R-   repr textFurierGd rddsd " 0 "" {MPLTEXT 1 0 14 "with(Mu ltInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1 -1/y)^n/x/y,[x,y]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tI0 0 0 3 0 0 } {CSTYLE "" -1 268 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{PSTYLE "N ormal" -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 "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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 74 " sym: checks whether a given func tion is symmetric w.r.t. given variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 256 18 "Calling Sequence :" }{TEXT 258 1 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 265 21 "sym(f,[v1,v2,...,vn])" }{TEXT 266 1 " " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{PARA 4 "" 0 "" {TEXT 2" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 269 9 "Examples :" }{TEXT 268 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "esp(2,[x,y,z]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(*&%\"xG\"\"\"%\"yGF&F&*&F%\"\"\"%\"zGF&F&*&F'F)F*F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "11 4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } 1 " " }{TEXT 260 13 "Description: " }{TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT 23 1 "T" }{TEXT 261 12 "he function " }{TEXT 263 18 "esp(n,[v1,...,vr])" }{TEXT 264 62 " outputs the elementary sym metric polynomial in the variables " }{TEXT 265 11 "[v1,...,vr]" } {TEXT 266 11 " of order n" }{TEXT 262 1 "." }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "W" }{TEXT 267 142 "henever there is a conflict b etween the function name esp and another name used in the same se ssion, use the long form MultInt['esp']. T 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 257 18 "Call ing Sequence: " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 270 19 "esp (n,[v1,...,vr]) " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 258 11 "Parameters:" }{TEXT 259 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 271 35 "n - a positive integer" }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 272 22 "v1,v2,... ,vr - names" }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 3 " " 0 "" {TEXT 23 1 " " }{TEXT 260 13 "Description: " }{TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT 23 1 "T" }{TEXT 261 12 "he function " }{TEXT 263 18 "esp(n,[v1,...,vr])" }{TEXT 264 62 " outputs the elementary sym metric polynomial in the variables " }{TEXT 265 11 "[v1,...,vr]" } {TEXT 266 11 " of order n" }{TEXT 262 1 "." }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "W" }{TEXT 267 142 "henever there is a conflict b etween the function name esp and another name used in the same se ssion, use the long form MultInt['esp']. 0 3 0 0 } {CSTYLE "" -1 268 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " 23 269 "times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Co urier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 271 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 272 "Courier" 0 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 "H eading 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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 256 68 "esp: generates t he elementary symmetric polynomials of a given order" }}{PARA 0 "" 0 " " {TEX²Ú<ibmintellinuxmaplinputcourimathtimeoutputcouriernormaleadingheadbulletitemespgeneratheelementarsymmetricpolynomialgivenorderingsequencvrparameterpositintegnamedescriptfunctsymmetricvariablhenevconflictetweenanothusedsamesessionuselongformmultintexamplwithxgygfzgf²Ú MultInt,esp³Ú MultInt´ÚMultInt,hypertorecdiffµÚMultInt,sumtorecdiff¶ÚMultInt,sumtointn·Ú MultInt,sym¸ÚMultInt,checkrecdiffMultInt³ÚMultInt,checkrecdiff¸Ú MultInt,esp²ÚMultInt,hypertorecdiff´ÚMultInt,sumtointn¶ÚMultInt,sumtorecdiffµÚ MultInt,sym·ÚµÚî({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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "C ourier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 257 "times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 258 "times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 259 "times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 260 "times" 1 14 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 " times" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 264 "courier" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 265 "times" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" 23 266 "times" 0 12 0 0 0 0 0 0 3 representai¶Ú representat µÚ¶Úrequir´Úreturn·Úrg ´ÚµÚriabl´Úrier´Úrm¶Úro´Úrue·Úsa µÚ·Úsame²Ú³Ú´Ú¶Ú¸Úsatisfi´ÚµÚ¸Úse ²Ú³Úsearch´Úsecond ´ÚµÚsequenc²Ú³Ú´ÚµÚ¶Ú·Ú¸Úsess³Ú´ÚµÚ¶Ú·Ú¸ÚsfµÚsgµÚshift´Úsion³Úsolv ´ÚµÚspecific³Ússion ²ÚµÚssum¶ÚstatisfiµÚsucces ´ÚµÚsum µÚ¶Úsumg µÚ¶Úsummat µÚ¶ÚsumtµÚsumtoµÚ sumtointn ³ÚµÚ¶Ú sumtorecdiff ³ÚµÚ¶Úsupport´Úsym ²Ú³Ú·Ú symmetric ²Ú·Úterm µÚ¶Útext ´ÚµÚtext ´ÚµÚcaseder?esentat=follow>hyp0item,nocoeffPperformQrepr textFurierGd rddsd](args)" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 275 17 " (args)" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 259 4 " " }{TEXT 261 2 " " }}{SECT 0 {PARA 257 "" 0 "" {TEXT 260 13 "Description: " }}{PARA 15 "" 0 "" {TEXT -1 28 "The functions available are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 23 2 " " }{TEXT 263 10 " " }{TEXT 264 1 " " } {HYPERLNK 17 "hypertorecdiff" 2 "MultInt/hypertorecdiff" "" }{TEXT 266 12 " " }{TEXT -1 2 " " }{TEXT 282 12 " " } {HYPERLNK 17 "checkrecdiff" 2 "MultInt/checkrecdiff" "" }{TEXT 281 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{TEXT 269 1 " " } {HYPERLNK 17 "sumtointn" 2 "MultInt/sumtointn" "" }{TEXT -1 7 " \+ " }{TEXT 270 1 " " }{TEXT -1 25 " " }{TEXT 268 3 " " }{TEXT 283 1 " " }{HYPERLNK 17 "esp" 2 "MultInt/esp" "" }} {PARA 0 "" 0 "" {TEXT 271 5 " " }{HYPERLNK 17 "sumtorecdiff" 2 "Mu ltInt/sumtorecdiff" "" }{TEXT 267 29 " " } {HYPERLNK 17 "sym" 2 "MultInt/sym" "" }{TEXT 272 2 " " }}{PARA 0 "" 0 "" {TEXT 273 12 " " }{TEXT 265 2 " " }{TEXT 262 7 " \+ " }}{PARA 15 "" 0 "" {TEXT 276 150 "To get help for a specific funct ion type either ?MultInt[] or ?MultInt, where is one from the above list. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 231 "These functions are par t of the MultInt package and so can be used in the form (arg uments) only after performing the command with(MultInt) or with(MultIn t, ). The functions can also be accessed in the long form " }{TEXT 277 26 "MultInt[](args)." }}{PARA 15 "" 0 "" {TEXT -1 120 "Whenever there is a conflict between a function name in MultIn t and another name used in the same session, use the form " }{TEXT 278 20 "MultInt[]." }}{PARA 15 "" 0 "" {TEXT -1 17 "This ver sion of " }{TEXT 279 7 "MultInt" }{TEXT 280 26 " is for Maple V Relea se 5." }}}{PARA 258 "" 0 "" {TEXT 23 4 " " }}}{MARK "4 2" 2 } {VIEWOPTS 1 1 0 1 1 1803 } ARA 0 "" 0 "" {TEXT 273 12 " " }{TEXT 265 2 " " }{TEXT 262 7 " \+ " }}{PARA 15 "" 0 ""²Ú³Ú´Ú µÚ@¶ÚH·ÚK7 39 "f \+ - an algebraic expression " }}{PARA 0 "" 0 "" {TEXT 23 4 " \+ " }{TEXT 268 20 "v1,...,vn - names" }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 260 13 "Descri ption: " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 261 72 "sym returns t rue, if f is symmetric w.r.t v1,..,vn, false otherwise. " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 262 139 "Whenever there is a confli ct between the function name sym and another name used in the sa me session, use the form MultInt['sym']." }{TEXT 263 2 " " }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 264 9 "Example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Mu ltInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1 -1/y)^n/x/y,[x,y]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {abov³Úaccess ³Ú´Úafter ³Ú´Úal´Ú algebraicµÚ¶Ú·Ú¸Úallow´Úalso³Ú´ÚµÚ¶Ú¸Úam´Úame´Úames µÚ¶Úanoth²Ú³Ú´ÚµÚ¶Ú·Ú¸Úansatz´Úarg³Úargs ³Ú´Úasks´Úat´Úation´Úavailabl³Úbetween³Ú´ÚµÚ¶Ú·Ú¸Úbi µÚ¶Ú binocoeff µÚ¶Ú binocoeffsgµÚbinomial µÚ¶Ú binomialg µÚ¶Úbound µÚ¶Úbullet²Ú³Ú´ÚµÚ¶Ú·Ú¸Úca´Úcall³Ú´ÚµÚ¶Ú·Ú¸Úcan ³Ú´Úcase´ÚÚncould´Únd ´Ú¶Únegat ´ÚµÚnewµÚng ´ÚµÚ¶Úngf´ÚµÚ¶Únk´Únm´Únocoeff µÚ¶ÚµÚ¶Úcdiff ´Ú¸Ú certificat´Úch¸Úcheck·Ú checkrecdiff ³Ú¸Úcheer ´ÚµÚchiµÚco²ÚµÚ·Úcoeff¶Ú coefficientµÚcommand ³Ú´Úconfli·Úconflict²Ú³Ú´ÚµÚ¶Ú¸Úconstant µÚ¶Úcou´ÚcoulµÚcourij²Ú³Ú´Ú#µÚ¶Ú·Ú¸Úcpu ´ÚµÚctµÚ¶Ú·Úction³Údash´Údefault´ÚdefinitµÚdenom´Ú denominator´ÚderµÚntergalµÚ intergral¶Úion³Úitem²Ú³Ú´ÚµÚ¶Ú·Ú¸ÚµÚprop ³Ú´Úption·Úration´Úrational´Úre¶ÚrecµÚrecdiffµÚrecurr´Ú recurrenc´ÚµÚ¸Úrelea³Úrepr¶Ú´Úibmintellinuxmaplinputcourimathtimehyperlinkoutputimestimesouritimescouriernormaltextheadbulletitemdashhypertorecdifffindnonzerorecurrencdifferntialwzequatsatisfigivenprophyperexponentialfunctioncallsequenchypertorecdifffnamxroptnmparameterexpressnamevariablintegrationfunctamevarioudesiroptiondescriptfnamformdiffpolynomialindependfreecasesearchwithpartorderatmostdefaultvaluhypertorecdiffmultiplfollowsupportnegatintegallowrequirecurrencequlistvariablintegerdenompolyguesdenominatortuplrationalfunctioncertificathypertoreccanalsonkansatzshiftmonomialhypertorecdiffusinggivenuserrationalmultintpackagcausedargsonlyafterperformcommandaccesslongwhenevconflictbetweenanothsamesessusehypertprecexampltryingrecurrncedifferentialequatisolvequationunknowncheersuccescpusecondngfgxgygfdiffgrgaskszerothmgfkgdoubltryigeteqncouldunknownsunknownsfindhy1²Ú2³Ú´Ú.µÚ/¶ÚJnds an integral representation of a summation expression of the fo rm:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$SumG6$%:(product~of~binocoeffs)~xG%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "i.e, finds a constant term (CT ) expression: CT(F(z1,...,zr)) of the given sum and ouputs F with the \+ integration variables z1,...,zr." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 56 "Whenever there is a conflict between the function name " }{TEXT 273 10 "msumtointn" }{TEXT -1 63 " and anoth er name used in the same session, use the long form " }{TEXT 274 20 "M ultInt['sumtoinÐôB@ôB@CT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 23 1 " " }{TEXT 270 9 "Examples:" }{TEXT 271 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "To find an intergral repr esentation of:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"kG\"\"\")-%)binom´Ú©F{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 "2D Math" -1 2 "Times" 0 1 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0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 16 3 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "times" 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 "" 0 257 1 {CSTYLE "" -1 -1 "times" 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 "" 0 258 1 {CSTYLE "" -1 -1 "times" 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 "" 0 259 1 {CSTYLE "" -1 -1 "times" 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 "" 0 260 1 {CSTYLE "" -1 -1 "times" 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 "" 0 261 1 {CSTYLE "" -1 -1 "times" 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 " " 0 262 1 {CSTYLE "" -1 -1 "times" 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 "" 0 263 1 {CSTYLE "" -1 -1 "time s" 0 14 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 "" 0 264 1 {CSTYLE "" -1 -1 "times" 0 14 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 "" 0 265 1 {CSTYLE "" -1 -1 "times" 1 14 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 "" 0 266 1 {CSTYLE "" -1 -1 "times" 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 "" 0 267 1 {CSTYLE "" -1 -1 "times" 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 "" 0 268 1 {CSTYLE "" -1 -1 "times" 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 " " 0 269 1 {CSTYLE "" -1 -1 "times" 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 "" 0 270 1 {CSTYLE "" -1 -1 "time s" 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 "" 0 271 1 {CSTYLE "" -1 -1 "times" 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 "" 0 272 1 {CSTYLE "" -1 -1 "times" 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 "" 0 273 1 {CSTYLE "" -1 -1 "times" 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 "" 3 274 1 {CSTYLE "" -1 -1 "times" 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 "" 0 275 1 {CSTYLE "" -1 -1 "times" 0 14 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 " " 0 276 1 {CSTYLE "" -1 -1 "courier" 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 4 "" 0 "" {TEXT 269 3 "hyp" }{TEXT 270 0 "" }{TEXT 256 13 "ertorecdiff: " }{TEXT 271 110 "finds a non-zero recurrence-differe ntial(WZ) equation satisfied by a given proper-hyperexponential func tion." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 " " {TEXT 261 0 "" }{TEXT 257 18 "Calling Sequences:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 0 "" }{TEXT 260 1 " " } {TEXT -1 0 "" }{TEXT 258 56 "hypertorecdiff(f, n, fnam(x1,x2,...,xr), \+ opt1, opt2,...)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0" "" }{TEXT 314 67 " hypertorecdiff(f, [n1,...,nm], fnam(x1,x 2,...,xr), opt1, opt2,...)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 262 0 "" }{TEXT -1 0 "" }{TEXT 23 1 " " }{TEXT 263 68 "f \+ - a proper-hyperexponential expression " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 23 2 " " }{TEXT 264 57 " n,n1,...,nm - names, the recurrence variables " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 23 10 " " }}{PARA 259 "" 0 "" {TEXT -1 65 " x1,x2,...,xr - names, the integr ation variables" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 " " {TEXT -1 58 " fnam - a name, the function n ame" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT -1 0 " " }{TEXT 23 2 " " }{TEXT 265 26 "opt1, opt2 ,... - " }{TEXT -1 24 "various desired options " }}{PARA 262 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " #}{TEXT 259 13 "Description: " } }{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 289 41 "hypertorecdiff(f, n, fn am(x1,x2,...,xr)) " }{TEXT 290 66 "finds a non-zero recurrence-differe ntial(WZ) equation of the form " }}{PARA 0 "" 0 "" {TEXT 306 82 "p1*fn am(n,x1,..xr) + p2*fnam(n+1,x1,...,xr)+... = Diff(R1*fnam(n,x1,...,xr) )+..., " }}{PARA 0 "" 0 "" {TEXT 307 8 "where p" }{TEXT 291 1 "1" } {TEXT 295 6 "p2,..." }{TEXT -1 41 " are polynomials independent (free ) of " }{TEXT 292 12 "x1, ..., xr." }}{PARA 0 "" 0 "" {TEXT 308 1 " \+ " }{TEXT 310 13 "In this case " }{TEXT 309 15 "hypertorecdiff " } {TEXT 293 86 "searches for a WZ-equation with recurrence part of order at most 6(the default value)." }}{PARA 15 "" 0 "" {TEXT 322 2 "hy" } {TEXT 315 49 "pertorecdiff(f, [n1,...,nm], fnam(x1,x2,...,xr)) " } {TEXT 316 50 "finds a non-zero multiple recurrence WZ-equation. " }} {PARA 15 "" 0 "" {TEXT -1 38 "The following options are supported: " }{TEXT 276 1 " " }}{PARA 16 "" 0 "" {TEXT 275 41 "hypertorecdiff(f, n, fnam(x$1,x2,...,xr), " }{TEXT 282 16 "recurrence_order" }{TEXT 283 1 " )" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 274 17 " recurrence_order " }{TEXT -1 25 " - non-negative integer" }}{PARA 0 "" 0 "" {TEXT -1 94 " This option allows one to input the required order of the r ecurrence part of the WZ-equ" }{TEXT 267 6 "ation." }}{PARA 0 "" 0 "" {TEXT -1 9 " If " }{TEXT 317 1 "n" }{TEXT -1 32 " is a list of va riables , then " }{TEXT 318 17 "recurrence_order " }{TEXT -1 35 "is a list of non-negative integers." }}{PARA 16 "" 0 "" {TEXT -1 0 "" } {TEXT 268 0 "" }{TEXT 277 72 "hypertorecdiff(f, n, fnam(x1, x2, ..., x r), recurrence_order, denom_poly" }{TEXT 266 2 ") " }{TEXT 278 1 " " } }{PARA 263 "" 0 "" {TEXT -1 4 " " }{TEXT 279 11 "denom_poly " } {TEXT -1 24 "- list of polynomials." }}{PARA 264 "" 0 "" {TEXT -1 65 " This option allows one to guess and input the denominators \+ " }{TEXT 280 10 "denom_poly" }{TEXT -1 88 " of the r-tuple of ration al functions(the certificates) of the r%equired WZ-equation." }} {PARA 0 "" 0 "" {TEXT -1 7 " If " }{TEXT 319 1 "n" }{TEXT -1 32 " i s a list of variables , then " }{TEXT 320 17 "recurrence_order " } {TEXT -1 35 "is a list of non-negative integers." }}{PARA 16 "" 0 "" {TEXT -1 0 "" }{TEXT 272 0 "" }{TEXT -1 0 "" }{TEXT 273 50 "hypertorec diff(f, n, fnam(x1, ..., xr), denom_poly" }{TEXT 281 2 ") " }}{PARA 0 "" 0 "" {TEXT 321 50 " n can also be a list of recurrence variables . " }}{PARA 16 "" 0 "" {TEXT -1 0 "" }{TEXT 284 0 "" }{TEXT -1 0 "" } {TEXT 285 72 "hypertorecdiff(f, [n1,n2,...,nk], fnam(x1, ..., xr), [N1 ,...,Nk], ansatz" }{TEXT 286 3 ") " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " n1,....,nk - recurrence variables" }}{PARA 0 "" 0 "" {TEXT -1 33 " N1,...Nk - shift variables" }}{PARA 0 "" 0 "" {TEXT -1 48 " ansatz - list of monomials in N1, ...,Nk" }} {PARA 0 "" 0 "" {TEXT -1 19 " In this case " }{TEXT 313 14 "hyper torecdiff" }{TEXT -1 55 " searches for a WZ-equation by using the giv en ansatz." &}}{PARA 16 "" 0 "" {TEXT -1 0 "" }{TEXT 287 0 "" }{TEXT -1 0 "" }{TEXT 288 84 "hypertorecdiff(f, [n1,n2,...,nk], fnam(x1, ..., xr), [N1,...,Nk], ansatz, denom_poly" }{TEXT 294 3 " ) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 " This option allows the user to \+ guess and input the denominators of the r-tuple of rational functions (the certificates) of the required WZ-equation." }}{PARA 15 "" 0 "" {TEXT -1 79 "This function is a part of the MultInt package, and so ca n be used in the form " }{TEXT 296 14 "hypertorecdiff" }{TEXT -1 41 "( args) only after performing the command " }{TEXT 297 13 "with(MultInt) " }{TEXT 298 4 " or " }{TEXT 299 30 " With(MultInt, hypertorecdiff)" } {TEXT 300 53 ". The function can also be accessed in the long form " } {TEXT 301 30 "MultInt[hypertorecdiff](args)." }}{PARA 15 "" 0 "" {TEXT -1 56 "Whenever there is a conflict between the function name \+ " }{TEXT 311 14 "hypertorecdiff" }{TEXT -1 57 " and another name used \+ in the same session, use the form " }{TE'XT 312 25 "MultInt['hypertprec diff']" }}}{PARA 265 "" 0 "" {TEXT 23 1 " " }}{PARA 266 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 303 9 "Examples:" }{TEXT 302 1 " " }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 0 50 "hypertorecdiff((1+x)^n*(1+y)^n/(1-x*y),n,f(x,y)); " } }{PARA 6 "" 1 "" {TEXT -1 81 " \n... trying to find a non-zero recurre nce-differential(WZ) equation with order 0" }}{PARA 6 "" 1 "" {TEXT -1 77 "\n could not find a non-zero recurrence-differential(WZ) equati on with order 0" }}{PARA 6 "" 1 "" {TEXT -1 81 " \n... trying to find \+ a non-zero recurrence-differential(WZ) equation 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 : .39 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/,&*&,&%\"nG\"\"%\"\"#\"\"\"F+-%\"fG6%F(%\"xG%\"yGF+F +*&,&F((!\"\"F3F+F+-F-6%,&F(F+F+F+F/F0F+F+,&-%%DiffG6$*&-%#_RG6$F/F0F+F ,\"\"\"F/F+-F96$*&-F=6$F0F/F+F,F?F0F+/F<,$*&,&F+F+F/F+F+,(*&F/F+F0F+F+ F/F*!\"$F+F+#F3F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 76 "The following call asks MultInt to find a non-ze ro WZ-equation of order 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "hypertorecdiff((1+x)^n*(1+y)^m/(1-x *y)^k,m,f(x,y),1);" }}{PARA 6 "" 1 "" {TEXT -1 41 " ... solving 19 equ ations for 13 unknowns" }}{PARA 6 "" 1 "" {TEXT -1 50 " Cheers! for th e success. CPU Time : .19 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"nG\"\"\"%\"mGF(\"\"# F(%\"kG!\"\"F(-%\"fG6%F)%\"xG%\"yGF(F(*&,(F+F(F)F,!\"#F(F(-F.6%,&F)F(F (F(F0F1F(F(,&-%%DiffG6$*&,&F(F(F0F(F(F-\"\"\"F0F(-F:6$,$*(F1F(,&F(F(F1 F(F(F-F>F,F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The following call asks MultInt to find a non-zero double recurrence WZ-equation." )}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "hypertorecdiff((1+x)^n*(1+y) ^m/(1-x*y)^k,[n,m],f(x,y));" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... tryi ng to get a recurrence eq with order [0, 0] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 91 "\ncould not find a non-zero recurrence-differential(WZ ) equation with order [0, 0] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... trying to get a recurrence eq with order [1, 0] in [n, m]" }} {PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 19 equations for 13 unknow ns" }}{PARA 6 "" 1 "" {TEXT -1 49 " Cheers! for the success. CPU Time : .33 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"nG\"\"\"% \"mGF(\"\"#F(%\"kG!\"\"F(-%\"fG6&F'F)%\"xG%\"yGF(F(*&,(F'F,!