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-1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle66" -1 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 240 "" 0 "" {TEXT 233 51 "Maple worksheet for tes ting the package infhsum.mpl" }}}{EXCHG {PARA 241 "" 0 "" {TEXT 234 62 "by Raimundas Vidunas, http://www.math.kyushu-u.ac.jp/~vidunas/" }} }{EXCHG {PARA 242 "" 0 "" {TEXT 235 28 "version of 30 November 2005" }}}{EXCHG {PARA 243 "" 0 "" }}{EXCHG {PARA 244 "> " 0 "" {MPLTEXT 1 236 25 "# currentdir(programdir):" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 237 90 "Read W. Koepf's package hsum9.mpl for hypergeometric summation , which can be obtained from" }}{PARA 245 "" 0 "" {TEXT 237 72 "http:/ /www.mathematik.uni-kassel.de/~koepf/Publikationen/index.html#down" }} }{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 238 17 "read \"hsum9.mpl\";" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#IVPackage~\"Hypergeometric~Summation \",~Maple~V~-~Maple~9G6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#IYCopyri ght~1998-2004,~Wolfram~Koepf,~University~of~KasselG6\"" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 237 64 "Read R. Vidunas' package infhsum.mpl, \+ which can be obtained from" }}{PARA 245 "" 0 "" {TEXT 237 51 "http://w ww.science.uva.nl/~thk/specfun/compalg.html" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 19 "read \"infhsum.mpl\";" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#IfnThis~is~a~Maple~package~for~computing~recurrence~rel ations,G6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#Ignclosed~form~express ions~and~uniformly~bounded~convergence~ofG6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#Ignnon-terminating~hypergeometric~series;~written~by~R. ~VidunasG6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#Q<~Version~4.29,~30-N ov-2005.6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#I;tested~in~Maple~7,~8 ~and~9G6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#ILSupported~by~NWO,~pro ject~number~613-06-565G6\"" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#ISThe~h elp~function~is~envoked~by~\"~infhsumhelp(~)~\"G6\"" }}}{SECT 1 {PARA 249 "" 0 "" {TEXT -1 13 "Documentation" }}{EXCHG {PARA 244 "> " 0 "" {MPLTEXT 1 236 14 "infhsumhelp();" }}{PARA 250 "" 1 "" {TEXT 240 70 "T his is a Maple 7, 8 and 9 package for computing recurrence relations," }{TEXT 240 61 "\n closed form expressions and uniformly bounded con vergence" }{TEXT 240 47 "\n of non-terminating hypergeometric ser ies" }{TEXT 240 44 "\n Version 4.29, 30-Nov-2005." } {TEXT 240 1 "\n" }{TEXT 240 59 "\n(c) Raimundas Vidunas, University of Amsterdam, 2000-2001;" }{TEXT 240 54 "\nSupported by the Dutch NWO, p roject number 613-06-565" }{TEXT 240 1 "\n" }{TEXT 240 59 "\nat presen t at Kyushu University, Hakozaki, Fukuoka, Japan;" }{TEXT 240 46 "\nUR L: http://www.math.kyushu-u.ac.jp/~vidunas/" }{TEXT 240 1 "\n" }{TEXT 240 63 "\nWe use Zeilberger's algorithm to compute recurrence relation s," }{TEXT 240 61 "\na modification of Wilf-Zeilberger's method to com pute closed" }{TEXT 240 45 "\nexpressions. We use W. Koepf's Maple pac kage" }{TEXT 240 58 "\n\"hsum9.mpl\", where similar procedures are imp lemented for" }{TEXT 240 74 "\nterminating hypergeometric series. Koep f's package can be downloaded from" }{TEXT 240 73 "\nhttp://www.mathem atik.uni-kassel.de/~koepf/Publikationen/index.html#down" }{TEXT 240 1 "\n" }{TEXT 240 53 "\nOur method is described in the following two pap ers:" }{TEXT 240 60 "\n[1] R. Vidunas and T.H. Koornwinder, \"Zeilberg er method for" }{TEXT 240 56 "\nnon-terminating hypergeometric series \", in preparation." }{TEXT 240 62 "\n[2] T.H. Koornwinder, \"Identiti es of nonterminating series by" }{TEXT 240 69 "\nZeilberger's algortih m\", J. Comput. Appl. Math., 99 (1998), 449-461." }{TEXT 240 1 "\n" } {TEXT 240 66 "\nThe commands: infsumrecursion, infclosedform, uniformc onvergence." }{TEXT 240 64 "\nFor more information please invoke infhs umhelp( command_name )." }{TEXT 240 63 "\nThe first two commands are t he analogues of Koepf's procedures" }{TEXT 240 60 "\n\"sumrecursion\" \+ and \"closedform\" for non-terminating series." }{TEXT 240 58 "\nOther reserved names of our package start with \"iHPG....\"" }}}{EXCHG {PARA 244 "> " 0 "" {MPLTEXT 1 236 29 "infhsumhelp(infsumrecursion);" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#I?infsumrecursion(term,~k,~f(n))G6 \"" }}{PARA 250 "" 1 "" {TEXT 240 55 "computes a recurrence relation o f hypergeometric series" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#I?f(n)=sum (hterm,~k=0..infinity)G6\"" }}{PARA 250 "" 1 "" {TEXT 240 57 "It uses \+ Zeilberger's algorithm and checks the convergence" }{TEXT 240 58 "\nof the relevant (Zeilberger's) telescoping sum and of the" }{TEXT 240 55 "\nhypergeometric series itself. This command is based on" }{TEXT 240 64 "\nKoepf's procedure \"sumrecursion\" (from his package \"hsum9 .mpl\")" }{TEXT 240 64 "\nwhich implements Zeilberger's algorithm for \+ terminating series." }{TEXT 240 65 "\nIf the relevant series does not \+ convergence, an error message is" }{TEXT 240 66 "\ndisplayed. If there are convergence conditions on the parameters," }{TEXT 240 66 "\nthey \+ are displayed as a warning, unless an option \"conditions\" is" } {TEXT 240 42 "\npresent. The command with all options is:" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#IXinfsumrecursion(term,~k,~f(n),~certificat e,~conditions)G6\"" }}{PARA 250 "" 1 "" {TEXT 240 61 "If \"conditions \" is present, then the convergence restrictions" }{TEXT 240 66 "\nare returned in the output sequence after the computed recurrence" } {TEXT 240 63 "\nIf \"certificate\" is present, then a so-called certif icate of a" }{TEXT 240 62 "\nZeilberger-type proof of the recurrence r elation is returned," }{TEXT 240 59 "\nafter the recursion and before \+ the convergence conditions." }}}{EXCHG {PARA 244 "> " 0 "" {MPLTEXT 1 