SQ  th" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 " Times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 264 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "Courier" 0 10version2n: " }}{PARA 15 "" 0 "" {TEXT -1 28 "The funct ions available are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 2 " " }{TEXT 262 10 " " }{TEXT 263 1 " " }{HYPERLNK 17 "hypertorecd iff" 2 "MultInt/hypertorecdiff" "" }{TEXT 265 12 " " } {TEXT -1 2 " " }{TEXT 280 12 " " }{HYPERLNK 17 "checkrecdi ff" 2 "MultInt/checkrecdiff" "" }{TEXT 279 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 " " }{TEXT 268 1 " " }{HYPERLNK 17 "sumtointn" 2 "MultInt/sumtointn" "" }{TEXT -1 7 " " }{TEXT 269 1 " " } {TEXT -1 26 " " }{TEXT 267 1 " " }{TEXT 281 1 " " }{HYPERLNK 17 "esp" 2 "MultInt/esp" "" }}{PARA 0 "" 0 "" {TEXT 270 5 " " }{HYPERLNK 17 "sumtorecdiff" 2 "MultInt/sumtorecdiff" " " }{TEXT 266 27 " " }{HYPERLNK 17 "sym" 2 "M ultInt/sym" "" }{TEXT 271 10 " " }{TEXT 264 2 " " }{TEXT 261 7 " " }}{PARA 15 "" 0 "" {TEXT 274 143 "To get help for a sp ecific function type either ?MultInt[] or ?MultInt,](arguments)" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 273 22 " (arguments)" }{TEXT -1 0 " " }{TEXT 258 4 " " }{TEXT 260 2 " " }}{SECT 0 {PARA 257 "" 0 "" {TEXT 259 13 "Descriptio 0 0 3 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 271 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 272 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Times" 0 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 275 "" 0 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 276 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 277 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 278 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Courier" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 280 " Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 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 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 €`%{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 256 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 257 "Times" 0 10 0 0 0 0 0 2 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 " Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Times" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Times" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 263 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Times " 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 268 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "Times" 1 14 0 0 0 0 0 0 0 0€` MultInt` MultInt,esp‚`MultInt,checkrecdiffƒ`MultInt,hypertorecdiff„`MultInt,sumtointn…`MultInt,sumtorecdiff†` MultInt,symlower „`…`map ƒ`†`mapl €``‚`ƒ`„`…`†`math`‚`ƒ`„`…`†`me…`mes…`method€`mgƒ`mgfƒ`mostƒ`msum…` msumtointn„`multi €`‚`multint,€``‚`ƒ` „`…`†`multiplƒ`mutlipl€`na…`name€``‚`ƒ`„`…`†`nceƒ`ncouldƒ`negat ƒ`…`new…`ng ‚`ƒ`„`…`ngfƒ`„`…`nkƒ`nmƒ`nocoeff„`non ƒ` …`normal€``‚`ƒ`„`…`†`nsƒ`nt €``‚`ƒ`„`…`†`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`MultInt€`MultInt,checkrecdiff‚` MultInt,esp`MultInt,hypertorecdiffƒ`MultInt,sumtointn„`MultInt,sumtorecdiff…` MultInt,sym†`escriptreturntruefalsotherwiswhenevconflictbetweenanothusedsamesessuseformultintexamplwithmultinttruegcheckrecdiff,MultInt‚` esp,MultInt`hypertorecdiff,MultIntƒ`sumtointn,MultInt„`sumtorecdiff,MultInt…` sym,MultInt†`cexprsionnamedescriptreturntruefalsotherwiswhenevconflictbetweenanothusedsamesessuseformultintexamplwithmultinttruegƒ`ϋ9{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Courie r" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Courier" 0 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 0 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Times" 0 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 262 "Courier" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Courier" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "Courier" 0 1 MultInt€`MultInt,checkrecdiff‚` MultInt,esp`MultInt,hypertorecdiffƒ`MultInt,sumtointn„`MultInt,sumtorecdiff…` MultInt,sym†`ypertorecdiffhypertorecdiffcheckrecdiffcheckrecdiffsumtointnespsumtorecdiffsymultintgethelpspecifictypeeithabovlistthespartmultipackagcanusedformonlyfterperformcommandwithalsoaccesslongwhenevconflictbetweennameanothsamesessusethversmapl`"ibmintelntmaplinputcourimathtimeoutputnormalheadbulletitemespgeneratelementarsymmetricpolynomialgivenordercallsequencvrparameterpositintegnamedescripthefunctvariablehenevconflictbetweenameanothusedsamesessuselongformmultintexamplwithxgygfzgfion, use the long form MultInt['esp']. " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " } }{SECT 0 {PARA 4 "" 0 "" {TEXT 268 9 "Examples:" }{TEXT 267 1 " " } {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%F&%\"zGF& F&*&F'F&F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } RA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 259 13 "Description: " }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "T " }{TEXT 260 12 "he