ansatz|any||| appropriat|||arg ||args ||||ariabl|ars|at |||denomina| denominato| denominator|||denompo|||denompol|desir||||diff||| differential |||||dixon|ds|each||||ec|ates|ator|automat || automatical|auxiliar|||availabl|axvar| | |betw|between+ ||||||||||binoco| binocoeff ||||binomial||||bo|bound|||but|||cal|call' |||||||||can||||case|||ce|certific| certificat |||||denomi|denomin|denomina| denominato| denominator|||denompo|||denompol|desir||||diff||| differential |||||dixon|ds|each||||ec|otherwis|ou|out|outp|output+||||||||||packag||||par|param| parameter' |||||||||part|||| particular|per|perform||||pi ||po|poly ||polynom| polynomial|||press|procedur|produ|product||||prov|pu|put|rati|rational|||rder|re ||||||||||findr|findrecM|||||finit|||form/||||||||||||{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -H1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 256 8 "FUNCTION" }{TEXT 257 66 ": Mint[checkrec]: takes a recurrence equation rec which involv es " }}{PARA 0 "" 0 "" {TEXT 23 76 " the symbols F_ or D_ o r R_, and returns True if f satisfies the " }}{PARA 0 "" 0 "" {TEXT 23 34 " recurrence equation. ," }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 258 19 "CALLING SEQUENC ES: " }}{PARA 0 "" 0 "" {TEXT 23 23 " checkrec(rec,f) " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 259 13 "PARAMETERS: " }}{PARA 0 "" 0 "" {TEXT 23 45 " rec \+ : recurrence equation " }}{PARA 0 "" 0 "" {TEXT 23 33 " \+ f : function " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 74 "- checkrec is useful in particular to verify the ou tput of the procedures " }}{PARA 0 "" 0 "" {TEXT 23 35 " findrec1, f indrec2 and findrecI3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 75 " - Whenever there is a conflict between the func tion name checkrec " }}{PARA 0 "" 0 "" {TEXT 23 77 " and another n ame used in the same session, use the form Mint['checkrec']. " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with( Mint): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "rec:=(n+1)^2*(n-1)^2*F_[n,x,y]-3*(2+3*n)*n^2*(1+3*n) *F_[n+1,x,y] = D_[x,-(1+x)*(x*y^2*n^3+x*y^2*n^2-x*y^2*n-x*y^2-6*n*y^2+ 10*n^3*y^2-72*n^3-60*n^2-4*n^2*y^2-12*n)/x/y^2*F_[n,x,y]]+D_[y,n*(1+y) *(12*x*y^2*n^2+22*n*y^2+6*y^2+2*x*y^2+20*n^2*y^2+10*x*y^2*n-39*n*y-33* n^2*y-9*n^2*x*y-3*n*x*y-12*y+3*n+3*n*x+9*x*n^2+9*n^2)/x^2/y^2*F_[n,x,y ]];" }}{PARA 2 "" 1 "" {TEXT -1 872 " 2 2 \+ 2\n rec := (n + 1) (n -J 1) F_[n, x, y] - 3 (2 \+ + 3 n) n (1 + 3 n) F_[n + 1, x, y] = D_[x, - ((1 + x)\n\n \+ 2 3 2 2 2 2 2 3 2 3 2 \+ 2 2\n (x y n + x y n - x y n - x y - 6 n y + 10 n y \+ - 72 n - 60 n - 4 n y - 12 n) F_[n, x, y])\n\n / 2 \+ 2 2 2 2 2 2 2 \+ 2\n / (x y )] + D_[y, n (1 + y) (12 x y n + 22 n y \+ + 6 y + 2 x y + 20 n y + 10 x y n - 39 n y\n /\n\n \+ 2 2 2 2 / 2 2\n - 33 n y - 9 n x y - 3 n x y - 12 y + 3 n + 3 n x + 9 x n + 9 n ) F_[n, x, y] / (x y )]\n \+ \+ /" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "checkrec(rec , (1+1/x)^n*(1+1/y)^(2*n)/x^n/y^n); " }}{PARA 2 "" 1 "" {TEXT -1 57 " \+ K true" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SEE ALSO: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Min t" 2 "Mint" "" }{TEXT 262 2 ", " }{HYPERLNK 17 "findrec1" 2 "Mint/find rec1" "" }{TEXT 261 2 ", " }{HYPERLNK 17 "findrec2" 2 "Mint/findrec2" "" }{TEXT 260 2 ", " }{HYPERLNK 17 "findrec3 " 2 "Mint/findrec3" "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}} {MARK "14 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } y + 10 x y n - 39 n y\n /\n\n \+ 2 2 2 2 / 2 2\n - 33 n y - 9 n x y - 3 n x y - 12 y + 3 n + 3 n x + 9 x n + 9 n ) F_[n, x, y] / (x y )]\n \+ \+ /" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "checkrec(rec , (1+1/x)^n*(1+1/y)^(2*n)/x^n/y^n); " }}{PARA 2 "" 1 "" {TEXT -1 57 " \+ Ktake|taken|||term||||ters|text+ ||||||||||th||||thes|time||||||tion|tointn ||tors|tput|trie|||true ||trying|||type|uenc|un|unct|und|unknown||||up|upper|||urrenc|us|use+ |||||||||||||rder|re ||used+ ||||||||||useful|using|ut|uxiliar|vari|variabl' |||||||||vars |||||ven|verif|vers|ving|warn ||whenev/ |||||||||||wheth|whos|||wi|wit|with+||||||||||xm|||zer|zero||||| mainorder|mainv|mainva ||mainvar||||mapl/ |||||||||||max ||mentar|mi|||min ||mint/U||||| | | ||||monomial|most ||msum||||| msumtointn |||||mtointn|multi|multipl|multisu|nam ||name/||||||||||||||| {VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT 23 63 " FUNCTION: esp: generates the ele mentary symmetric polynomials " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }} {PARA 4 "" 0 "" {TEXT 23 20 " CALLING SEQUENCES: " }}{PARA 0 "" 0 "" {TEXT 23 16 " esp(n,v) " }}{PARA 0 "" 0 ""P {TEXT 23 2 " " }} {PARA 4 "" 0 "" {TEXT 23 14 " PARAMETERS: " }}{PARA 0 "" 0 "" {TEXT 23 71 " n : integer(the order of the elementary sym. f unction)" }}{PARA 0 "" 0 "" {TEXT 23 39 " v : list of \+ variables " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 4 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT 23 106 " - the funct ion esp(order,vars) outputs the elementary symmetric polynomial in the variables v of order n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 468 " \+ / n \\\n \+ ----- | --------' |\n \+ \\ |' | | |\n \+ ) | | | v[i.j]|\n \+ / | | | |\n \+ ----- | | | |\n Q\+ i.1<... " 0 "" {MPLTEXT 1 0 11 "with(Mi nt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "esp(2,[x,y,z]); \+ " }}{PARA 7 "" 1 "" {TEXT -1 31 "Warning, new definition for Chi" }} {PARA 2 "" 1 "" {TEXT -1 58 " \+ x y + x z + y z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 } \+ ----- | | | |\n Q| {VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE " " -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 " Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 256 9 "FUNCTION:" } {TEXT 257 63 " Mint[sym]: checks whether a given function is symmetric w.r.t " }}{PARA 0 "" 0 "" {TEXT 23 40 " S given v ariables. " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 258 18 "CALLING SEQ UENCE: " }}{PARA 0 "" 0 "" {TEXT 23 20 " sym(f,vars) " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 259 13 "PARAMETERS: " }}{PARA 0 "" 0 "" {TEXT 23 45 " f \+ : an algebraic expression " }}{PARA 0 "" 0 "" {TEXT 23 39 " \+ vars : list of variables " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }} {SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 68 "- sym returns true, if f is symmetric w.r. t. vars, false otherwise. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 74 "- Whenever there is a conflict between the \+ function name sym and " }}{PARA 0 "" 0 "" {TEXT 23 70 " another name used in the same session, use the form Mint['sym']. " }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0T {PARA 3 "" 0 "" {TEXT 23 11 " EXAMPLES: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(Mint):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1-1/y)^n/x/y, [x,y]) ;" }}{PARA 2 "" 1 "" {TEXT -1 38 " \+ true" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }}}{MARK "0 2" 61 }{VIEWOPTS 1 1 0 1 1 1803 } {SECT 0 {PARA 3 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 68 "- sym returns true, if f is symmetric w.r. t. vars, false otherwise. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 74 "- Whenever there is a conflict between the \+ function name sym and " }}{PARA 0 "" 0 "" {TEXT 23 70 " another name used in the same session, use the form Mint['sym']. " }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0Trec ||||recurr ||recurren| recurrenc||||||releas|rential|return || rgeometric|rgs|rs ||same/ |||||||||||satisf|se|search|||secon|second||||seq|sequ|sequenc' |||||||||sess+ ||||||||||shavar|shift|shvar|solv|||||sought|||specific|ssed|ssum|||||st|su|succes|||||sum ||sumbound ||||summat|||| sumtointn|||||sumva|sumvar ||||sym |||symbol| symmetric ||synopsi/ ||||||||||| 0 0 3 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" -1 271 "times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 272 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 267 9 "HELP FOR:" } {TEXT 268 37 " a package for multiple integration " }}{PARA 0 "" 0 " " {TEXT 23 3 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 269 18 "CAL LING SEQUENCE: " }{TEXT 270 1 " " }}{PARA 0 "" 0 "" {TEXT 23 27 " Mi nt[](args) " }}{PARA 0 "" 0 "" {TEXT 23 20 " (a rgs) " }}{PARA 0 "" 0 "" {TEXT 256 5 " " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT 271 11 "SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT 23 77 " - This package contains functions to