236 27 "infhsumhelp(infclosedform);" }}{PARA 247 "" 1 "" {XPPMATH -1 " 6#I:infclosedform(term,~k,~n)G6\"" }}{PARA 250 "" 1 "" {TEXT 240 63 "c omputes a closed form expression for the hypergeometric series" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#I?f(n)=sum(hterm,~k=0..infinity)G6\"" }}{PARA 250 "" 1 "" {TEXT 240 61 "This command is based on Koepf's pr ocedure \"closedform\" (from" }{TEXT 240 62 "\nhis package \"hsum9.mpl \"). The procedure is successful only if" }{TEXT 240 62 "\n\"infsumrec ursion(term,k,s(n))\" finds a first order recurrence" }{TEXT 240 64 " \nrelation (or a recurrence relation with two homogeneous terms)," } {TEXT 240 62 "\nand the series is uniformly bounded (with respect to n ) by an" }{TEXT 240 64 "\nabsolutely convergent series. For a single \+ \"initial\" evaluation" }{TEXT 240 63 "\nwe use the limit n->infinity \+ of the series. If the series does" }{TEXT 240 64 "\nnot converge unifo rmly (as described above) an error message is" }{TEXT 240 65 "\nreturn ed. If there are convergence conditions on the parameters," }{TEXT 240 61 "\nthese conditions are returned as a warning unless the option " }{TEXT 240 57 "\n\"conditions\" is present. The command with all opt ions is" }{TEXT 240 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#Ijninfcl osedform(term,~k,~n=0,~certificate,~conditions,~ucrelaxed)G6\"" }} {PARA 250 "" 1 "" {TEXT 240 61 "If \"conditions\" is present, then the convergence restrictions" }{TEXT 240 66 "\nare returned in the output sequence after the computed recurrence" }{TEXT 240 64 "\nThe optional substitution \"n=0\" (instead of just a name \"n\") is" }{TEXT 240 65 "\nperformed after the computation; then the discrete parameter \"n \"" }{TEXT 240 64 "\nis not seen in the output. (General substitutions are allowed)." }{TEXT 240 63 "\nIf \"certificate\" is present, then a so-called certificate of a" }{TEXT 240 66 "\nZeilberger-type proof of the hypergeometric identity is returned." }{TEXT 240 60 "\nIf \"ucrel axed\" is present, then in the case that the series" }{TEXT 240 65 "\n can be majorized by a sum of O(k^p) with -1<=p<0, no error but a" } {TEXT 240 62 "\nwarning message is generated, and a conjectural expres sion is" }{TEXT 240 10 "\nreturned." }}}{EXCHG {PARA 244 "> " 0 "" {MPLTEXT 1 236 32 "infhsumhelp(uniformconvergence);" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#I?uniformconvergence(term,~k,~n)G6\"" }}{PARA 250 "" 1 "" {TEXT 240 40 "checks whether the hypergeometric series" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#I=f(n)=sum(term,k=0..infinity)G6\"" }} {PARA 250 "" 1 "" {TEXT 240 57 "is uniformly bounded (with respect to \+ n) by an absolutely" }{TEXT 240 64 "\nconvergent series. If the series does not converge uniformly in" }{TEXT 240 66 "\nthis sense, an error message is returned. Otherwise the output is" }{TEXT 240 65 "\nthe li mit n->infinity of the hypergeometric series. If there are" }{TEXT 240 63 "\nconvergence conditions on the parameters, these conditions a re" }{TEXT 240 65 "\nreturned as a warning unless the option \"conditi ons\" is present." }{TEXT 240 32 "\nThe command with all options is" } {TEXT 240 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#IVuniformconvergen ce(term,~k,~n,~conditions,~ucrelaxed)G6\"" }}{PARA 250 "" 1 "" {TEXT 240 61 "If \"conditions\" is present, then the convergence restriction s" }{TEXT 240 66 "\nare returned in the output sequence after the comp uted recurrence" }{TEXT 240 60 "\nIf \"ucrelaxed\" is present, then in the case that the series" }{TEXT 240 65 "\ncan be majorized by a sum \+ of O(k^p) with -1<=p<0, no error but a" }{TEXT 240 65 "\nbut a warning message is generated, and a conjectural expression" }{TEXT 240 13 "\n is returned." }}}}{SECT 1 {PARA 251 "" 0 "" {TEXT 241 18 "Classical th eorems" }}{SECT 1 {PARA 252 "" 0 "" {TEXT 242 13 "Gauss theorem" }} {SECT 0 {PARA 253 "" 0 "" {TEXT 243 19 "Recurrence relation" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 60 " infsumrecursion( hyperterm( [a ,b], [c+n], 1, k), k, s(n) );" }}{PARA 254 "" 1 "" {TEXT 218 81 "Warni ng, The condition(s) for uniformly bounded convergence are: 0 < Re(-a- b+c+n)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,&I \"cG6\"\"\"\"I\"nGF(F)F),*I\"aGF(!\"\"I\"bGF(F-F'F)F*F)F)-I\"sGF(6#F*F )F)*(,(F'F)F*F)F.F-F),(F'F)F*F)F,F-F)-F06#,&F*F)F)F)F)F-\"\"!" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 71 "infsumrecursion( hyperte rm( [a,b], [c+n], 1, k), k, s(n), conditions );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6$/,&*(,&I\"cG6\"\"\"\"I\"nGF(F)F),*I\"aGF(!\"\"I\"bGF(F- F'F)F*F)F)-I\"sGF(6#F*F)F)*(,(F'F)F*F)F.F-F),(F'F)F*F)F,F-F)-F06#,&F*F )F)F)F)F-\"\"!2F8-I#ReGI*protectedGF<6#F+" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 69 "infsumrecursion( hyperterm( [a,b], [c], 1, k), k , f(c), conditions );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6$/,&*(,(I\"aG 6\"\"\"\"I\"bGF(F)I\"cGF(!\"\"F)F+F)-I\"fGF(6#F+F)F)*(,&F*F)F+F,F),&F+ F,F'F)F)-F.6#,&F+F)F)F)F)F)\"\"!2-I#ReGI*protectedGF:6#F&F6" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 70 "infsumrecursion( hyperte rm( [a,b], [c], 1, k), k, f(a), certificate );" }}{PARA 254 "" 1 "" {TEXT 218 80 "Warning, The condition(s) for uniformly bounded converge nce are: Re(a+b-c) <= -1" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6$/,&*&,(I\"cG6\"!\"\"I\"aGF(\"\"\"F+F+F+-I\"fGF(6#F*F+F+ *&,*F*F+I\"bGF(F+F'F)F+F+F+-F-6#,&F*F+F+F+F+F)\"\"!,$*(,(F'F+I\"kGF(F+ F)F+F+F9F+F*F)F)" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 59 "inf sumrecursion( hyperterm( [a+n,b], [c], 1, k), k, f(n) );" }}{PARA 254 "" 1 "" {TEXT 218 82 "Warning, The condition(s) for uniformly bounded \+ convergence are: Re(a+n+b-c) <= -1" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*&,*I\"nG6\"\"\"\"I\"cGF(!