function " }{TEXT 262 18 "esp(n,[v1,...,vr])" } {TEXT 263 62 " outputs the elementary symmetric polynomial in the vari ables " }{TEXT 264 11 "[v1,...,vr]" }{TEXT 265 11 " of order n" } {TEXT 261 1 "." }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "W" } {TEXT 266 142 "henever there is a conflict between the function n ame esp and another name used in the same sess he elementary symmetric polynomials of a given order" }{TEXT -1 0 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 256 18 "Calling Sequence: " }{TEXT 274 0 "" }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 269 19 "esp (n,[v1,...,vr]) " }{TEXT 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " } {TEXT 257 11 "Parameters:" }{TEXT 258 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 270 35 "n - a positive integer" }} {PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 271 22 "v1,v2,...,vr - name s" }{TEXT 23 3 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 259 13 "Description: " }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "T " }{TEXT 260 12 "he function " }{TEXT 262 18 "esp(n,[v1,...,vr])" } {TEXT 263 62 " outputs the elementary symmetric polynomial in the vari ables " }{TEXT 264 11 "[v1,...,vr]" }{TEXT 265 11 " of order n" } {TEXT 261 1 "." }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "W" } {TEXT 266 142 "henever there is a conflict between the function n ame esp and another name used in the same sess rier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" 23 268 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 270 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 271 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 272 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 274 "Times" 0 10 0 0 0 0 0 2 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 275 " Times" 0 10 0 0 0 0 0 2 0 0 0 0 3 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT 272 7 " esp: " }{TEXT 275 63 "generates t `τ {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 " Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 263 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 264 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Times " 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "Cou€``‚` ƒ`„`:…`I†`Rourier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 " Times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 264 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "Courier" 0 10‚`%{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 " Times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 263 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 264 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 " Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "T€`Σibmintelnthyperlinkcouritimenormalheadbullitemmultintpackagemutliplintegratprophyperexponentialfunctionhewzmethodcallsequencfunctargumentdescriptionsavailablhypertorecdiffhypertorecdiffcheckrecdiffcheckrecdiffsumtointnespsumtorecdiffsymultintgethelpspecifictypeeithabovlistthespartmultipackagcanusedformonlyfterperformcommandwithalsoaccesslongwhenevconflictbetweennameanothsamesessusethversmapl`"ibmintelntmaplinputcourimathtimeoutputnormalheadbulletitemespgeneratelementarsymmetricpolynomialgivenordercallsequencvrparameterpositintegnamedescripthefunctvariablehenevconflictbetweenameanothusedsamesessuselongformmultintexamplwithxgygfzgfDbiEconflict8equat,fgFformhypertor0longormalpuHsame9 sumtorecdiffGurrenc/€@Ξ€LΛ€DΛ€hΛ€`Λ€„Λ€|Λ€ Λ€˜Λ€ΈΛ€°Λ€ΤΛ€ΜΛ€πΛ€θΛ€ Μ€Μ€(Μ€ Μ€LΜ€DΜ€hΜ€`Μ€€Μ€xΜ€XŽœΜ€”Μ€ΌΜ€΄Μ€ΨΜ€ΠΜ€πΜ€θΜ€0ŽΝ€Ν€$Ν€Ν€$Ž›ŽόšŽHŽ ›Ž›Ž<›Ž4›ŽX›ŽP›Žp›Žh›ŽŒ›Ž„›Ž¨›Ž ›ŽŽΘ›Žΐ›Žδ›Žά›ŽœŽό›ŽœŽœŽ4œŽ,œŽLœŽDœŽdœŽ\œŽ€œŽxœŽ œŽ˜œŽΐœŽΈœŽΨœŽΠœŽτœŽμœŽŽŽ0Ž(ŽPŽHŽpŽhŽˆŽ€Ž Ž˜ŽΌŽ΄Ž€‘ŽΨŽΠŽτŽμŽΨŽžŽžŽ,žŽ$žŽHžŽ@žŽdžŽ\žŽ€žŽxžŽdŽ žŽ˜žŽΌžŽ΄žŽΤžŽΜžŽΤΜπθ0(H@`X|tœ”Ό΄ΨΠŽτμ $h‘ŽD<\Txp`Ž" 1 m n > where is one from the above list. " }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 231 "These functions are part of the MultI nt package and so can be used in the form (arguments) only a fter performing the command with(MultInt) or with(MultInt, ) . The functions can also be accessed in the long form " }{TEXT 275 30 "MultInt[](arguments)" }{TEXT 282 1 "." }}{PARA 15 "" 0 "" {TEXT -1 120 "Whenever there is a conflict between a function name in \+ MultInt and another name used in the same session, use the form " } {TEXT 276 20 "MultInt[]." }}{PARA 15 "" 0 "" {TEXT -1 17 "Th is version of " }{TEXT 277 7 "MultInt" }{TEXT 278 16 " is for Maple 6 ." }}}{PARA 258 "" 0 "" {TEXT 23 4 " " }}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } XT 271 10 " " }{TEXT 264 2 " " }{TEXT 261 7 " " }}{PARA 15 "" 0 "" {TEXT 274 143 "To get help for a sp ecific function type either ?MultInt[] or ?MultInt, " 0 "" {MPLTEXT 1 0 15 "with(MultI nt): " }}}{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#>%$recG/,&*(),&%\"nG\"\"\"F+F+ \"\"#F+),&F*F+!\"\"F+F,F+-%\"fG6%F*%\"xG%\"yGF+F+**,&F,F+F*\"\"$F+)F*F ,F+,&F+F+F*F7F+-F16%F)F3F4F+!