\"#F(F+F(F (-F.6&,&F'F(F(F(F)F0F1F(F(,&-%%DiffG6$,$*(F0F(,&F(F(F0F(F(F-\"\"\"F,F0 F(-F:6$*&,&F(F(F1F(F(F-F?F1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "hypertorecdiff((1+x)^n*(1+y)^m/(1-x*y)^k,[n,m],f(x,y),[0,1]);" } }{PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 19 equations for 13* unkno wns" }}{PARA 6 "" 1 "" {TEXT -1 49 " Cheers! for the success. CPU Tim e: .24 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"nG\"\"\"% \"mGF(\"\"#F(%\"kG!\"\"F(-%\"fG6&F'F)%\"xG%\"yGF(F(*&,(F+F(F)F,!\"#F(F (-F.6&F',&F)F(F(F(F0F1F(F(,&-%%DiffG6$*&,&F(F(F0F(F(F-\"\"\"F0F(-F:6$, $*(F1F(,&F(F(F1F(F(F-F>F,F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "The following call asks MultInt to fi nd a non-zero WZ-equation of with the given ansatz." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "hypertore cdiff(1/(1-x-x^2-y-y^2)/x^(m+1)/y^(n+1),[n,m],f(x,y),[N,M],[N,M,N*M,M^ 2]); " }}{PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 29 equations for \+ 22 unknowns" }}{PARA 6 "" 1 "" {TEXT -1 50 " Cheers! for the success. \+ CPU Time : .44 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*&,(\"\")\"\"\"%\"mG\"\"%%\"nGF*F(- %\"fG6&F+F)%\"xG%\"yGF(F(*&,&F+\"\"#F3F(F(-F-6&,&F+F(F(F(F)F/F0F(F(*&F 6F(-F-6&F6,&F)F(F(F(+F/F0F(F(*&,(F'F(F+F3F)F*F(-F-6&F+F:F/F0F(F(*&,&!#5 F(F)!\"&F(-F-6&F+,&F3F(F)F(F/F0F(F(,&-%%DiffG6$,$*&*&,(*$)F/F3\"\"\"F* F/F*FBF(F(F,FPFPF/!\"\"!\"\"F/F(-FH6$,$*&*(,&F(F(F0F3F(,&F(F(F/F3F(F,F PFPF/FQFRF0F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 270 "" 0 "" {TEXT 23 4 " " }}{PARA 271 "" 0 "" {TEXT 23 1 " \+ " }}{PARA 272 "" 0 "" {TEXT 23 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 305 10 "See Also:" }{TEXT 304 1 " " }}{PARA 273 "" 0 "" {TEXT -1 0 " " }{HYPERLNK 17 "MultInt" 2 "MultInt" "" }}}{PARA 274 "" 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 "16 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } " Cheers! for the success. \+ CPU Time : .44 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*&,(\"\")\"\"\"%\"mG\"\"%%\"nGF*F(- %\"fG6&F+F)%\"xG%\"yGF(F(*&,&F+\"\"#F3F(F(-F-6&,&F+F(F(F(F)F/F0F(F(*&F 6F(-F-6&F6,&F)F(F(F(+jg µÚ¶Újgf µÚ¶Úkg´ÚµÚ¶Úkm 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0 -1 0 }{PSTYLE ""5 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 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 263 14 "sumtorecdiff -" }{TEXT 264 1 " " }{TEXT 265 104 "finds a non-zero recurrence-different ial (WZ) equation for the Constant Term expression of a given sum." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 257 17 "Calling Sequences" }{TEXT 258 2 ": " }} {PARA 0 "" 0 "" {TEXT 23 4 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{TEXT 289 41 "sumtorecdiff(binocoeffs, x, k, n, [l,u]) "6 }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 291 69 " sumtorecdiff(binocoeffs, x, [k1 ,...,km], n, [[l1,u1],...,[lm,um]]) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 292 54 " sumtorecdiff(binocoeffs, x, k, n, [l,u], rec_order) \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 3 " " }{TEXT 290 78 "sum torecdiff(binocoeffs, x, [k1,...,km], n, [[l1,u1],...,[lm,um]], rec_or der) " }}{PARA 0 "" 0 "" {TEXT 279 2 " " }{TEXT -1 0 "" }{TEXT 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 259 11 "Parameters:" } {TEXT 260 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 280 54 "bi nocoeffs - finite product of binomial coefficients" }}{PARA 0 "" 0 " " {TEXT 23 4 " " }{TEXT 281 36 "x - algebraic expression " }}{PARA 0 "" 0 "" {TEXT 23 6 " k," }{TEXT 282 43 "k1,...,km - n ames, the summation variables" }}{PARA 0 "" 0 "" {TEXT 23 4 " " } {TEXT 283 44 "n - name, the recurrence variable" }}{PARA 0 "" 0 "" {TEXT 23 9 " l,u,l" }{TEXT 284 73 "1,u1,... - expressions, \+ lower and upper b7ounds of the summation variables" }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 285 64 "rec_order - non-negative integer, \+ the order of the recurrence" }}{PARA 0 "" 0 "" {TEXT 23 3 " " }} {SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 267 0 "" }{TEXT 268 0 " " }{TEXT 269 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 42 "Given a summation expression of the form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"SG6#%\"nG-%$SumG6$*&-%(ProductG6 #%+binocoeffsG\"\"\"%\"xGF0%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 239 "sumt orecdiff first finds an intergal representation (constant term expre ssion) of the given sum : S(n) = CT(f(n,z1,...,zr)) and then finds a \+ non-zero WZ-equation statisfied by f(n,z1,..,zr) and outputs the WZ-eq uation and f(n,z1,..,zn)." }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 276 63 "Whenever there is a conflict between the function name \+ " }{TEXT 286 4 "msum" }{TEXT 287 63 " and another name used in the sa me session, use the long form " }{TEXT 288 238 "MultInt['sumtorecdiff' ]" }{TEXT 277 2 ". " }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }{TEXT -1 0 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 271 0 "" }{TEXT 272 9 "Examples:" }{TEXT 270 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "To find a non-zero WZ-equation satisfied \+ by the CT expresion of Dixon's sum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*&)-%)binomialG6$,$%\"nG\"\"# %\"kG\"\"$\"\"\")!\"\"F.