\"\"I\"aGF(F)F)F)F)-I \"fGF(6#F'F)F)*&,,F,F)F'F)I\"bGF(F)F*F+F)F)F)-F.6#,&F'F)F)F)F)F+\"\"!" }}}{EXCHG {PARA 255 "> " 0 "" }}}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 11 "Closed form" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infcl osedform( hyperterm( [a,b], [c+n], 1, k), k, n );" }}{PARA 254 "" 1 "" {TEXT 218 81 "Warning, The condition(s) for uniformly bounded converg ence are: 0 < Re(-a-b+c+n)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#**-I&GAMMAG6$I*protectedGF'I(_syslibG6\"6#,(I\"bGF)!\" \"I\"cGF)\"\"\"I\"nGF)F/F--F%6#,(F0F/F.F/I\"aGF)F-F--F%6#,&F.F/F0F/F/- F%6#,*F4F-F,F-F.F/F0F/F/" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 69 "infclosedform( hyperterm( [a,b], [c+n], 1, k), k, n=0, certifi cate );" }}{PARA 254 "" 1 "" {TEXT 218 79 "Warning, The condition(s) f or uniformly bounded convergence are: 0 < -Re(a+b-c)" }{TEXT 218 1 "\n " }}{PARA 247 "" 1 "" {XPPMATH -1 "6$**-I&GAMMAG6$I*protectedGF'I(_sys libG6\"6#,&I\"bGF)!\"\"I\"cGF)\"\"\"F--F%6#,&F.F/I\"aGF)F-F--F%6#F.F/- F%6#,(F3F-F,F-F.F/F/,$*&,&F.F/I\"nGF)F/F/I\"kGF)F/F-" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 50 "infclosedform( hyperterm( [a,b], [c], 1, k), k, c," }{MPLTEXT 1 239 14 "\nconditions );" }}{PARA 247 "" 1 " " {XPPMATH -1 "6$**-I&GAMMAG6$I*protectedGF'I(_syslibG6\"6#,&I\"bGF)! \"\"I\"cGF)\"\"\"F--F%6#,&F.F/I\"aGF)F-F--F%6#F.F/-F%6#,(F3F-F,F-F.F/F /2-I#ReGF'6#,(F3F/F,F/F.F-\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform( hyperterm( [a+n,b], [c], 1, k), k, a );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconvergence) \+ The series does not converge for large n, unless it is terminating" } {TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 56 "infc losedform( hyperterm( [a+n,b], [c+n], 1, k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 81 "Error, (in iHPGsumOkpower) The series does not conv erge, unless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 58 "infclosedform( hyperterm( [a+n,b], [c+2*n ], 1, k), k, n );" }}{PARA 254 "" 1 "" {TEXT 218 81 "Warning, The cond ition(s) for uniformly bounded convergence are: 0 < Re(-a-b+c+n)" } {TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#**-I&GAMMAG6$I*pro tectedGF'I(_syslibG6\"6#,&I\"cGF)\"\"\"I\"nGF)\"\"#F--F%6#,(I\"bGF)!\" \"F,F-F.F/F4-F%6#,(F.F-F,F-I\"aGF)F4F4-F%6#,*F8F4F3F4F,F-F.F-F-" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform( hyperterm ( [a,b], [c-n], 1, k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Erro r, (in iHPGuniconvergence) The series does not converge for large n, u nless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 56 "infclosedform( hyperterm( [a-n,b], [c+n], 1, k), k, n );" }}{PARA 254 "" 1 "" {TEXT 218 94 "Warning, The condition(s) \+ for uniformly bounded convergence are: 0 < Re(-b+2*n+c-a), Re(b) < 0" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#**-I&GAMMAG6$I*pr otectedGF'I(_syslibG6\"6#,*I\"bGF)!\"\"I\"nGF)\"\"#I\"cGF)\"\"\"I\"aGF )F-F1-F%6#,(F2F-F.F/F0F1F--F%6#,(F,F-F0F1F.F1F--F%6#,&F0F1F.F1F1" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 56 "infclosedform( hyperterm ( [a+n,b], [c-n], 1, k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Er ror, (in iHPGuniconvergence) The series does not converge for large n, unless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 58 "infclosedform( hyperterm( [a+n,b], [c-2*n], 1, k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconv ergence) The series does not converge for large n, unless it is termin ating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 60 "infclosedform( hyperterm( [a+2*n,b], [c+3*n], 1, k), k, n );" }} {PARA 254 "" 1 "" {TEXT 218 81 "Warning, The condition(s) for uniforml y bounded convergence are: 0 < Re(-a-b+c+n)" }{TEXT 218 1 "\n" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#*4)\"\"$I\"bG6\"\"\"\"-I&GAMMAG6$I*pr otectedGF,I(_syslibGF'6#,*F&#!\"\"F%#F(F%F(I\"nGF'F(I\"cGF'F2F1-F*6#,* F&F0#\"\"#F%F(F3F(F4F2F1-F*6#,(F&F0F3F(F4F2F1-F*6#,(F3F(F4F(I\"aGF'F1F 1-F*6#,(F4F2F3F(F2F(F(-F*6#,(F4F2F3F(F8F(F(-F*6#,*F@F1F&F1F4F(F3F(F(-F *6#,&F4F2F3F(F(" }}}{EXCHG {PARA 255 "> " 0 "" }}}}{SECT 0 {PARA 252 " " 0 "" {TEXT 242 16 "Kummer's theorem" }}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 20 "Recurrence relations" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 66 "infsumrecursion( hyperterm([a+2*n,b],[1+a+2*n-b],-1 ,k), k, s(n) );" }}{PARA 254 "" 1 "" {TEXT 218 76 "Warning, The condit ion(s) for uniformly bounded convergence are: Re(b) < 1/2" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,*I\"bG6\"!\"\"I\"aGF( \"\"\"I\"nGF(\"\"#F+F+F+,*F'F)F*F+F,F-F-F+F+-I\"sGF(6#F,F+F+*(,(F*F+F, F-F+F+F+,*F,F-F*F+F'!\"#F-F+F+-F06#,&F,F+F+F+F+F)\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 62 "infsumrecursion( hyperterm([a,b +n],[1+a-b-n],-1,k), k, s(n) );" }}{PARA 254 "" 1 "" {TEXT 218 76 "War ning, The condition(s) for uniformly bounded convergence are: Re(b+n) \+ < 0" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*&,(I\"aG 6\"!\"\"I\"bGF(\"\"#I\"nGF(F+\"\"\"-I\"sGF(6#F,F-F-*&,(F*F-F,F-F'F)F-- F/6#,&F,F-F-F-F-!\"#\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 62 "infsumrecursion( hyperterm([a+n,b-n],[1+a-b],-1,k), k, s(n) ); " }}{PARA 254 "" 1 "" {TEXT 218 76 "Warning, The condition(s) for unif ormly bounded convergence are: Re(b) < 1/2" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,(**,&I\"bG6\"\"\"\"I\"nGF(F)F),(F'!\"\"F* F)F)F)F),*\"\"$F)F'F,F*\"\"#I\"aGF(F)F)-I\"sGF(6#F*F)F)*(,4*$F*F/F.*&F *F)F'F)!\"$*&F0F)F*F)F.F*\"\"'*$F0F/F)F0F.*&F0F)F'F)!\"#F'F=F/F)F),*F' F,F0F)F*F/F/F)F)-F26#,&F*F)F)F)F)F=**,(F0F)F*F)F)F)F),*F'F=F*F)F/F)F0F )F),*F'F,F0F)F*F/F)F)F)-F26#,&F*F)F/F)F)F)\"\"!" }}}{EXCHG {PARA 255 " > " 0 "" }}}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 11 "Closed form" }} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 61 "infclosedform( hyperterm ([a+2*n,b],[1+a+2*n-b],-1,k), k, n );" }}{PARA 254 "" 1 "" {TEXT 218 74 "Warning, The condition(s) for uniformly bounded convergence are: R e(b) < 0" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*.)\"\" #,$I\"aG6\"!