\"$,&-%%DiffG6$,$*&*(,&F+F+F3F+F+,6*(F3F+ )F4F,F+)F*F7F+F+*(F3F+FGF+F8F+F+*(F3F+FGF+F*F+F/*&F3F+FGF+F/*&F*F+FGF+ !\"'*&FHF+FGF+\"#5*$FHF+!#s*$F8F+!#g*&F8F+FGF+!\"%F*!#7F+F0F+F+*&F3F+) F4F,F+F/F/F3F+-F?6$*&**F*F+,&F+F+F4F+F+,@FI\"#7FL\"#A*$FGF+\"\"'FKF,FT \"#?FJFO*&F*F+F4F+!#R*&F8F+F4F+!#L*(F8F+F3F+F4F+!\"**(F*F+F3F+F4F+F " 0 "" {MPLTEXT 1 0 52 "checkrecdiff(rec, (1+1/x) ^n*(1+1/y)^(2*n)/x^n/y^n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 " " 0 "" {TEXT 23 1 " " }{TEXT 266 10 "See Also: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "MultInt" 2 "MultInt" "" }{TEXT 258 2 ", \+ " }{HYPERLNK 17 "hypertorecdiff" 2 "MultInt/hypertorecdiff" "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 2" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ,&F+F+F*F7F+-F16%F)F3F4F+!\"$,&-%%DiffG6$,$*&*(,&F+F+F3F+F+,6*(F3F+ )F4F,F+)F*F7F+F+*(F3F+FGF+F8F+F+*(F3F+FGF+F*F+F/*&F3F+FGF+F/*&F*F+FGF+ !\"'*&FHF+FGF+\"#5*$FHF+!#s*$F8F+!#g*&F8F+FGF+!\"%otherwis†`ouput„`output`‚`ƒ`„`…`†`pack€`packag €`ƒ` parameter`‚`„`…`†` paremeterƒ`part €`ƒ`perform €`ƒ` pertorecdiffƒ`pf…`pigf…`polyƒ`polynƒ` polynomial `ƒ`posit`pportƒ` presentat„`procedur‚`product „`…`productg…`prop €`ƒ`puƒ`‚`ƒ`…`xg `‚`ƒ`„`‡`xgf…`xr ƒ` ygf`‚`ƒ`‡` ypertorecdiffƒ`zero ƒ` …`zgf `‡`zn…`zr „`…`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`fp ƒ`…`fpf ƒ`…`freeƒ`frf…`frfgof‚`fs…`ft‚`fter€`ftf…`funct€` `‚`ƒ`„`…`†`function €`ƒ`fvf‚`fx…`generat`get €`ƒ`gf „`…`given `‚`ƒ`„`…`†`gsƒ`guesƒ`he€``‚`head 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"T imes" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 4" "" 0 "" {TEXT -1 15 "hypertorecdiff:" }{TEXT 256 109 " finds a non-zero recurrence-differential(WZ) equation staisfied by a \+ given proper-hyperexponential function." }}{PARA 4 "" 0 "" {TEXT -1 18 "Calling Sequences:" }}{PARA 0 "" 0 "" {TEXT 257 61 " hypertorecdi ff(f, n, fnam(x1, x2, ..., xr), opt1, opt2,...)" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 258 72 "hypertorecdiff(f, [n1 , ..., nm], fnam(x1, x2, ..., xr), opt1, opt2,...)" }{TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT -1 11 "Paremeters:" }}{PARA 0 "" 0 "" {TEXT -1 68 " f - a proper-hyperexponential expressio ns" }}{PARA 0 "" 0 "" {TEXT -1 57 " n, n1, ...,nm - names, th e recurrence variables" }}{PARA 0 "" 0 "" {TEXT -1 58 " x1, x2,..., xr - names, the integration variables" }}{PARA 0 "" 0 "" {TEXT -1 52 " fnam - a name, the function name" }}{PARA 0 "" 0 "" {TEXT -1 47 " opt1, opt2, ... - various desired options " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PAR#A 4 "" 0 "" {TEXT -1 11 "Description" }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 259 41 "hyper torecdiff(f, n, fnam(x1,x2,...,xr)) " }{TEXT 260 66 "finds a non-zero \+ recurrence-differential(WZ) equation of the form " }}{PARA 0 "" 0 "" {TEXT 261 86 " p1*fnam(n,x1,..xr) + p2*fnam(n+1,x1,...,xr)+... = Di ff(R1*fnam(n,x1,...,xr))+..., " }}{PARA 0 "" 0 "" {TEXT 294 11 " whe re p" }{TEXT 281 1 "1" }{TEXT 285 6 "p2,..." }{TEXT -1 41 " are polyn omials independent (free) of " }{TEXT 282 12 "x1, ..., xr." }}{PARA 0 "" 0 "" {TEXT 295 1 " " }{TEXT 297 13 "In this case " }{TEXT 296 15 "hypertorecdiff " }{TEXT 283 86 "searches for a WZ-equation with recur rence part of order at most 6(the default value)." }}{PARA 15 "" 0 "" {TEXT 308 2 "hy" }{TEXT 301 49 "pertorecdiff(f, [n1,...,nm], fnam(x1,x 2,...,xr)) " }{TEXT 302 50 "finds a non-zero multiple recurrence WZ-eq uation. " }}{PARA 15 "" 0 "" {TEXT -1 38 "The following options are su pported: " }{TEXT 268 1 " " }}{PARA 16 "" 0 "" {TEXT 267 41 "hypertor ecd$iff(f, n, fnam(x1,x2,...,xr), " }{TEXT 274 16 "recurrence_order" } {TEXT 275 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 266 17 " rec urrence_order" }{TEXT -1 25 " - non-negative integer" }}{PARA 0 "" 0 "" {TEXT -1 100 " This option allows one to input the required order of the recurrence part of the WZ-equation." }}{PARA 0 "" 0 "" {TEXT -1 9 " If " }{TEXT 303 1 "n" }{TEXT -1 32 " is a list of va riables , then " }{TEXT 304 17 "recurrence_order " }{TEXT -1 35 "is a list of non-negative integers." }}{PARA 16 "" 0 "" {TEXT -1 0 "" } {TEXT 263 0 "" }{TEXT 269 72 "hypertorecdiff(f, n, fnam(x1, x2, ..., x r), recurrence_order, denom_poly" }{TEXT 262 2 ") " }{TEXT 270 1 " " } }{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 271 11 "denom_poly " } {TEXT -1 24 "- list of polynomials." }}{PARA 256 "" 0 "" {TEXT -1 65 " This option allows one to guess and input the denominators \+ " }{TEXT 272 10 "denom_poly" }{TEXT -1 88 " of the r-tuple of ration al functions(the certificates) of the %required WZ-equation." }} {PARA 0 "" 0 "" {TEXT -1 7 " If " }{TEXT 305 1 "n" }{TEXT -1 32 " i s a list of variables , then " }{TEXT 306 17 "recurrence_order " } {TEXT -1 35 "is a list of non-negative integers." }}{PARA 16 "" 0 "" {TEXT -1 0 "" }{TEXT 264 0 "" }{TEXT -1 0 "" }{TEXT 265 50 "hypertorec diff(f, n, fnam(x1, ..., xr), denom_poly" }{TEXT 273 2 ") " }}{PARA 0 "" 0 "" {TEXT 307 4 " " }{TEXT 309 44 "n can also be a list