\"\"\"F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sumto recdiff(binomial(2*n,k)^3,(-1)^k,k,n,[0,infinity],1);" }}{PARA 6 "" 1 "" {TEXT -1 41 " ... solving 77 equations for 73 unknowns" }}{PARA 7 " " 1 "" {TEXT -1 31 "Warning, new definition for Chi" }}{PARA 6 "" 1 " " {TEXT -1 51 " Cheers! for the success. CPU Time : 3.97 seconds." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #7%/,&*(9,&%\"nG\"\"$\"\"\"F*F*,&\"\"#F*F(F)F*-%#_FG6%F(%$z_1G%$z_2GF*F )*&),&F(F*F*F*F,\"\"\"-F.6%F4F0F1F*F*,&-%%DiffG6$*&-%#_RG6$F0F1F*F-F5F 0F*-F:6$*&-F>6$F1F0F*F-F5F1F*/F=,$*&*&,&F0F*F*F*F*,`tF*F*F(F)*&)F0F)F5 )F1F)F5!\"%*()F0F,F5FMF5F(F*\"\"%*(FPF5FMF5)F(F,F5FQ*(FPF5)F1F,F5F(F5 \"#UF0\"\"(*(FLF5F1F*F(F5\"\"**(FLF5F1F5FSF5FN*(FLF5F1F5)F(F)F5!#7*(FP F5F1F5FSF5FQ*(FPF5F(F5F1F5!\"**(FPF5F1F5FfnF5\"#7*&F0F*FUF5!#5*(F0F5F( F5FUF5!#M*&FPF5F1F5FN*&FLF5F1F5FQ*&F0F5F1F5FQ*&FPF5FUF5\"#5*&FLF5)F1FQ F5!\"\"*$FSF5F,*&F0F5FgoF5\"\")*&F0F5F(F5\"#@*(F0F5FUF5FfnF5F\\o*(F0F5 FgoF5FfnF5!#K*(F0F5FgoF5FSF5Feo*(F0F5FgoF5)F(FQF5!#C*(F0F5FgoF5F(F5\"# E*(FfnF5F0F5F1F5Fgn*(FSF5F0F5F1F5FN*(FLF5FgoF5FSF5!\"#*(F0F5FUF5FSF5!# =*&FMF5FfnF5F\\o*$FUF5F,*$FMF5FN*&FPF5FgoF5F**(FPF5FgoF5FSF5\"#C*(FPF5 FgoF5FfnF5F\\o*(FPF5FgoF5F(F5\"#8*(FLF5FMF5F(F5F\\q*$FPF5!\")*&FLF5FUF 5Fjp*&F(F5F1F5!\"$*&F0F5FSF5F,*&FUF5FSF5\"#A*(FLF5FgoF5F(F5F[r*(FfnF5F PF5FUF5Fgn*(FcpF5FPF5FUF5Fdp*(F0F5FMF5FSF5!#I*(F0F5FMF5FfnF5F\\o*(F0F5 FMF5FcpF5Fbq*(F0F5FMF5F(F5!#D*(FLF5:FUF5FSF5!#A*(FLF5FUF5FfnF5F[p*(FLF5 FUF5FcpF5Fbq*(FLF5FUF5F(F5!#9*(FLF5FMF5FSF5!#E*(FLF5FMF5FfnF5Fgn*&F0F5 FcpF5Fdp*&FfnF5F1F5Fbq*(FSF5FPF5FUF5\"#W*&F0F5FMF5FN*&F0F5FfnF5!#O*(F( F5F0F5F1F5FY*&FPF5FSF5F^o*&FcpF5F1F5Fbq*&FUF5F(F5\"#9*&FUF5FfnF5Fhq*&F UF5FcpF5Fdp*&FMF5FSF5FQ*&FMF5F(F5Fjn*&FPF5FfnF5\"#K*&FPF5FcpF5Fbq*&FPF 5F(F5F_sF*F5*()F1\"\"#F5,&F(F,F*F*\"\"\"F0\"\"\"!\"\"#FhoFQ/F-,$*&*()* &FIF5F0F[u,$F(F,F*)*&,&F1F*F*F*F5F1F[uFcuF*),&F*F*FcoFhoFcuF*F5*(F0\" \"\"F1\"\"\")%#PiG\"\"#F5F[uF\\u" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "To find a non-zero WZ-equation for the CT form of " } {TEXT 278 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**-%)binomialG6$,&%\"iG\"\"\"%\"jGF,F+F,-F(6$% \"nGF+F,-F(6$F0F-F,)!\"\"F*F,%$i,jG" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(MultInt): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "sumtorecdiff(binomial(i+j,i)*binomial(n,i)*binomial( n,j),(-1)^(i+j),[i,j];,n,[[0,infinity],[0,infinity]]); " }}{PARA 6 "" 1 "" {TEXT -1 81 " \n... trying to find a non-zero recurrence-differen tial(WZ) equation with order 0" }}{PARA 6 "" 1 "" {TEXT -1 77 "\n coul d not find a non-zero recurrence-differential(WZ) equation with order \+ 0" }}{PARA 6 "" 1 "" {TEXT -1 81 " \n... trying to find a non-zero rec urrence-differential(WZ) equation 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 : .15 seconds." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$/,&-%#_FG6$%\"nG%$z_1G\"\"\"-F'6$,&F)F+F+F+F*!\"\"-%%DiffG6$\"\"!F* /F&,$*&*(%\"IGF+),$F*F/F)F+),$*&\"\"\"F>F*!\"\"F/F)F+F>*&F*\"\"\"%#PiG \"\"\"F?#F/\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 274 0 "" }{TEXT 275 9 "See Also:" }{TEXT 273 1 " " }}{PARA 0 "" <0 "" {TEXT -1 0 "" }{HYPERLNK 17 "MultInt" 2 "MultInt" "" }{TEXT 262 1 "," } {TEXT 261 1 " " }{HYPERLNK 17 "sumtointn" 2 "MultInt/sumtointn" "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}}{MARK "26" 0 }{VIEWOPTS 1 1 0 1 1 1803 } XT -1 81 " \n... trying to find a non-zero rec urrence-differential(WZ) equation 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 : .15 seconds." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7$/,&-%#_FG6$%\"nG%$z_1G\"\"\"-F'6$,&F)F+F+F+F*!\"\"-%%DiffG6$\"\"!F* /F&,$*&*(%\"IGF+),$F*F/F)F+),$*&\"\"\"F>F*!\"\"F/F)F+F>*&F*\"\"\"%#PiG \"\"\"F?#F/\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 274 0 "" }{TEXT 275 9 "See Also:" }{TEXT 273 1 " " }}{PARA 0 "" <esp ²Ú³Úetween²Úexampl²Ú´ÚµÚ¶Ú·Ú¸ÚexprµÚexpresµÚexpress ´ÚµÚ¶Ú·Ú¸Úfals·Úfbf´ÚfbqµÚ fcofhofcufµÚfcpfµÚfdpµÚfeoµÚffnf µÚ¸Úfg´ÚµÚ¸ÚfgnµÚfgof µÚ fh´Úfhf¸ÚfhofqµÚfhqµÚfi´Úfif µÚ¸Úfind´Ú µÚ¶Úfinit µÚ¶ÚfirstµÚfj¸ÚfjnµÚfjpµÚfkfp¸Úflf µÚ¸Úfm¸ÚfmfµÚfn ´ÚµÚfnam ´Ú fo¶Úfollow´Ú´Úncould´Únd ´Ú¶Únegat ´ÚµÚnewµÚng ´ÚµÚ¶Úngf´ÚµÚ¶Únk´Únm´Únocoeff µÚ¶Úform²Ú³Ú´ÚµÚ¶Ú·Ú¸ÚfpµÚfpfµÚfpfpf´ÚfqµÚfqfµÚfqfrf´Úfree´Úfsf µÚ¸Úfu¸ÚfufµÚfun³Úfunc ´Ú·Úfunct²Ú³Ú ´ÚµÚ¶Ú·Ú¸Úfunction ³Ú´Úfwf¸ÚfyµÚge³Úgenerat²Úget ³Ú´Úgf 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"Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 276 "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 "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 "H eading 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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0B 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT 259 68 " sumtointn: finds an integral re presentation of a given binomial sum" }}{PARA 0 "" 0 "" {TEXT 23 2 " \+ " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 257 19 "Calling Sequences: \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{TEXT 275 34 "sumtointn(bi nocoeffs, x, k, [l,u])" }{TEXT 276 1 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 262 63 "sumtointn(binocoeffs, x, [k1,k2,...,km], [[l1,u1 ],...,[lm,um]])" }{TEXT 263 1 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }} {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 260 11 "Parameters:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 256 4 " " }{TEXT 264 49 "binocoeffs - finite product of binomial coeffs " }}{PARA 0 "" 0 "" {TEXT 23 4 " \+ " }{TEXT 265 36 "x - algebraic expression " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 7 " k, " }{TEXT 266 43 "k1,...,km - n amCes, the summation variables" }}{PARA 0 "" 0 "" {TEXT -1 8 " \+ " }{TEXT 267 29 "l,u,l1,u1,... - expressions, " }{TEXT 261 57 "lower a nd upper bounds of each of the summation variables" }}{PARA 0 "" 0 "" {TEXT 23 5 " " }}{PARA 0 "" 0 "" {TEXT 23 11 " " }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 268 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 269 9 "sumtointn" }{TEXT -1 73 " finds an integral representation of a summation expression of the fo rm:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$SumG6$%:(product~of~binocoeffs)~xG%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "i.e, finds a constant term (CT ) expression: CT(F(z1,...,zr)) of the given sum and ouputs F with the \+ integration variables z1,...,zr." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 56 "Whenever there is a conflict between the function name " }{TEXT 273 10 "msumtointn" }{TEXT -1 63 " and anoth er name used in the same session, Duse the long form " }{TEXT 274 20 "M ultInt['sumtointn']" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 23 1 " " }{TEXT 270 9 "Examples:" }{TEXT 271 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "To find an intergral repr esentation of:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"kG\"\"\")-%)binomialG6$%\"nGF)\"\"$ \"\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(MultInt): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sumtointn(binomial(n,k)^3,(-1)^k,k,[0,infinity]); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*()*&,&%$z_1G\"\"\"F(F(\"\"\"F'!\" \"%\"nGF()*&,&%$z_2GF(F(F(F)F/F*F+F(),&F(F(*&F'F(F/F(!\"\"F+F(7$F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "To find a CT representai on of :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**)!\"\",&%\"iG\"\"\"%\"jGF+F+-%)binomialG6$F)F*F+-F.6$ %\"nGF*FE+-F.6$F2F,F+%$i,jG" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "sumtointn(binomial(i+j,i)*binomial (n,i)*binomial(n,j),(-1)^(i+j),[i,j],[[0,infinity],[0,infinity]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$*&),$%$z_1G!\"\"%\"nG\"\"\"),$*&\"\"\" F-F&!\"\"F'F(F)7#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " } {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 272 10 " See Also: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "MultInt" 2 "MultInt" "" }{TEXT 258 2 ", " }{HYPERLNK 17 "sumtorecdiff" 2 "MultInt /ssum" "" }}}}{MARK "17" 0 }{VIEWOPTS 1 1 0 1 1 1803 } 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "To find a CT representai on of :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**)!\"\",&%\"iG\"\"\"%\"jGF+F+-%)binomialG6$F)F*F+-F.6$ %\"nGF*FEtfµÚth ³Ú´Úthes³Úti ³Ú´ÚtialµÚtim´Útimeh²Ú ³Ú´Ú'µÚ¶Ú·Ú¸Ú tion ´Ú·Ú torecdiff ´ÚµÚtrueg ·Ú¸Útryi´Útrying ´ÚµÚtupl´Útype³ÚuationµÚufµÚufcufµÚultint¶Úum µÚ¶Úument³Úunkno´Úunknow´Úunknown ´ÚµÚupper µÚ¶Úurier²ÚµÚ·ÚÚfj¸ÚfjnµÚfjpµÚfkfp¸Úflf µÚ¸Úfm¸ÚfmfµÚfn ´ÚµÚfnam ´Ú fo¶Úfollow´ÚurrencµÚuse²Ú³Ú´ÚµÚ¶Ú·Ú¸Úused ²Ú³Ú´ÚµÚ¶Ú·Ú¸Úuseful¸Úuser´Úusing´Úva´Úvalu´Úvariabl²Ú´ÚµÚ¶Ú·Úvariou´Úver ³Ú¸Úvn·Úvr²ÚwarnµÚwhenev³Ú´ÚµÚ¶Ú·Ú¸Úwheth ·Ú¸Úwith²Ú³Ú´Ú µÚ¶Ú·Ú¸Úwns´Úwz³Ú´ÚµÚ¸Úxg²Ú´Ú¶Ú¸ÚxgfµÚxr ´Ú ygf²Ú´Ú¸Úze´Úzero ´Ú µÚzgf²ÚznµÚzr µÚ¶ÚÚ¶Ú·Ú¸Úcan ³Ú´Úcase´ÚÚncould´Únd ´Ú¶Únegat ´ÚµÚnewµÚng ´ÚµÚ¶Úngf´ÚµÚ¶Únk´Únm´Únocoeff µÚ¶Ú1 -1àôB@ôB@ ·ÚÝ {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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 257 "time s" 1 14 0 0 0 0 0 1 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 "times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 261 "times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 262 "times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "Co urier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 264 "times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 265 "Courier" 0 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" 0 10 0 0 0 0 0 0 0 rueG" }}} {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 "12 4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 } {TEXT 23 3 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 260 13 "Descri ption: " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 261 72 "sym returns t rue, if f is symmetric w.r.t v1,..,vn, false otherwise. " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 262 139 "Whenever there is a confli ct between the function name sym and another name used in the sa me session, use the form MultInt['sym']." }{TEXT 263 2 " " }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 264 9 "Example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Mu ltInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1 -1/y)^n/x/y,[x,y]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tI·Ú<ibmintellinuxmaplinputcourimathtimeoutputcourierormalheadbulletitemsymcheckwhethgivenfunctionsymmetricvariablcallsequencvnparameteralgebraicexpressnamedescriptionreturnruefalsotherwiswhenevconflictbetweenfunctanothusedsamesessuseformmultintexamplwithmultinttrueg¸ÚÓibmintellinuxmaplinputcourimathtimehyperlinkoutputimesnormalheadbulletitemcheckrecdiffcheckswhethgivenwzequatsatisfifunctcallsequenceqparameterrecurrencdifferentialalgebraicexpressdescriptusefulparticularverifyprocedurhypertorecdiffmultintwhenevconflictbetweenfunctionnameanothusedsamesessuseformmultintexamplwithrecdiffrecgngfgxgygfdiffgfhffiffjfmflffkfpfwffsfpfffnftruegalsohypertorcdiff¸Ú{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 256 "t imes" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 }{CSTYLE "" -1 257 "times" 1 14 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 "times" 1 14 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 "times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 262 "times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 263 "time s" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 264 "times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 265 "times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 266 "times" 1 14 0 0 0 0 0 0 0 0 L0 0 3 0 0 } {CSTYLE "" -1 267 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE " " -1 268 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 269 " Courier" 0 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 "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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 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 4 "" 0 "" {TEXT 23 1 " " }{TEXT 259 80 "Mcheckrecdiff: ch ecks whether a given WZ-equation satisfied by a given function." }} {PARA 256 "" 0 "" {TEXT 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " } {TEXT 256 18 "Calling Sequence: " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 267 19 "checkrecdiff(eq,f) " }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 257 11 "Parameters:" }{TEXT 260 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 268 51 "eq \+ - a recurrence-differential(WZ) equation " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 269 31 "f - algebraic expression" }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " } {TEXT 262 11 "Description" }{TEXT 261 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 263 79 "checkrecdiff is useful, in particular, to ver ify the output of the procedure " }{HYPERLNK 17 "hypertorecdiff" 2 "Mu ltInt/hypertorecdiff" "" }{TEXT 264 1 "." }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 265 157 "Whenever there is a conflict between the fu nction name cheNckrecdiff and another name used in the same session, \+ use the form MultInt['checkrecdiff']. " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Example:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(MultInt): " }}}{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) = Diff(-(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 ),x)+Diff(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),y);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$recG/,&*() ,&%\"nG\"\"\"F+F+\"\"#\"\"\"),&F*F+!\"\"F+F,F--%\"fG6%F*%\"xG%\"yGF+F+ **,&F,F+F*\"\"$F+)F*F,F-,&F+F+F*F8F+-F26%F)F4F5F+!\"$,&-%%DiffG6$,$*&* (,&F+F+F4F+F+,6*(F4F+)F5F,F-)F*F8F-F+*(F4F-FHF-F9F-F+*(F4F-FHF-F*F+F0* &F4F-FHF-F0*&F*F-FHF-!\"'*&FIF-FHF-\"#5*$FIF-!#sO*$F9F-!#g*&F9F-FHF-!\" %F*!#7F+F1F-F-*&F4\"\"\")F5\"\"#F-!\"\"F0F4F+-F@6$*&**F*F-,&F+F+F5F+F+ ,@FJ\"#7FM\"#A*$FHF-\"\"'FLF,FU\"#?FKFP*&F*F-F5F+!#R*&F9F-F5F-!#L*(F9F -F4F-F5F-!\"**(F*F-F4F-F5F-F=F5FWF*F8*&F*F-F4F-F8*&F4F-F9F-\"\"*FSF[pF +F1F-F-*&)F4\"\"#F-)F5\"\"#F-FfnF5F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "checkrecdiff(rec, (1+1/x)^n*(1+1/y)^(2*n)/x^n/y^n); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " } {TEXT 266 10 "See Also: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "MultInt" 2 "MultInt" "" }{TEXT 258 2 ", " }{HYPERLNK 17 "hypertore cdiff" 2 "MultInt/hypertorecdiff" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}}{MARK "11 6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } fG6%F*%\"xG%\"yGF+F+ **,&F,F+F*\"\"$F+)F*F,F-,&F+F+F*F8F+-F26%F)F4F5F+!\"$,&-%%DiffG6$,$*&* (,&F+F+F4F+F+,6*(F4F+)F5F,F-)F*F8F-F+*(F4F-FHF-F9F-F+*(F4F-FHF-F*F+F0* &F4F-FHF-F0*&F*F-FHF-!\"'*&FIF-FHF-\"#5*$FIF-!#sOnon ´Ú µÚnormal²Ú³Ú´ÚµÚ¶Ú¸Úns´Úntial´Úonly ³Ú´Úopt´Úoption´Úorder²Ú´ÚµÚorecdiffµÚormal·Úotherwis·Úouput¶Úouri´ÚµÚ¶Úoutput²Ú´ÚµÚ¶Ú·Ú¸Úpacka³Úpackag ³Ú´Úpar³Ú parameter²Ú´ÚµÚ¶Ú·Ú¸Úpart´Ú particular¸Úperform ³Ú´Ú¶Ú ertorecdiff´Úes´Úesentat¶ÚµÚfmfµÚfn ´ÚµÚfnam ´Ú fo¶Úfollow´Ú³Úname²Ú³Ú´ÚµÚ¶Ú·Únce´Úncould´Únd ´Ú¶Únegat ´ÚµÚnewµÚng ´ÚµÚ¶Úngf´ÚµÚ¶Únk´Únm´Únocoeff µÚ¶Ú pertorecdiff´Úpf µÚ¸Úpfpf´ÚpigµÚpoly´Ú polynomial ²Ú´Úposit²Ú presentat¶Úprocedur¸Úproduct µÚ¶ÚproductgµÚprop ³Ú´Úption·Úration´Úrational´Úre¶Úrec µÚ¸ÚrecdiffµÚrecg¸Úrecurr´Ú recurrenc´ÚµÚ¸Úrelea³Úrepr¶Úgues´Úhe²Úhead ²Ú³Ú´ÚµÚ¶Ú·Ú¸Úhelp³Úhenev²Úhy´Úhyp´Ú