\"\"\"\"\")\"\"%,$I\"nGF(F)F*I#PiGI*protectedGF0#F*F%-I&GA MMAG6$F0I(_syslibGF(6#,*I\"bGF(F)F'F*F.F%F*F*F*-F36#,(F'F1F.F*F1F*F)-F 36#,*F.F*F'F1F8F)F*F*F)" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 113 "infclosedform( hyperterm([a+n,b],[1+a+n-b],-1,k), k, n, condition s ); # The recurrence relation has second order!" }}{PARA 247 "" 1 "" {XPPMATH -1 "6$*,)\"\"#,&I\"nG6\"!\"\"I\"aGF(F)\"\"\"I#PiGI*protectedG F-#F+F%-I&GAMMAG6$F-I(_syslibGF(6#,*F+F+F*F+F'F+I\"bGF(F)F+-F06#,(F*F. F'F.F.F+F)-F06#,*F'F.F*F.F5F)F+F+F)2-I#ReGF-6#F5\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 53 "infclosedform( hyperterm([a,b], [1+a-b],-1,k), k, b );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in i HPGuniconvergence) The series does not converge for large n, unless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 61 "infclosedform( hyperterm( [a+n,b+n], [1+a-b], -1, k ), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconver gence) The series does not converge for large n, unless it is terminat ing" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 61 "infclosedform( hyperterm( [a,b+n], [1+a-b-n], -1, k), k, n );" }} {PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconvergence) The ser ies does not converge for large n, unless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 61 "infclosedf orm( hyperterm( [a,b-n], [1+a-b+n], -1, k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 94 "Error, (in iHPGoutconditions) The series does not con verge uniformly, unless it is terminating" }{TEXT 217 1 "\n" }}} {EXCHG {PARA 255 "> " 0 "" }}}}{SECT 0 {PARA 257 "" 0 "" {TEXT 244 31 "Bailey's and Gauss series for " }{XPPEDIT 18 0 "z = 1/2;" "6#/%\"zG* &\"\"\"F&\"\"#!\"\"" }}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 20 "Recurre nce relations" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 61 "infsumr ecursion( hyperterm( [a,1-a], [c], 1/2, k), k, f(c) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(I\"cG6\"\"\"\",&F&F(F(F(F(-I\"fGF'6#F&F(F(*( ,&I\"aGF'F(F&F(F(,(F&!\"\"F1F(F/F(F(-F+6#,&F&F(\"\"#F(F(F(\"\"!" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 61 "infsumrecursion( hyperte rm( [a,1-a], [c], 1/2, k), k, f(a) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*&,(I\"cG6\"!\"\"\"\"\"F*I\"aGF(F*F*-I\"fGF(6#F+F*F**&,&F+F*F 'F*F*-F-6#,&F+F*\"\"#F*F*F*\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 62 "infsumrecursion( hyperterm( [a+2*n,b], [(1+b+a)/2+n ], 1/2, k)," }{MPLTEXT 1 239 13 "\n k, f(n) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*&,*\"\"\"F'I\"bG6\"F'I\"aGF)F'I\"nGF)\"\"#F'-I\"fGF )6#F+F'F'*&,(F*F'F+F,F'F'F'-F.6#,&F+F'F'F'F'!\"\"\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 60 "infsumrecursion( hyperterm( [a+ n,b-n], [(1+b+a)/2], 1/2, k)," }{MPLTEXT 1 239 24 "\n k, f(n), conditi ons );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*&,(\"\"\"F'I\"bG6\"!\" \"I\"nGF)F'F'-I\"fGF)6#F+F'F'*&,(I\"aGF)F'F+F'F'F'F'-F-6#,&F+F'\"\"#F' F'F'\"\"!" }}}{EXCHG {PARA 255 "> " 0 "" }}}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 11 "Closed form" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 56 "infclosedform( hyperterm( [a,1-a], [c], 1/2, k), k, c );" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#*,I#PiGI*protectedGF%#\"\"\"\"\"#)F(, &F'F'I\"cG6\"!\"\"F'-I&GAMMAG6$F%I(_syslibGF,6#F+F'-F/6#,(F+F&F&F'I\"a GF,#F-F(F--F/6#,&F6F&F+F&F-" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 56 "infclosedform( hyperterm( [a,1-a], [c], 1/2, k), k, a );" }} {PARA 256 "" 1 "" {TEXT 217 80 "Error, (in iHPGuniconvergence) The lim it series does not exist/does not converge" }{TEXT 217 1 "\n" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 68 "infclosedform( hyperterm ( [a+2*n,b], [(1+b+a)/2+n], 1/2, k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconvergence) The series does not conve rge for large n, unless it is terminating" }{TEXT 217 1 "\n" }}} {EXCHG {PARA 255 "> " 0 "" }}}}{SECT 0 {PARA 252 "" 0 "" {TEXT 242 16 "Other 2F1 series" }}{PARA 258 "" 0 "" {TEXT 245 34 "Evaluation of two Gosper's series:" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 63 "inf closedform( hyperterm( [-a,1/2], [2*a+3/2], 1/4, k), k, a );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,$*,)\"\"%,$I\"aG6\"!\"\"\"\"\"I#PiGI*prote ctedGF-#F+\"\"#-I&GAMMAG6$F-I(_syslibGF)6#,&F(F/#\"\"$F/F+F+-F16#,&#\" \"(\"\"'F+F(F+F*-F16#,&F(F+#\"\"&F=F+F*#F/F7" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 63 "infclosedform( hyperterm( [a,1/2], [-2*a+3/2] , 1/4, k), k, a );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGu niconvergence) The series does not converge for large n, unless it is \+ terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 63 "infclosedform( hyperterm( [-a,1/2], [2*a+5/2], 1/4, k), k, a );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,$**)\"\"%,$I\"aG6\"!\"\"\"\" \"I#PiGI*protectedGF-#F+\"\"#-I&GAMMAG6$F-I(_syslibGF)6#,&F(F/#\"\"&F/ F+F+-F16#,&F(F+#\"\"$F/F+!\"##F+F<" }}}{PARA 259 "" 0 "" {TEXT 246 21 "Evaluation of 2F1(3-2" }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 246 8 ") series" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 68 "infclosedform( hyperterm( [3/2-c,1/2], [c], 3-2*sqrt(2), k), k, c \+ );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*.I#PiGI*protectedGF%#\"\"\"\" \"#,&\"\"%F'*$F(F&!\"##!\"\"F()F(,&F'F'I\"cG6\"F.F'-I&GAMMAG6$F%I(_sys libGF26#F1F'-F46#,&F1F&#\"\"$\"\")F'F.-F46#,&F1F&#F'F=F'F." }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 72 "infclosedform( hyperterm( [3/2- c-n,1/2], [c+n], 3-2*sqrt(2), k), k, c );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*.I#PiGI*protectedGF%#\"\"\"\"\"#,&\"\"%F'*$F(F&!\"##! \"\"F()F(,(I\"cG6\"F.F'F'I\"nGF2F.F'-I&GAMMAG6$F%I(_syslibGF26#,&F1F'F 3F'F'-F56#,(F3F&F1F&#\"\"$\"\")F'F.