of recu rrence variables" }{TEXT 310 2 ". " }}{PARA 16 "" 0 "" {TEXT -1 0 "" } {TEXT 276 0 "" }{TEXT -1 0 "" }{TEXT 277 72 "hypertorecdiff(f, [n1,n2, ...,nk], fnam(x1, ..., xr), [N1,...,Nk], ansatz" }{TEXT 278 3 ") " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " n1,....,nk - recur rence variables" }}{PARA 0 "" 0 "" {TEXT -1 33 " N1,...Nk - shif t variables" }}{PARA 0 "" 0 "" {TEXT -1 48 " ansatz - list of m onomials in N1, ...,Nk" }}{PARA 0 "" 0 "" {TEXT -1 19 " In this c ase " }{TEXT 300 14 "hypertorecdiff" }{TEXT -1 55 " searches for a WZ -equa&tion by using the given ansatz." }}{PARA 16 "" 0 "" {TEXT -1 0 " " }{TEXT 279 0 "" }{TEXT -1 0 "" }{TEXT 280 84 "hypertorecdiff(f, [n1, n2,...,nk], fnam(x1, ..., xr), [N1,...,Nk], ansatz, denom_poly" } {TEXT 284 3 " ) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 " T his 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 can be used in the form " }{TEXT 286 14 "h ypertorecdiff" }{TEXT -1 41 "(args) only after performing the command \+ " }{TEXT 287 13 "with(MultInt)" }{TEXT 288 4 " or " }{TEXT 289 30 " Wi th(MultInt, hypertorecdiff)" }{TEXT 290 53 ". The function can also be accessed in the long form " }{TEXT 291 30 "MultInt[hypertorecdiff](ar gs)." }}{PARA 15 "" 0 "" {TEXT -1 56 "Whenever there is a conflict bet ween the function name " }{TEXT 298 14 "hypertorecdiff" }{TEXT -1 57 " and another name used in the 'same session, use the form " }{TEXT 299 25 "MultInt['hypertprecdiff']" }{TEXT -1 0 "" }}}{PARA 256 "" 0 " " {TEXT 23 3 " " }{TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 317 9 "Examples:" }{TEXT 316 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 256 "> " 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 12 "" 1 "" {XPPMATH 20 "6#7$/,&*&,&%\"nG\"\"%\"\"#\"\"\"F+-%\"fG6%F(%\"xG%\"y(GF+F +*&,&F(!\"\"F3F+F+-F-6%,&F(F+F+F+F/F0F+F+,&-%%DiffG6$*&-%#_RG6$F/F0F+F ,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-zero W Z-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$*&,&F(F(F0F(F(F-F(F0F(-F:6$,$*(F1F(,&F(F(F1F(F( F-F(F,F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The following call asks MultInt to find a non-zero doub le recurrence WZ-equatio)n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "hypertorecdiff((1+x)^n*(1+y) ^m/(1-x*y)^k,[n,m],f(x,y));" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... tryi ng to get a recurrence eq with order [0, 0] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 91 "\ncould not find a non-zero recurrence-differential(WZ ) equation with order [0, 0] in [n, m]" }}{PARA 6 "" 1 "" {TEXT -1 62 "\n... trying to get a recurrence eq with order [1, 0] in [n, m]" }} {PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 19 equations for 13 unknow ns" }}{PARA 6 "" 1 "" {TEXT -1 49 " Cheers! for the success. CPU Time : .33 seconds." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*&,*%\"nG\"\"\"% \"mGF(\"\"#F(%\"kG!\"\"F(-%\"fG6&F'F)%\"xG%\"yGF(F(*&,(F'F,!\"#F(F+F(F (-F.6&,&F'F(F(F(F)F0F1F(F(,&-%%DiffG6$,$*(F0F(,&F(F(F0F(F(F-F(F,F0F(-F :6$*&,&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* unknow ns" }}{PARA 6 "" 1 "" {TEXT -1 49 " Cheers! for the success. CPU Time : .24 seconds." }}{PARA 12 "" 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$*&,&F(F(F0F(F(F-F(F0F(-F:6$,$*(F 1F(,&F(F(F1F(F(F-F(F,F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 88 "The following call asks MultInt to find a non-zero WZ-equation of with the given ansatz." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "hypertorecd iff(1/(1-x-x^2-y-y^2)/x^(m+1)/y^(n+1),[n,m],f(x,y),[N,M],[N,M,N*M,M^2] ); " }}{PARA 6 "" 1 "" {TEXT -1 42 "\n ... solving 29 equations for 22 unknowns" }}{PARA 6 "" 1 "" {TEXT -1 50 " Cheers! for the success. C PU Time : .44 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*&,(\"\")\"\"\"%\"mG\"\"%%\"nGF*F(- %\"fG6&F+F)%\"xG%\"yGF(F(*&,&F+\"\"#F3F(F(-F-6&,&F+F(F(F(F)F/F0F(F(*&F 6F(-F-6&F6,&F)F(F(F(F/F0F(F(+*&,(F'F(F+F3F)F*F(-F-6&F+F:F/F0F(F(*&,&!#5 F(F)!\"&F(-F-6&F+,&F3F(F)F(F/F0F(F(,&-%%DiffG6$,$*&*&,(*$)F/F3F(F*F/F* FBF(F(F,F(F(F/!\"\"FPF/F(-FH6$,$*&*(,&F(F(F0F3F(,&F(F(F/F3F(F,F(F(F/FP FPF0F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 256 " " 0 "" {TEXT 315 1 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 312 10 "See Al so:" }{TEXT 311 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "MultInt" 2 "MultInt" "" }}}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 23 0 "" }{TEXT -1 0 "" }}}{MARK "11 25 4" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } f(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. C PU 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(+equatiƒ`equation ƒ`…`erƒ`„`…`es ƒ`…`esentat„`esp €``exampl`‚`ƒ`„`…`†`expr …`†`expres…`express ‚`„`…` expressioƒ`fals†`faq…`fb…`fbfƒ`fbofgof…`fbpf…`fcp…`fdo…`fen…`fenf …` ff €`ƒ`ffn…`ffof …` fg‚`ƒ`…`‚`ƒ`„`…`ngfƒ`„`…`nkƒ`nmƒ`nocoeff„`non ƒ` …`normal€``‚`ƒ`„`…`†`nsƒ`nt €``‚`ƒ`„`…`†`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`„`4ibmintelntmaplinputcourimathtimehyperlinkoutputercouriernormalheadbulletitemsumtointnfindintegralrepresentatgivenbinomialsumcallsequencbinocoeffbinocoeffkmlmumparameterfinitproductcoeffalgebraicexpressnamesummatvariabllowerupperboundeachdescriptrepresentatformsumgxgkgconstanttermctzrouputwithintegratwhenevconflictbetweenfunctmsumtointnanothusedsamesessuselongformultintexamplintergralrepresentatbinomialgngfmultintinfinitgfrepresentaigjgfjgbinomialngalsosumtorecdiffssum…`žibmintelntmaplinputcourimathtimehyperlinkoutputtimescouriernormaltexterheadwarnbulletitemsumtorecdifffindnonzerorecurrencdifferialwzequatconstanttermexpressgivensumcallsequencbinocoeffsumtorecdiffkmlmumrecorderparameterbinocoeffsfinitproductbinomialcoefficientalgebraicnamessummatvariablnamelowerupperboundnegatintegdescriptformsgngsumgproductgbinocoeffsgxgfkgfirstintergalrepresentatexprssionctzrstatisfiequationznwhenevconflictbetweenfunctmsumanothusedsamesessuselongmultintexamplsatisfiexpresdixonbinomialgwithinfinitsolvequationunknownnewdefinitchicheersuccescpusecondfggfdiffgrgflffoffpuffkffrffmfenfftffpfffoffdoffnfipfjqfbpffcpfenfaqfjoenffxfgqfinfbpfsffgofgofphffgoffbofgofpigffgofgtjgfngfjgtryingrecurrencedifferentialsecondsigfalsoultintsumtointnointnuse€``‚`ƒ`„`…`†`used €``‚`ƒ`„`…`†`useful‚`userƒ`usingƒ`vaƒ`valuƒ`vari`variabl ƒ`„`…`†`variouƒ`verif‚`vers€`vn†`vr`warn…`weenƒ`wheƒ`whenev€`‚`ƒ`„`…`†`whet†`wheth‚`wiƒ`with€``‚`ƒ` „`…`†`wz€`‚`ƒ`…`xg`‚`ƒ`„`xgf…`xr ƒ` ygf`‚`ƒ` ypertorecdiffƒ`zero ƒ` …`zgf`zn…`zr „`…``†`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ` hypertorecƒ` hypertorecd €`ƒ` hypertorecdiƒ`hypertorecdiff€`‚`ƒ`hypertprecdiffƒ`ial…`ibm€``‚`ƒ`„`…`†`iff €`ƒ`ig„`igf…`imesƒ`independƒ`infinit „`…`input `‚`ƒ`„`…`†`integ`ƒ`…`integerƒ`integral„`integrat€`ƒ`„`intel€``‚`ƒ`„`…`†`intergal…` intergral„`ions€`item€``‚`ƒ`„`…`†`jg „`…`jgf „`…`kgƒ`„`…`km „`…`krecdiff‚`le ƒ`†`list €`ƒ`lm „`…`long€``ƒ`„`…`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`kgdoubtryigetncouldunknowhypertorecdiffpumgngffbffpffhfpesormalheadtextmaplebulletitemtimesdashnormalhypertorecdifffindnonzerorecurrencdifferentialwzequatstaisfigivenprophyperexponentialfunctcallsequenchypertorecdifffnamxroptnmparemeterexpressionsnamethvariablintegratvarioudesiroptiondescripthypertorecdiffformdiwherepolynomialindependfreecasesearchwithrecurrencpartorderatmostdefaultvaluhypertorecdiffmultiplequationfollowsupporthypertorecdiffrecurrencnegatintegallowrequirlistvariablintegerdenompolypolynomialguesdenominatortuplrationalfunctioncertificathypertorecdiffcanalsorecurrencnkansatzshifonomialaseusinghisuserrationalmultintpackagusedypertorecdiffargsonlyafterperformcommandwiaccesslongargswhenevconflictbetweenanothsamesessusehypertprecdiffexampltryingrecurrnceequatisolvequationunknowncheersuccescpusecondngfgxgygfdiffgrgasksequationmgf1‚`Οibmintelntmaplinputcourimathtimehyperlinkoutputnormalheadbullitemcheckrecdiffcheckwhethgivenwzequatsatisfifunctcallingsequenccheckrecdiffeqparameterrecurrencdifferentialuationalgebraicexpressdescriptusefularticularverifprocedurhypertorecdiffmultinthypertorecdiffwhenevconflictbetweennameanothusedhesamesessuseformexamplwithmultirecdiffrecgfgxgygfdiffgfgffhffiflfkfftfjfofvffrfgoftruegalsoorecdiffmultiplequationfollowsupporthypertorecdiffrecurrencnegatintegallowrequirlistvariablintegerdenompolypolynomialguesdenominatortuplrationalfunctioncertificathypertorecdiffcanalsorecurrencnkansatzshifonomialaseusinghisuserrationalmultintpackagusedypertorecdiffargsonlyafterperformcommandwiaccesslongargswhenevconflictbetweenanothsamesessusehypertprecdiffexampltryingrecurrnceequatisolvequationunknowncheersuccescpusecondngfgxgygfdiffgrgasksequationmgf1 -1 266 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "Cou rier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 271 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 272 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 273 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 274 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 276 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{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 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 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 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Map4le Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 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 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 50 "binocoeffs \+ - finite product of binomial coeffs " }}{PARA 0 "" 0 "" {TEXT 23 4 " \+ " }{TEXT 265 37 "x - algebraic expression " }}{PARA 0 " " 0 "" {TEX5T -1 0 "" }{TEXT 23 7 " k, " }{TEXT 266 44 "k1,...,km \+ - names, the summation variables" }}{PARA 0 "" 0 "" {TEXT -1 8 " \+ " }{TEXT 267 29 "l,u,l1,u1,... - expressions, " }{TEXT 261 57 "lower and 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 "msumt6ointn" }{TEXT -1 63 " and anoth er name used in the same session, use 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*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(F'!