prove identities which involves pu re " }}{PARA 0 "" 0 "" {TEXT 23 55 " multiple integrals with Hype rgeometric integrand. " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 0 "" 0 "" {TEXXT 23 36 " - The functions available are: " }}{PARA 0 " " 0 "" {TEXT 23 13 " " }{HYPERLNK 17 "findrec1" 2 "Mint/fi ndrec1" "" }{TEXT 257 10 " " }{HYPERLNK 17 "sumtointn" 2 "Min t/sumtointn" "" }{TEXT 261 12 " " }{HYPERLNK 17 "sym" 2 "Mi nt/sym" "" }{TEXT 265 3 " " }}{PARA 0 "" 0 "" {TEXT 23 13 " \+ " }{HYPERLNK 17 "findrec2" 2 "Mint/findrec2" "" }{TEXT 258 10 " \+ " }{HYPERLNK 17 "msumtointn" 2 "MInt/msumtointn" "" }{TEXT 262 11 " " }{HYPERLNK 17 "esp" 2 "Mint/esp" "" }{TEXT 266 3 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 13 " " }{HYPERLNK 17 "findrec 3" 2 "Mint/findrec3" "" }{TEXT 259 10 " " }{HYPERLNK 17 "ssum " 2 "Mint/ssum" "" }{TEXT 263 1 " " }}{PARA 0 "" 0 "" {TEXT 23 13 " \+ " }{HYPERLNK 17 "checkrec" 2 "Mint/checkrec" "" }{TEXT 260 9 " " }{HYPERLNK 17 "msum" 2 "Mint/msum" "" }{TEXT 264 5 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 75 " - T o get help for a specific funYction type either ?Mint[] or \+ " }}{PARA 0 "" 0 "" {TEXT 23 74 " ?Mint, where is one from the above list. " }}{PARA 0 "" 0 "" {TEXT 272 3 " " }}{PARA 0 "" 0 "" {TEXT 23 77 " - These functions are part of th e Mint package and so can be used in the " }}{PARA 0 "" 0 "" {TEXT 23 74 " form findrec1(..), findrec2(..) ...only after performing the command " }}{PARA 0 "" 0 "" {TEXT 23 70 " with(Mint) or with(Mint ). The functions can always be " }}{PARA 0 "" 0 "" {TEXT 23 56 " accessed in the long form Mint[](args) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 69 " - Whenev er there is a conflict between a function name in Mint and " }}{PARA 0 "" 0 "" {TEXT 23 78 " another name used in the same session u se the form Mint[]. " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }} {PARA 0 "" 0 "" {TEXT 23 41 " - The following variables are global: \+ " }}{PARA 0 "" 0 "" {TEXT 23 37 " F_, D_,Z R_, z_ \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 51 " - Th is version of Mint is for Maple V Release 5. " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}}{MARK "0 2" 35 }{VIEWOPTS 1 1 0 1 1 1803 } 77 " - These functions are part of th e Mint package and so can be used in the " }}{PARA 0 "" 0 "" {TEXT 23 74 " form findrec1(..), findrec2(..) ...only after performing the command " }}{PARA 0 "" 0 "" {TEXT 23 70 " with(Mint) or with(Mint ). The functions can always be " }}{PARA 0 "" 0 "" {TEXT 23 56 " accessed in the long form Mint[](args) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 69 " - Whenev er there is a conflict between a function name in Mint and " }}{PARA 0 "" 0 "" {TEXT 23 78 " another name used in the same session u se the form Mint[]. " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }} {PARA 0 "" 0 "" {TEXT 23 41 " - The following variables are global: \+ " }}{PARA 0 "" 0 "" {TEXT 23 37 " F_, D_,Zfunc|funct/'||||||||||| functi|functio|function|generat|get ||gi|given||||||global|grat|guess|||ha|having|head+ ||||||||||help|hose|hype| hypergeom|hypergeometric || hyperlink' |||||||||ial|ibm/ |||||||||||ical ||identit|ihilator|inator ||ind|index ||indic ||indrec|||infinit||||input+ ||||||||||int|inte|integ||||integer|integral| integrand||||integrat||||||shift||ibmintellinuxmaplinputcourinormaltextoutputheadfunctmintsymcheckwhethgivensymmetricariablcallsequencvarsparameteralgebraicexpresslistvariablsynopsireturntruefalsotherwiswhenevconflictbetweennameanothusedsamesessuseformexamplwith|ibmintellinuxhyperlinkcouritimenormalhelppackagmultiplintegratcallingsequencmintfunctargsrgssynopsicontainfunctionprovidentitinvolvpureintegralwithhypergeometricintegrandavailablfindrecmintfindrecsumtointnminsymmsumtointnespssumcheckrecmsumgetspecifictypeeithabovlistthespartthcanusedformonlyafterperformcommandalwayaccesslongwheneverconflictbetweennameanothsamesesssefollowvariablglobalversmaplreleasA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 4 "" 0 "" {TEXT 23 11 " SYNOPSIS: " }}{PARA 0 "" 0 "" {TEXT 23 106 " - the funct ion esp(order,vars) outputs the elementary symmetric polynomial in the variables v of order n." }}{PARA 0 "" 0 "" {TEXT -1 0 ""