-F56#,(F3F&F1F&#F'F?F'F." }}}{PARA 258 "" 0 "" {TEXT 245 72 "We can even change the sign of n (i.e., the direction of the recursion)" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 72 "infclosedform( hyperterm( [3/2-c+n,1/2], [c-n], 3-2*sqrt(2), k ), k, c );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*.I#PiGI*protectedGF%# \"\"\"\"\"#,&\"\"%F'*$F(F&!\"##!\"\"F()F(,(I\"nG6\"F'I\"cGF2F.F'F'F'-I &GAMMAG6$F%I(_syslibGF26#,&F3F'F1F.F'-F56#,(#F'\"\")F'F3F&F1F-F.-F56#, (F1F-#\"\"$F>F'F3F&F." }}}{EXCHG {PARA 255 "> " 0 "" }}}{SECT 0 {PARA 252 "" 0 "" {TEXT 242 17 "Saalschutz series" }}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 20 "Recurrence relations" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 73 "infsumrecursion( hyperterm([ a,b,c+n], [e,a+b+c+n-e +1], 1, k), k, s(n) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,*I\"n G6\"\"\"\"I\"eGF(!\"\"I\"cGF(F)F)F)F),.I\"aGF(F)I\"bGF(F)F,F)F'F)F*F+F )F)F)-I\"sGF(6#F'F)F)*(,,F/F)F,F)F'F)F*F+F)F)F),,F.F)F,F)F'F)F*F+F)F)F )-F16#,&F)F)F'F)F)F+,$*,-I&GAMMAG6$I*protectedGF>I(_syslibGF(6#F*F)-F< 6#F/F+-F<6#F.F+-F<6#,.F.F)F/F)F,F)\"\"#F)F'F)F*F+F)-F<6#,(F,F)F)F)F'F) F+F+" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 58 "infsumrecursion ( hyperterm([ a,b,c], [e,a+b+c-e+1], 1, k)," }{MPLTEXT 1 239 12 "\n k, f(c) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,,I\"aG6\"\"\"\"I\"b GF(F)I\"cGF(F)I\"eGF(!\"\"F)F)F),(F,F)F+F-F-F)F)-I\"fGF(6#F+F)F)*(,*F* F)F+F)F,F-F)F)F),*F'F)F)F)F,F-F+F)F)-F06#,&F+F)F)F)F)F)*,-I&GAMMAG6$I* protectedGF " 0 "" {MPLTEXT 1 239 58 "i nfsumrecursion( hyperterm([ a,b,c], [e,a+b+c-e+1], 1, k)," }{MPLTEXT 1 239 12 "\n k, f(e) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*,I\"eG 6\"\"\"\",(F&!\"\"I\"cGF'F(I\"bGF'F(F(,(F&F*F+F(I\"aGF'F(F(,(F&F*F.F(F ,F(F(-I\"fGF'6#F&F(F(*,,&F+F*F&F(F(,&F&F*F,F(F(,&F&F*F.F(F(,*F.F(F,F(F +F(F&F*F(-F16#,&F&F(F(F(F(F**.,*F.F(F+F(F,F(F&!\"#F(-I&GAMMAG6$I*prote ctedGFAI(_syslibGF'6#,,F.F(F,F(F+F(F&F*F(F(F(-F?6#F,F*-F?6#F.F*-F?6#F+ F*-F?F9F(" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 58 "infsumrecu rsion( hyperterm([ a,b,c], [e,a+b+c-e-1], 1, k)," }{MPLTEXT 1 239 12 " \n k, f(c) );" }}{PARA 256 "" 1 "" {TEXT 217 93 "Error, (in iHPGsumrec ursion) The series does not converge uniformly, unless it is terminati ng" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 73 "infsumrecursion( hyperterm([ a+n,b-n,c], [e,a+b+c-e+1], 1, k), k, s(n ) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,*I\"nG6\"\"\"\"I\"eGF(F )I\"cGF(!\"\"I\"bGF(F,F),*I\"aGF(F)F'F)F)F)F*F,F)-I\"sGF(6#F'F)F)*(,(F -F,F'F)F*F)F),,F/F)F+F)F'F)F*F,F)F)F)-F16#,&F)F)F'F)F)F,,$*.,*F-F,F/F) F'\"\"#F)F)F)-I&GAMMAG6$I*protectedGF@I(_syslibGF(6#F*F)-F>6#,,F/F)F-F )F+F)F*F,F)F)F)-F>6#,&F-F)F'F,F,-F>6#F+F,-F>6#,(F/F)F)F)F'F)F,F," }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 58 "sumrecursion( hyperterm( [ a+n, b], [-5], z, k), k, s(n) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6 #/,(*&,(I\"nG6\"\"\"\"I\"aGF(F)\"\"'F)F)-I\"sGF(6#F'F)F)*&,0!\"(F)*&I \"zGF(F)I\"bGF(F)!\"\"*&F3F)F*F)F)F*!\"#F3F)*&F3F)F'F)F)F'F7F)-F-6#,&F )F)F'F)F)F)*(,(F*F)F)F)F'F)F),&F3F)F5F)F)-F-6#,&F'F)\"\"#F)F)F5\"\"!" }}}{EXCHG {PARA 255 "> " 0 "" }}}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 11 "Closed form" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 67 "infcl osedform( hyperterm([a,b,c+n], [e,a+b+c+n-e+1], 1, k), k, n );" }} {PARA 254 "" 1 "" {TEXT 218 79 "Warning, The condition(s) for uniforml y bounded convergence are: Re(-e+a+b) < 0" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,&*2-I&GAMMAG6$I*protectedGF(I(_syslibG6\"6 #I\"eGF*\"\"\"-F&6#,(F,F-I\"aGF*!\"\"I\"bGF*F2F--F&6#,&F,F-F1F2F2-F&6# ,&F,F-F3F2F2-F&6#,,F1F-I\"cGF*F-I\"nGF*F-F,F2F-F-F2-F&6#,,F>F-F-F-F,F2 F3F-F=F-F2-F&6#,*F>F-F=F-F,F2F-F-F--F&6#,.F1F-F3F-F=F-F>F-F,F2F-F-F-F- *0F%F--F&6#F3F2-F&6#F1F2-F&6#,(F=F-F-F-F>F-F2FDF2-I*HypergeomGF*6%7%F< FAF-7$,*\"\"#F-F>F-F=F-F,F2FOF-F-FEF-F-" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 69 "infclosedform( hyperterm([a,b,c+n], [e,a+b+c+n-e+1 ], 1, k), k, n=0 );" }}{PARA 254 "" 1 "" {TEXT 218 79 "Warning, The co ndition(s) for uniformly bounded convergence are: Re(-e+a+b) < 0" } {TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,&*2-I&GAMMAG6$I*p rotectedGF(I(_syslibG6\"6#I\"eGF*\"\"\"-F&6#,(F,F-I\"aGF*!\"\"I\"bGF*F 2F--F&6#,&F,F-F1F2F2-F&6#,&F,F-F3F2F2-F&6#,*F1F-I\"cGF*F-F-F-F,F2F2-F& 6#,*F-F-F,F2F3F-F=F-F2-F&6#,(F=F-F,F2F-F-F--F&6#,,F1F-F3F-F=F-F,F2F-F- F-F-*0F%F--F&6#F3F2-F&6#F1F2-F&6#,&F=F-F-F-F2FCF2-I*HypergeomGF*6%7%F@ F " 0 "" {MPLTEXT 1 239 63 "infclosedform( hyperterm([a,b,c], [e,a+b+c-e+1], 1, k), k, c );" }}{PARA 254 "" 1 "" {TEXT 218 79 "Warning, The condition (s) for uniformly bounded convergence are: Re(-e+a+b) < 0" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,&*2-I&GAMMAG6$I*protectedGF (I(_syslibG6\"6#I\"eGF*\"\"\"-F&6#,(F,F-I\"aGF*!\"\"I\"bGF*F2F--F&6#,& F,F-F1F2F2-F&6#,&F,F-F3F2F2-F&6#,*F1F-I\"cGF*F-F-F-F,F2F2-F&6#,*F-F-F, F2F3F-F=F-F2-F&6#,(F=F-F,F2F-F-F--F&6#,,F1F-F3F-F=F-F,F2F-F-F-F-*0-I*H ypergeomGF*6%7%F " 0 "" {MPLTEXT 1 239 67 "infclosedform( hyperterm([a,b,c-n], [e,a+b+c-n-e+1] , 1, k), k, n );" }}{PARA 254 "" 1 "" {TEXT 218 79 "Warning, The condi tion(s) for uniformly bounded convergence are: Re(-e+a+b) < 0" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,&*2-I&GAMMAG6$I*protect edGF(I(_syslibG6\"6#I\"eGF*\"\"\"-F&6#,(F,F-I\"aGF*!\"\"I\"bGF*F2F--F& 6#,&F,F-F1F2F2-F&6#,&F,F-F3F2F2-F&6#,,F1F2F3F2I\"cGF*F2I\"nGF*F-F,F-F2 -F&6#,(F=F2F>F-F,F-F2-F&6#,*F>F-F,F-F=F2F3F2F--F&6#,*F>F-F1F2F,F-F=F2F -F-*2F%F--F&6#,.F1F-F3F-F=F-F>F2F,F2F-F-F--F&6#F3F2-F&6#,&F=F-F>F2F2-F &6#F1F2FDF2FGF2-I*HypergeomGF*6%7%FA,(F>F-F=F2F-F-F-7$,,F-F-F>F-F,F-F= F2F3F2,,F-F-F>F-F1F2F,F-F=F2F-F-F2" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 63 "infclosedform( hyperterm([a,b,c], [e,a+b+c-e+1], 1, k), k, e );" }}{PARA 254 "" 1 "" {TEXT 218 78 "Warning, The condition (s) for uniformly bounded convergence are: Re(a+b+c) < 1" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,&*2-I&GAMMAG6$I*protectedGF (I(_syslibG6\"6#,*I\"aGF*!\"\"I\"bGF*F.I\"cGF*F.I\"eGF*\"\"\"F.