\"\"% \"nGF()*&,&%$z_2GF(F(F(F(F.F)F*F(),&F(F(*&F'F(F.F(F)F*F(7$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 representaion of \+ :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$**)!\"7\",&%\"iG\"\"\"%\"jGF+F+-%)binomialG6$F)F*F+-F.6$%\"nGF* F+-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)*bin omial(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'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" "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "14 17 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } PARA 0 "" 0 "" {TEXT -1 32 "To find a CT representaion of \+ :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$**)!\"7constant „`…`cou „`…`courip€`` ‚`ƒ`&„`…`†` cpu ƒ`…`ct „`…`dashƒ`defaultƒ`definit…`denomƒ` denominatorƒ`descript€``‚`ƒ`„`…`†`desirƒ`diƒ`diff ‚`ƒ`differ…` differential ‚`ƒ`…`diffg‚`ƒ`…`dixon…`doubƒ`ds…`each„`ecdiffƒ`ecific€`eith€` elementar`enf…`eq‚`ƒ`…`equƒ`equat‚`ƒ` …`€``‚`ƒ`„`…`†`nsƒ`nt €``‚`ƒ`„`…`†`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`satisfi ‚`…`searchƒ`secon…`second ƒ`…`sequenc€``‚`ƒ`„`…`†`sess€``‚`ƒ`„`…`†`sf…`sg…`shifƒ`sion†`solv ƒ`…`sp€`ssion…`ssum„`staisfiƒ`statisfi…`suƒ`succes ƒ`…`sum „`…`sumg „`…`summat „`…`sumt…` sumtointn €`„`…` sumtorecdiff €`„`…``ƒ`„`‡`xgf…`xr ƒ` ygf`‚`ƒ`‡` ypertorecdiffƒ`zero ƒ` …`zgf `‡`zn…`zr „`…`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`…`Ε'{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 "Times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Times " 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 263 "Times" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 264 "Courier" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "Times" 0 12 0 0 0 0 0 2 0 0 0 0 3 0 0 1 }{CSTYLE "" 23 266 "Times" 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 ;"Courier" 1 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 268 "Courier" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "Times" 0 12 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 271 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 272 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 273 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 274 "Courier" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 275 "Tim es" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 276 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 277 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 278 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 280 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 281 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 282 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 283 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 284 "Cou rier" 0 10 0 0 0 0 0 0 0< 0 0 0 3 0 0 1 }{CSTYLE "" -1 285 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 286 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 287 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 288 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 289 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 290 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 291 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 292 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{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 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Couri er" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 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 0 -1 8 2 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 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 0 -1 -1 -1 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 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 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 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." } {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 266 0 "" }{TEXT 23 1 " " }{TEXT 257 17 "Calling Sequences" }{TEXT 258 2 ": " }{TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 289 41 "sumtorecdiff(binocoeffs, x , k, n, [l,u]) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 291 69 " sumt orecdiff(>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 2 " " }{TEXT 290 81 "sumtorecdiff(binocoeffs, x, [k1,...,km], n, [ [l1,u1],...,[lm,um]], rec_order) " }{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 "binocoef fs - 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 - na mes, 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 bounds 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 23 "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 " " }{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\"\"$\"\"\") !\"\"F.F0F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sumtorecdiff(binomial(2*n,k)^3,(-1)^k,k,n,[0,infinity ],1);" }}{PARA 6 "" 1 "" {TEXT -1 41 " ... solving 77 equations for 73 unknowns" }}{PARA 7 "" 1 "" {TEXT -1 31 "Warning, new definition for \+ Chi" }}{PARA 6 "" 1 "" {TEXT -1 51 " Cheers! for the success. CPU Tim e : 3.97 seconds." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7%/,&*(,&%\"nG\"\"$\"\"\"F*F*,&\"\"#F*F(F)F*-%#_ FG6%F(%$z_1G%$z_2GF*F)*&),&F(F*F*F*F,F*-F.6%F4F0F1F*F*,&-%%DiffG6$*&-% #_RG6$F0F1F*F-F*F0F*-F96$*&-F=6$F1F0F*FA-F*F1F*/F<,$*&*&,&F0F*F*F*F*,`t F*F*F(F)*&)F0F)F*)F1F)F*!\"%*()F0F,F*FLF*F(F*\"\"%*(FOF*FLF*)F(F,F*FP* (FOF*)F1F,F*F(F*\"#UF0\"\"(*(FKF*F1F*F(F*\"\"**(FKF*F1F*FRF*FM*(FKF*F1 F*)F(F)F*!#7*(FOF*F1F*FRF*FP*(FOF*F(F*F1F*!\"**(FOF*F1F*FenF*\"#7*&F0F *FTF*!#5*(F0F*F(F*FTF*!#M*&FOF*F1F*FM*&FKF*F1F*FP*&F0F*F1F*FP*&FOF*FTF *\"#5*&FKF*)F1FPF*!\"\"*$FRF*F,*&F0F*FfoF*\"\")*&F0F*F(F*\"#@*(F0F*FTF *FenF*F[o*(F0F*FfoF*FenF*!#K*(F0F*FfoF*FRF*Fdo*(F0F*FfoF*)F(FPF*!#C*(F 0F*FfoF*F(F*\"#E*(FenF*F0F*F1F*Ffn*(FRF*F0F*F1F*FM*(FKF*FfoF*FRF*!\"#* (F0F*FTF*FRF*!#=*&FLF*FenF*F[o*$FTF*F,*$FLF*FM*&FOF*FfoF*F**(FOF*FfoF* FRF*\"#C*(FOF*FfoF*FenF*F[o*(FOF*FfoF*F(F*\"#8*(FKF*FLF*F(F*F[q*$FOF*! \")*&FKF*FTF*Fip*&F(F*F1F*!\"$*&F0F*FRF*F,*&FTF*FRF*\"#A*(FKF*FfoF*F(F *Fjq*(FenF*FOF*FTF*Ffn*(FbpF*FOF*FTF*Fcp*(F0F*FLF*FRF*!#I*(F0F*FLF*Fen F*F[o*(F0F*FLF*FbpF*Faq*(F0F*FLF*F(F*!#D*(FKF*FTF*FRF*!#A*(FKF*FTF*Fen F*Fjo*(FKF*FTF*FbpF*Faq*(FKF*FTF*F(F*!#9*(FKF*FLF*FRF*!#E*(FKF*FLF*Fen F*Ffn*&F0F*FbpF*Fcp*&FenF*F1F*Faq*(FRF*FOF*FTF*\"#W*&F0F*FLF*FM*&BF0F*F enF*!#O*(F(F*F0F*F1F*FX*&FOF*FRF*F]o*&FbpF*F1F*Faq*&FTF*F(F*\"#9*&FTF* FenF*Fgq*&FTF*FbpF*Fcp*&FLF*FRF*FP*&FLF*F(F*Fin*&FOF*FenF*\"#K*&FOF*Fb pF*Faq*&FOF*F(F*F^sF*F**()F1F,F*,&F(F,F*F*F*F0F*Fgo#FgoFP/F-,$*&*()*&F HF*F0Fgo,$F(F,F*)*&,&F1F*F*F*F*F1FgoF^uF*),&F*F*FboFgoF^uF*F**(F0F*F1F *)%#PiGF,F*FgoFgt" }}}{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 ":" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**-%)binomialG6$,&%\"i G\"\"\"%\"jGF,F+F,-F(6$%\"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 recurren ce-differential(WZ) equation with order 0" }}{PARA 6 "" 1 "" {TEXT -1 77 "\n could not find a Cnon-zero recurrence-differential(WZ) equation \+ with order 0" }}{PARA 6 "" 1 "" {TEXT -1 81 " \n... trying to find a n on-zero recurrence-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 secon ds." }}{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+F+*&F*F+%# PiGF+F/#F/\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 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 "M ultInt" "" }{TEXT 262 2 ", " }{HYPERLNK 17 "sumtointn" 2 "MultInt/sumt ointn" "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } t find a Cable`abov€`access €`ƒ`afterƒ`age€`alƒ` algebraic‚`„`…`†`allowƒ`also€`‚`ƒ`„`…`ame`anoth€``‚`ƒ`„`…`†`ansatzƒ`arƒ`argsƒ`argument€` articular‚`aseƒ`asksƒ`atƒ`ationƒ`availabl€`betƒ`between€``‚`„`…`†`bi„`command €`ƒ`conflict€``‚`ƒ`„`…`†`yperexponential €`ƒ` hyperlink€`‚`ƒ`„`…`hypertorƒ`zn…`zr „`…`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`bin„`binocoef…` binocoeff „`…` binocoeffsg…`binomial „`…` binomialg „`…`bound „`…`bull €`‚`bullet`ƒ`„`…`†`call€``ƒ`„`…`†`calli‚`can €`ƒ`caseƒ`ce…` certificatƒ`chec‚`check ‚`†` checkrecdi€` checkrecdiff €`‚`cheer ƒ`…`chi…`coeff„` coefficient…`command €`ƒ`conflict€``‚`ƒ`„`…`†`yperexponential €`ƒ` hyperlink€`‚`ƒ`„`…`hypertorƒ`zn…`zr „`…`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`fgf‚`fgo…`fgof…`fgofgt…`fgofp…`fgq…`fhƒ`fhf‚`fi‚`fin…`findƒ` „`…`finit „`…`fip…`first…`fjfo‚`fjo…`fjq…`fkf ‚`…`fl‚`flf…`fm…`fnam ƒ` fo„`fof…`followƒ`form€``‚`ƒ`„`…`†`intergal…` intergral„`ions€`item€``‚`ƒ`„`…`†`jg „`…`jgf „`…`kgƒ`„`…`km „`…`krecdiff‚`le ƒ`†`list €`ƒ`lm „`…`long€``ƒ`„`…`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`sym €`†` symmetric `†`term „`…`text ƒ`…`th €`ƒ`thes€`tim ƒ`…`time^€``‚` ƒ`„`…` †` torecdiffƒ`true†`trueg ‚`†`tryiƒ`trying ƒ`…`tuplƒ`type€`uation‚`ƒ`…`uf…`ultint€`„`…`†`um „`…`unknowƒ`unknown ƒ`…`upper „`…`urrencƒ`€`eith€` elementar`enf…`eq‚`ƒ`…`equƒ`equat‚`ƒ` …`erlink€`‚`ƒ`„`…`hypertorƒ`ƒ`†`list €`ƒ`lm „`…`long€``ƒ`„`…`‡`omialƒ`optƒ`optionƒ`order`ƒ`…`‡`orecdiff…`ormalƒ`rationƒ`rationalƒ`re ƒ`„`rec‚`ƒ`…`recg‚`recuƒ`recurƒ`recurrƒ`recurren…` recurrenc‚`ƒ`…`rencƒ`repr„` representa„` representat „`…`requirƒ`return†`rg ƒ`…`riablƒ`rier „`…`rm„`rrencƒ` rtorecdiff‚`sa…`same€``‚`ƒ`„`†``‚`ƒ`…`xg `‚`ƒ`„`‡`xgf…`xr ƒ` ygf`‚`ƒ`‡` ypertorecdiffƒ`zero ƒ` …`zgf `‡`zn…`zr „`…`ointn…`omial ƒ`„`only €`ƒ`onomialƒ`optƒ`optionƒ`order`ƒ`…`orecdiff…`ormalƒ`†`ε {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Times" 0 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 " Times" 1 14 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 1 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 264 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Courier " 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "CJourier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 268 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 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 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 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 0 -1 8 2 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 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 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 " sym: " }{TEXT 269 68 "checks whet her a given function is symmetric w.r.t. given variables." }{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(fK,[v1,v2,...,vn])" }{TEXT 266 4 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 257 11 "Parameters:" }{TEXT 259 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 267 39 "f - an algebraic expres sion " }}{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 "Description: " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 261 72 "sym returns true, if f is symmetric w. r.t v1,..,vn, false otherwise. " }}{PARA 15 "" 0 "" {TEXT -1 0 "" } {TEXT 262 139 "Whenever there is a conflict between the function name sym and another name used in the same session, use the form M ultInt['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: " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1-1/y)^n/x/y,[x,Ly]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 1 " \+ " }}}{MARK "12" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } EXT 268 20 "v1,...,vn \+ - names" }}{PARA 0 "" 0 "" {TEXT 23 3 " " }}{SECT 0 {PARA 4 "" 0 " " {TEXT 23 1 " " }{TEXT 260 13 "Description: " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{TEXT 261 72 "sym returns true, if f is symmetric w. r.t v1,..,vn, false otherwise. " }}{PARA 15 "" 0 "" {TEXT -1 0 "" } {TEXT 262 139 "Whenever there is a conflict between the function name sym and another name used in the same session, use the form M ultInt['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: " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "with(MultInt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sym((1-1/x)^n*(1-1/y)^n/x/y,[x,L†`,ibmintelntmaplinputcourimathtimeoutputnormalheadmaplebulletitemsymcheckwhethergivenfunctsymmetricvariablcallsequencvnparameteralgebraicexprsionnamedescriptreturntruefalsotherwiswhenevconflictbetweenanothusedsamesessuseformultintexamplwithmultinttrueg‡`τ {VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 256 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 257 " Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 260 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 262 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 263 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 264 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 265 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "Times " 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 267 "CouON" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" 23 268 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 270 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 271 "Courier" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 272 "Times" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " " -1 274 "Times" 0 10 0 0 0 0 0 2 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 275 " Times" 0 10 0 0 0 0 0 2 0 0 0 0 3 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 4 "" 0 "" {TEXT 272 7 " esp: " }{TEXT 275 63 "generates tPOlementary symmetric polynomials of a given order" }{TEXT -1 0 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 256 18 "Calling Sequence: " }{TEXT 274 0 "" }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 269 19 "esp (n,[v1,...,vr]) " }{TEXT 23 2 " " }}{PARA 4 "" 0 "" {TEXT 23 1 " " } {TEXT 257 11 "Parameters:" }{TEXT 258 2 " " }}{PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 270 35 "n - a positive integer" }} {PARA 0 "" 0 "" {TEXT 23 4 " " }{TEXT 271 22 "v1,v2,...,vr - name s" }{TEXT 23 3 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 259 13 "Description: " }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "T " }{TEXT 260 12 "he function " }{TEXT 262 18 "esp(n,[v1,...,vr])" } {TEXT 263 62 " outputs the elementary symmetric polynomial in the vari ables " }{TEXT 264 11 "[v1,...,vr]" }{TEXT 265 11 " of order n" } {TEXT 261 1 "." }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "W" } {TEXT 266 142 "henever there is a conflict between the function n ame esp and another name used in the same sessQP use the long form MultInt['esp']. " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 2 " " } }{SECT 0 {PARA 4 "" 0 "" {TEXT 268 9 "Examples:" }{TEXT 267 1 " " } {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%F&%\"zGF& F&*&F'F&F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } RA 4 "" 0 "" {TEXT 23 1 " " }{TEXT 259 13 "Description: " }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "T " }{TEXT 260 12 "he function " }{TEXT 262 18 "esp(n,[v1,...,vr])" } {TEXT 263 62 " outputs the elementary symmetric polynomial in the vari ables " }{TEXT 264 11 "[v1,...,vr]" }{TEXT 265 11 " of order n" } {TEXT 261 1 "." }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 23 1 "W" } {TEXT 266 142 "henever there is a conflict between the function n ame esp and another name used in the same sessQ