-F&6#,& F0F.F1F2F.-F&6#,&F1F2F/F.F.-F&6#,&F1F2F-F.F.-F&6#,(F-F.F1F2F0F.F2-F&6# F1F2-F&6#,(F1F2F-F.F/F.F2-F&6#,(F1F2F0F.F/F.F2F2*6,*F-F2F/F2F0F2F1!\"# F2-I*HypergeomGF*6%7',,F-#F.\"\"#F/FOF0FOF1F2F2F2F;F8F5F27&,*F0F.F-F.F 1F2F2F2,*F1F2F2F2F-F.F/F.,*F-FOF/FOF0FOF1F2,*F0F.F1F2F2F2F/F.F.F2,(F0F 2F1F.F/F2F.,(F0F2F-F2F1F.F.,(F1F.F-F2F/F2F.-F&6#F0F.-F&6#,,F-F2F/F2F0F 2F1F.F2F2F2-F&6#F/F.F?F2-F&6#F-F.F2" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 71 "infclosedform( hyperterm([a+2*n,b,c], [e,a+2*n+b+c- e+1], 1, k), k, n );" }}{PARA 254 "" 1 "" {TEXT 218 78 "Warning, The c ondition(s) for uniformly bounded convergence are: Re(c-e+b) < 0" } {TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#,&*2-I&GAMMAG6$I*p rotectedGF(I(_syslibG6\"6#,,I\"nGF*\"\"#\"\"\"F/I\"bGF*F/I\"aGF*F/I\"e GF*!\"\"F3-F&6#,*F-F.F/F/F1F/F2F3F/-F&6#,,F-F.F/F/F1F/I\"cGF*F/F2F3F3- F&6#,(F2F/F:F3F0F3F/-F&6#,&F:F3F2F/F3-F&6#F2F/-F&6#,&F2F/F0F3F3-F&6#,. F1F/F-F.F0F/F:F/F2F3F/F/F/F/*4,H*&F0F/F:F/F/F0F/*&F1F/F0F/F/*&F-F/F0F/ F.*&F0F/F2F/F3\"\"$F/F-\"#5F1\"\"&F2!\"$F:F/*$F1F.F.*&F1F/F-F/\"\")*&F 1F/F:F/F/*&F1F/F2F/FR*$F-F.FU*&F-F/F:F/F.*&F-F/F2F/!\"'*&F:F/F2F/F3*$F 2F.F/F/-I*HypergeomGF*6%7),,F-F/F/F/F1#F/F.F:F]oF2#F3F.,,F-F/F/F/F0F]o F1F]oF2F^o,,F-F/F]oF/F1F]oF:F]oF2F^o,0F0#F/FUF1F]oF-F/F:FboF2#FRFU#\"# 8FUF/*$,6F/F/*$F0F.F/F0F.F2FenF:F.*$F:F.F/FKFenFNF.FfnF.FgnF/F]o#F3FU, 0F0FboF1F]oF-F/F:FboF2FcoFdoF/FfoFbo,,F-F/F]oF/F0F]oF1F]oF2F^oF/7(,(F1 F]oF-F/F/F/,*F-F/#FOF.F/F1F]oF2F^o,0F0FboF1F]oF-F/F:FboF2Fco#FQFUF/Ffo Fjo,0F0FboF1F]oF-F/F:FboF2FcoFbpF/FfoFbo,(F1F]oF-F/F`pF/,*F-F/F.F/F1F] oF2F^oF/F/,*F-F.F.F/F1F/F2F3F3F6F3FFF/-F&6#,(F1F/F-F.F.F/F3-F&6#F:F3-F &6#F0F3FAF/\"#;" }}}{EXCHG {PARA 255 "> " 0 "" }}}}{SECT 0 {PARA 252 " " 0 "" {TEXT 242 15 "Dixon's theorem" }}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 20 "Recurrence relations" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 66 "infsumrecursion( hyperterm([a+2*n,b,c],[1+a+2*n-b,1 +a+2*n-c],1,k)," }{MPLTEXT 1 239 12 "\n k, f(n) );" }}{PARA 254 "" 1 " " {TEXT 218 87 "Warning, The condition(s) for uniformly bounded conver gence are: -2 < Re(2*n+a-2*b-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*.,,I\"nG6\"\"\"#I\"aGF(\"\"\"F)F+I\"bGF(!\"#I\"c GF(F-F+,*F,!\"\"F*F+F'F)F+F+F+,*F,F0F*F+F'F)F)F+F+,*F+F+F*F+F'F)F.F0F+ ,*F)F+F*F+F'F)F.F0F+-I\"fGF(6#F'F+F+*.,(F*F+F'F)F+F+F+,*F.F-F)F+F*F+F' F)F+,*F,F-F*F+F'F)F)F+F+,,F,F0F)F+F*F+F'F)F.F0F+,,F,F0F+F+F*F+F'F)F.F0 F+-F56#,&F'F+F+F+F+F0\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 65 "infsumrecursion( hyperterm([a,b,c],[1+a-b,1+a-c],1,k), k, f(b) );" }}{PARA 254 "" 1 "" {TEXT 218 87 "Warning, The condition(s) for u niformly bounded convergence are: -2-Re(a-2*b-2*c) <= -2" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,&I\"bG6\"!\"#I\"aGF(\" \"\"F+,(F*F+F'!\"\"I\"cGF(F-F+-I\"fGF(6#F'F+F+*(,&F*F+F'F-F+,(F*F+F'F) F.F)F+-F06#,&F'F+F+F+F+F-\"\"!" }}}{EXCHG {PARA 255 "> " 0 "" }}} {SECT 0 {PARA 253 "" 0 "" {TEXT 243 11 "Closed form" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 64 "infclosedform( hyperterm([a+2*n,b,c], [1+a+2*n-b,1+a+2*n-c],1,k)," }{MPLTEXT 1 239 9 "\n k, n );" }}{PARA 254 "" 1 "" {TEXT 218 87 "Warning, The condition(s) for uniformly boun ded convergence are: -2 < Re(2*n-2*b+a-2*c)" }{TEXT 218 1 "\n" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#*6)\"\"#,$I\"aG6\"!\"\"\"\"\")\"\"%,$ I\"nGF(F)F*I#PiGI*protectedGF0#F*F%-I&GAMMAG6$F0I(_syslibGF(6#,,I\"bGF (F)F*F*F'F*F.F%I\"cGF(F)F)-F36#,*F.F*F'F1F8F)F*F*F)-F36#,(F'F1F.F*F1F* F)-F36#,,F.F*F'F1F*F*F8F)F9F)F*-F36#,*F9F)F*F*F'F1F.F*F)-F36#,*F8F)F'F *F.F%F*F*F*-F36#,*F*F*F'F*F.F%F9F)F*" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 60 "infclosedform( hyperterm([a,b,c],[1+a-b,1+a-c],1,k) , k, b );" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconverg ence) The series does not converge for large n, unless it is terminati ng" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 255 "> " 0 "" }}}}{SECT 0 {PARA 252 "" 0 "" {TEXT 242 29 "Watson's and Whipple's series" }}{SECT 0 {PARA 253 "" 0 "" {TEXT 243 20 "Recurrence relations" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 67 "infsumrecursion( hyperterm([a,b,c],[( a+b+1)/2,2*c],1,k), k, f(c) );" }}{PARA 254 "" 1 "" {TEXT 218 80 "Warn ing, The condition(s) for uniformly bounded convergence are: Re(a+b-2* c) < 1" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,&I \"cG6\"\"\"#\"\"\"F*F*,*I\"aGF(F*I\"bGF(F*F'!\"#!\"\"F*F*-I\"fGF(6#F'F *F**(,(F-F*F'F.F/F*F*,(F'F.F/F*F,F*F*-F16#,&F'F*F*F*F*F*\"\"!" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 62 "infsumrecursion( hyperte rm([a+2*n,b,c],[(a+b+1)/2+n,2*c],1,k)," }{MPLTEXT 1 239 12 "\n k, f(n) );" }}{PARA 254 "" 1 "" {TEXT 218 95 "Warning, The condition(s) for u niformly bounded convergence are: -1/2+1/2*Re(a+2*n+b-2*c) <= -1" } {TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,*I\"aG6\"\" \"\"I\"nGF(\"\"#F)F)I\"cGF(!\"#F),*F)F)I\"bGF(F)F'F)F*F+F)-I\"fGF(6#F* F)F)*(,(F'F)F*F+F)F)F),,F,F-F)F)F/F)F'F)F*F+F)-F16#,&F*F)F)F)F)!\"\"\" \"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 67 "infsumrecursion( hyperterm([a,1-a,(e+f-1)/2],[e,f],1,k), k, f(a) );" }}{PARA 254 "" 1 "" {TEXT 218 76 "Warning, The condition(s) for uniformly bounded conve rgence are: 1 < Re(e+f)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*(,(I\"aG6\"\"\"\"F)F)I\"eGF(!\"\"F),(F'F+I\"fGF(F)F +F)F)-F-6#F'F)F)*(,&F'F)F*F)F),&F'F)F-F)F)-F-6#,&F'F)\"\"#F)F)F)\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 60 "infsumrecursion( hyp erterm([a,1-a,(e+f-1)/2],[e+n,f-n],1,k)," }{MPLTEXT 1 239 12 "\n k, f( n) );" }}{PARA 254 "" 1 "" {TEXT 218 76 "Warning, The condition(s) for uniformly bounded convergence are: 1 < Re(e+f)" }{TEXT 218 1 "\n" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*,,(I\"eG6\"\"\"\"I\"nGF(F)F)F)F), &F'F)F*F)F),*I\"aGF(F)!\"#F)I\"fGF(F)F*!\"\"F),*F-F0F0F)F/F)F*F0F)-F/6 #F*F)F)*,,*F'F)F*F)F)F)F-F0F),(F*F)F-F)F'F)F),(F0F)F/F)F*F0F),(F.F)F/F )F*F0F)-F/6#,&F*F)\"\"#F)F)F0\"\"!" }}}{EXCHG {PARA 255 "> " 0 "" }}} {SECT 0 {PARA 253 "" 0 "" {TEXT 243 11 "Closed form" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 62 "infclosedform( hyperterm([a,b,c],[(a+ b+1)/2,2*c],1,k), k, c );" }}{PARA 254 "" 1 "" {TEXT 218 80 "Warning, \+ The condition(s) for uniformly bounded convergence are: Re(a+b-2*c) < \+ 1" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*2I#PiGI*prote ctedGF%#\"\"\"\"\"#-I&GAMMAG6$F%I(_syslibG6\"6#,(F&F'I\"bGF-F&I\"aGF-F &F'-F*6#,&F&F'F1F&!\"\"-F*6#,&F&F'F0F&F5-F*6#,(I\"cGF-F'F&F'F1#F5F(F5- F*6#,(F0F=F " 0 "" {MPLTEXT 1 239 60 "infclosedform( hyperterm ([a,b,c],[1+a-b,1+a-c],1,k), k, a );" }}{PARA 254 "" 1 "" {TEXT 218 100 "Warning, The condition(s) for uniformly bounded convergence are: \+ -2 < Re(a-2*b-2*c), -1 < Re(-b-c+a)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*4)\"\"#,$I\"aG6\"!\"\"\"\"\"I#PiGI*protectedGF,#F *F%-I&GAMMAG6$F,I(_syslibGF(6#,*I\"cGF(F)F*F*F'F*I\"bGF(F)F)-F/6#,(F*F *F'F*F5F)F*-F/6#,&F-F*F'F-F)-F/6#,(F*F*F'F*F4F)F*-F/6#,*F4F)F*F*F'F-F5 F)F*-F/6#,(F*F*F'F-F4F)F)-F/6#,(F*F*F'F-F5F)F)" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 70 "infclosedform( hyperterm([a,1+a/2,b,c],[a/2 ,1+a-b,1+a-c],1,k), k, a );" }}{PARA 254 "" 1 "" {TEXT 218 104 "Warnin g, The condition(s) for uniformly bounded convergence are: -1 < Re(a-2 *b-2*c), (-1)/2 < Re(-b-c+a)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*4)\"\"#,$I\"aG6\"!\"\"\"\"\"I#PiGI*protectedGF,#F*F%-I &GAMMAG6$F,I(_syslibGF(6#,*I\"cGF(F)F*F*F'F*I\"bGF(F)F)-F/6#,(F*F*F'F* F5F)F*-F/6#,(F*F*F'F*F4F)F*-F/6#,*F-F*F'F-F4F)F5F)F*-F/6#,(F'F-F-F*F5F )F)-F/6#,(F'F-F4F)F-F*F)-F/6#,&F*F*F'F-F)" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 39 "infclosedform( hyperterm([a,1+a/2,b,c]," } {MPLTEXT 1 239 33 "\n[a/2,1+a-b,1+a-c],-1,k), k, a );" }}{PARA 254 "" 1 "" {TEXT 218 84 "Warning, The condition(s) for uniformly bounded con vergence are: -1 < Re(-2*b+a-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#**-I&GAMMAG6$I*protectedGF'I(_syslibG6\"6#,&I\"aGF) \"\"\"F-F-!\"\"-F%6#,*I\"cGF)F.F-F-F,F-I\"bGF)F.F.-F%6#,(F-F-F,F-F3F.F --F%6#,(F-F-F,F-F2F.F-" }}}{PARA 258 "" 0 "" {TEXT 245 25 "Maple knows this as well:" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 34 "simpli fy( hypergeom([a,1+a/2,b,c]," }{MPLTEXT 1 239 26 "\n[a/2,1+a-b,1+a-c], -1 ) );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#**-I&GAMMAG6$I*protectedGF 'I(_syslibG6\"6#,&I\"aGF)\"\"\"F-F-!\"\"-F%6#,*I\"cGF)F.F-F-F,F-I\"bGF )F.F.-F%6#,(F-F-F,F-F3F.F--F%6#,(F-F-F,F-F2F.F-" }}}{EXCHG {PARA 255 " > " 0 "" }}}}}{PARA 260 "" 0 "" }{SECT 1 {PARA 251 "" 0 "" {TEXT 241 19 "Unidentified 6F5(1)" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform(hyperterm([a,a/2+1,b,c,1+a/2+d,1+a/2-d]," }{MPLTEXT 1 239 42 "\n[a/2,1+a-b,1+a-c,a/2+d,a/2-d],-1,k),k,a);" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, The condition(s) for uniformly bounded con vergence are: 1 < Re(-2*b+a-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 " " {XPPMATH -1 "6#*.,(*$I\"aG6\"\"\"#\"\"\"*&I\"bGF'F)I\"cGF'F)!\"%*$I \"dGF'F(F-F)-I&GAMMAG6$I*protectedGF3I(_syslibGF'6#,&F&F)F)F)!\"\"-F16 #,*F,F7F)F)F&F)F+F7F7-F16#,(F)F)F&F)F+F7F)-F16#,(F)F)F&F)F,F7F),&F%F)F .F-F7" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform( hyperterm([a,a/2+1,b,c,1+a/2+d,1+a/2-d]," }{MPLTEXT 1 239 42 "\n[a/2,1 +a-b,1+a-c,a/2+d,a/2-d],-1,k),k,c);" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconvergence) The series does not converge for large n, unless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform(hyperterm([a,a/2+1,b,c,1+a/2+ d,1+a/2-d]," }{MPLTEXT 1 239 42 "\n[a/2,1+a-b,1+a-c,a/2+d,a/2-d],-1,k) ,k,d);" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, The condition(s) fo r uniformly bounded convergence are: 1 < Re(-2*b+a-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*.,(*$I\"aG6\"\"\"#\"\"\"*&I\" bGF'F)I\"cGF'F)!\"%*$I\"dGF'F(F-F)-I&GAMMAG6$I*protectedGF3I(_syslibGF '6#,&F&F)F)F)!\"\"-F16#,*F,F7F)F)F&F)F+F7F7-F16#,(F)F)F&F)F+F7F)-F16#, (F)F)F&F)F,F7F),&F%F)F.F-F7" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform(hyperterm([a,a/2+1,b,c,1+a/2+d,1+a/2-d]," } {MPLTEXT 1 239 41 "\n[a/2,1+a-b,1+a-c,a/2+d,a/2-d],1,k),k,a);" }} {PARA 254 "" 1 "" {TEXT 218 101 "Warning, The condition(s) for uniform ly bounded convergence are: 1 < Re(-2*b+a-2*c), 1/2 < Re(-b+a-c)" } {TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*8)\"\"#,&!\"\"\" \"\"I\"aG6\"F'F(,6*$F)\"\"$F(*$F)F%F'*&F)F%I\"cGF*F(!\"#*&F)F%I\"bGF*F (F1*&F)F(I\"dGF*F%!\"%*(F)F(F3F(F0F(\"\"%*$F5F%F8*&F0F(F5F%\"\")*&F3F( F5F%F;*&F3F(F0F(F8F(I#PiGI*protectedGF?#F(F%-I&GAMMAG6$F?I(_syslibGF*6 #,(F(F(F)F(F3F'F(-FB6#,(F(F(F)F(F0F'F(-FB6#,*F0F'F(F(F)F(F3F'F'-FB6#,* F)F@F3F'F0F'#F'F%F(F(-FB6#,(F)F@F0F'F@F(F'-FB6#,(F)F@F@F(F3F'F'-FB6#,& F(F(F)F@F',&F.F(F9F6F'" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform(hyperterm([a,a/2+1,b,c,1+a/2+d,1+a/2-d]," }{MPLTEXT 1 239 41 "\n[a/2,1+a-b,1+a-c,a/2+d,a/2-d],1,k),k,b);" }}{PARA 256 "" 1 "" {TEXT 217 97 "Error, (in iHPGuniconvergence) The series does not \+ converge for large n, unless it is terminating" }{TEXT 217 1 "\n" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform(hyperterm( [a,a/2+1,b,c,1+a/2+d,1+a/2-d]," }{MPLTEXT 1 239 41 "\n[a/2,1+a-b,1+a-c ,a/2+d,a/2-d],1,k),k,d);" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, T he condition(s) for uniformly bounded convergence are: 1 < Re(-2*b+a-2 *c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*8)\"\"#,&! \"\"\"\"\"I\"aG6\"F'F(,6*$F)\"\"$F(*$F)F%F'*&F)F%I\"cGF*F(!\"#*&F)F%I \"bGF*F(F1*&F)F(I\"dGF*F%!\"%*(F)F(F3F(F0F(\"\"%*$F5F%F8*&F0F(F5F%\"\" )*&F3F(F5F%F;*&F3F(F0F(F8F(I#PiGI*protectedGF?#F(F%-I&GAMMAG6$F?I(_sys libGF*6#,(F(F(F)F(F3F'F(-FB6#,(F(F(F)F(F0F'F(-FB6#,*F0F'F(F(F)F(F3F'F' -FB6#,*F)F@F3F'F0F'#F'F%F(F(-FB6#,(F)F@F0F'F@F(F'-FB6#,(F)F@F@F(F3F'F' -FB6#,&F(F(F)F@F',&F.F(F9F6F'" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 105 "infsumrecursion(hyperterm([-a,-a/2+1,b,-c,1-a/2+d,1-a/2-d], [-a/2,1-a-b,1-a+c,-a/2-d,-a/2+d],1,k),k,s(a));" }}{PARA 254 "" 1 "" {TEXT 218 86 "Warning, The condition(s) for uniformly bounded converge nce are: 1+Re(a+2*b-2*c) <= -2" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&*4I\"aG6\"\"\"\",&F&F(I\"dGF'!\"#F(,&F&F(F*\"\"#F(,( F&F(F(F(I\"cGF'F+F(,(F(F(F&F(I\"bGF'F-F(,*F&F(F(F(F1F(F/!\"\"F(,(F&F(F 1F(F/F3F(,B*$F&\"\"$F(*&F&F-F1F(F-*&F&F-F/F(F+*$F&F-\"\"(*(F&F(F1F(F/F (!\"%*&F&F(F/F(!\")*&F&F(F1F(\"\")F&\"#;*&F&F(F*F-F=*&F/F(F*F-FAF/F?*$ F*F-!#7F1FA\"#7F(*&F1F(F/F(F=*&F1F(F*F-F?F(-I\"sGF'6#F&F(F(*4,(F(F(F&F (F/F3F(,&F&F(F/F3F(,(F&F(F-F(F*F-F(,(F-F(F&F(F*F+F(,(F&F(F(F(F1F(F(,&F &F(F1F(F(,*F&F(F/F+F1F-F7F(F(,6F6F(F8F-F9F+F:F(FCF=F " 0 "" {MPLTEXT 1 239 105 "infsumrecursion(hyperterm([-a,-a/2+1,b,-c,1-a/2+d, 1-a/2-d],[-a/2,1-a-b,1-a+c,-a/2-d,-a/2+d],1,k),k,s(b));" }}{PARA 254 " " 1 "" {TEXT 218 86 "Warning, The condition(s) for uniformly bounded c onvergence are: 1+Re(a+2*b-2*c) <= -2" }{TEXT 218 1 "\n" }}{PARA 247 " " 1 "" {XPPMATH -1 "6#/,&**,(\"\"\"F'I\"aG6\"F'I\"bGF)\"\"#F',(F(F'F*F 'I\"cGF)!\"\"F',:*$F(\"\"$F'*&F(F+F-F'!\"#*&F(F+F*F'F+*$F(F+F1*&F(F'I \"dGF)F+!\"%*(F(F'F*F'F-F'F8*&F(F'F-F'F8*&F*F'F7F+!\")*&F*F'F-F'\"\"%* &F-F'F7F+\"\")*$F7F+!#7F-F>F'-I\"sGF)6#F*F'F'**,&F(F'F*F'F',*F(F'F-F3F *F+F1F'F',6F0F'F4F+F2F3F5F'F6F8F9F8FAF8F=F>F;F " 0 "" {MPLTEXT 1 239 105 "infsumrecur sion(hyperterm([-a,-a/2+1,b,-c,1-a/2+d,1-a/2-d],[-a/2,1-a-b,1-a+c,-a/2 -d,-a/2+d],1,k),k,s(c));" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, T he condition(s) for uniformly bounded convergence are: Re(a+2*b-2*c) < -1" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#/,&**,(!\"\" \"\"\"I\"aG6\"F(I\"cGF*F'F(,*I\"bGF*\"\"#F)F(F(F(F+!\"#F(,:*$F)\"\"$F( *&F)F.F+F(F/*&F)F.F-F(F.*$F)F.F'*(F)F(F-F(F+F(!\"%*&F)F(I\"dGF*F.F7*&F )F(F-F(F7*&F-F(F9F.!\")*&F-F(F+F(\"\"%*&F+F(F9F.\"\")*$F9F.F>F-F>F(-I \"sGF*6#F+F(F(**,(F)F(F'F(F+F/F(,*F'F(F)F(F-F(F+F'F(,6F1F(F4F.F3F/F5F( F8F7F6F7FAF7F=F>F;F " 0 "" {MPLTEXT 1 239 105 "infsumrecursion(hyperterm([-a,-a/2+1 ,b,-c,1-a/2+d,1-a/2-d],[-a/2,1-a-b,1-a+c,-a/2-d,-a/2+d],1,k),k,s(d));" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, The condition(s) for unifo rmly bounded convergence are: Re(a+2*b-2*c) < -1" }{TEXT 218 1 "\n" }} {PARA 247 "" 1 "" {XPPMATH -1 "6#/,&**,&I\"aG6\"\"\"\"I\"dGF(!\"#F),&F 'F)F*\"\"#F),F*$F'F-F)*&F'F-I\"bGF(F)F-*&F'F-I\"cGF(F)F+*$F'\"\"$F)!\" %F)F1!\")F3\"\")F'F6*$F*F-F6*&F1F)F*F-F7*&F3F)F*F-F8*&F'F)F*F-F6F*F7*& F1F)F*F)!#;*&F3F)F*F)\"#;*&F'F)F*F)F7*(F'F)F1F)F3F)F6*&F1F)F3F)\"\"%F) -I\"sGF(6#F*F)F)**,(F+F)F'F)F*F+F),(F'F)F-F)F*F-F),6F4F)F0F-F2F+F/F)F< F6FBF6F9F6FCFDF:F7F;F8F)-FF6#,&F)F)F*F)F)!\"\"\"\"!" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 54 "infclosedform(hyperterm([a,a/2+1,b,c, 1+a/2+d,1+a/2-d]," }{MPLTEXT 1 239 42 "\n[a/2,1+a-b,1+a-c,a/2+d,a/2-d] ,-1,k),k,a);" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, The condition (s) for uniformly bounded convergence are: 1 < Re(-2*b+a-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#*.,(*$I\"aG6\"\"\"#\"\" \"*&I\"bGF'F)I\"cGF'F)!\"%*$I\"dGF'F(F-F)-I&GAMMAG6$I*protectedGF3I(_s yslibGF'6#,&F&F)F)F)!\"\"-F16#,*F,F7F)F)F&F)F+F7F7-F16#,(F)F)F&F)F+F7F )-F16#,(F)F)F&F)F,F7F),&F%F)F.F-F7" }}}{EXCHG {PARA 255 "> " 0 "" }} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 56 "infsumrecursion(hyperter m([a,a/2+1,b,c,1+a/2+d,1+a/2-d]," }{MPLTEXT 1 239 47 "\n[a/2,1+a-b,1+a -c,a/2+d,a/2-d],1,k),k,s(a))[1];" }}{PARA 254 "" 1 "" {TEXT 218 83 "Wa rning, The condition(s) for uniformly bounded convergence are: 1 < Re( -2*b+a-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#&/,& *4,B*$I\"aG6\"\"\"$\"\"\"*&F)\"\"#I\"bGF*F,!\"#*$F)F.\"\"&*&F)F.I\"cGF *F,F0*&F)F,I\"dGF*F.!\"%*(F)F,F/F,F4F,\"\"%F)\"\")*&F)F,F/F,!\")*&F)F, F4F,FF9F,-FL6#FOF,FD\"\"!6#F," }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 99 "infsumrecursion(hyperterm([a,a/2+1,b,c,1+a/2+d,1+a/2-d],[a/2,1 +a-b,1+a-c,a/2-d,a/2+d],1,k),k,s(a));" }}{PARA 254 "" 1 "" {TEXT 218 83 "Warning, The condition(s) for uniformly bounded convergence are: 1 < Re(-2*b+a-2*c)" }{TEXT 218 1 "\n" }}{PARA 247 "" 1 "" {XPPMATH -1 " 6#/,&*4,B*$I\"aG6\"\"\"$\"\"\"*&F(\"\"#I\"bGF)F+!\"#*$F(F-\"\"&*&F(F-I \"cGF)F+F/*&F(F+I\"dGF)F-!\"%*(F(F+F.F+F3F+\"\"%F(\"\")*&F(F+F.F+!\")* &F(F+F3F+F;F.F;F3F;*&F.F+F3F+\"#7*&F.F+F5F-F9*&F3F+F5F-F9*$F5F-F6F8F+F +,*F.F/!\"\"F+F(F+F3F/F+,(F-F+F(F+F3FCF+,(F-F+F(F+F.FCF+,(F+F+F(F+F.FC F+,(F+F+F(F+F3FCF+,&F(F+F5F-F+,&F(F+F5F/F+-I\"sGF)6#F(F+F+*4,&F(F+F-F+ F+,(F(F+F-F+F5F-F+,(F(F+F-F+F5F/F+,(F(F+F+F+F3F/F+,(F+F+F(F+F.F/F+,*F( F+F-F+F.FCF3FCF+,*F3FCF+F+F(F+F.FCF+,6F'F+F0FCF2F/F,F/F4F6F7F8FAF8F@F9 F?F9F=F8F+-FK6#FNF+FC\"\"!" }}}{EXCHG {PARA 255 "> " 0 "" }}}{PARA 260 "" 0 "" }{SECT 1 {PARA 251 "" 0 "" {TEXT 241 19 "Uniform convergen ce" }}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 35 "uniformconvergence ( 2^(-k), k, n );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#\"\"#" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 37 "uniformconvergence( n*2^ (-k), k, n );" }}{PARA 256 "" 1 "" {TEXT 217 95 "Error, (in iHPGunicon vergence) The series does not converge uniformly, unless it is termina ting" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 43 "uniformconvergence( n/(n+1)*2^(-k), k, n );" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#\"\"#" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 38 "uniformconvergence( n*2^(-k), k, n ); " }}{PARA 256 "" 1 "" {TEXT 217 95 "Error, (in iHPGuniconvergence) The series does not converge un iformly, unless it is terminating" }{TEXT 217 1 "\n" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 239 12 "op(1,[%]);a;" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#\"\"#" }}{PARA 247 "" 1 "" {XPPMATH -1 "6#I\"aG6\"" }}} {EXCHG {PARA 261 "> " 0 "" }}}{PARA 262 "" 0 "" }{PARA 263 "" 0 "" } {PARA 264 "" 0 "" }{PARA 264 "" 0 "" }{PARA 264 "" 0 "" }{PARA 265 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }