{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 } {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 "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 "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 279 "This Maple worksheet acco mpanies the paper \n\nWolfram Koepf, Dieter Schmersau: Positivity and \+ Monotony Properties of the de Branges Functions (2003)\n\nand contains the Maple computations for some of the theorems in the paper. Details on the algorithms used can be found in the book" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Wolfram Koepf: Hypergeome tric Summation. Vieweg, Braunschweig/Wiesbaden, 1998." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "with(sumtools);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 7,%)HypersumG%+SumtohyperG%0extended_gosperG%'gosperG%/hyperrecursionG %)hypersumG%*hypertermG%)simpcombG%-sumrecursionG%+sumtohyperG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "tauterm:=y^k*binomial(n+k+1, 2*k+1)*hyperterm([k+1/2,n+k+2,k,k-n],[k+1,2*k+1,k+3/2],y,j);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(tautermG*8)%\"yG%\"kG\"\"\"-%)binomialG6$ ,(%\"nGF)F(F)F)F),&*&\"\"#F)F(F)F)F)F)F)-%+pochhammerG6$,&F(F)#F)F1F)% \"jGF)-F36$,(F.F)F(F)F1F)F7F)-F36$F(F7F)-F36$,&F(F)F.!\"\"F7F))F'F7F)- F36$,&F(F)F)F)F7F@-F36$F/F7F@-F36$,&F(F)#\"\"$F1F)F7F@-%*factorialG6#F 7F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Lambdaterm:=y^k*bino mial(n+k+1,2*k+1)*hyperterm([k+1/2,n+k+2,k-n],[2*k+1,k+3/2],y,j);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%+LambdatermG*4)%\"yG%\"kG\"\"\"-%)bi nomialG6$,(%\"nGF)F(F)F)F),&*&\"\"#F)F(F)F)F)F)F)-%+pochhammerG6$,&F(F )#F)F1F)%\"jGF)-F36$,(F.F)F(F)F1F)F7F)-F36$,&F(F)F.!\"\"F7F))F'F7F)-F3 6$F/F7F>-F36$,&F(F)#\"\"$F1F)F7F>-%*factorialG6#F7F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "The next computations use Zeilberger's algorit hm to deduce the recurrence equations presented in Theorem 6." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "RE1:=sumrecursion(tauterm,j, tau(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RE1G,,**,&*&\"\"#\"\"\" %\"nGF*F*F*!\"\"F*,(F+F*%\"kGF*F)F,F*,(F+F*F.F,F)F,F*-%$tauG6#,&F+F*\" \"%F,F*F**(F)F*,4*(F4F*%\"yGF*)F+\"\"$F*F**&F4F*F9F*F,*&\"#:F*)F+F)F*F **(\"#7F*F8F*F>F*F,*(\"#6F*F8F*F+F*F**&\"#F*F**& F:F*F>F*F,*(\"\")F*F8F*F+F*F,*&\"\"'F*F+F*F*FFF,F)F,*&F:F*F8F*F*F*-F16 #,&F+F*F)F,F*F,*(F)F*,4*(F4F*F8F*F9F*F**&F4F*F9F*F,*(F@F*F8F*F>F*F,*& \"\"*F*F>F*F**&FEF*F+F*F,*(FBF*F8F*F+F*F**&F:F*F8F*F,FFF,F*F*F*-F16#FM F*F***,&F:F,*&F)F*F+F*F*F*,&F+F*F.F,F*,&F+F*F.F*F*-F16#F+F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "# very time and memory consu ming!\nTIME:=time(): \nRE2:=sumrecursion(tauterm,j,tau(k));\ntime()-TI ME;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RE2G,,*.%\"yG\"\"\",&%\"kGF( F(!\"\"F(,&*&\"\"#F(F*F(F(\"\"$F+F(,(%\"nGF(F*F(F.F+F(,(F1F(F*F+\"\"%F (F(-%$tauG6#,&F*F(F3F+F(F(**F.F(F)F(,6*&F3F()F*F/F(F(*(F3F(F'F(F;F(F+* (\"#HF(F'F()F*F.F(F(*&\"#GF(F?F(F+*(\"#nF(F'F(F*F(F+*&\"#jF(F*F(F(*(\" \"'F(F'F(F1F(F(*(F/F(F'F()F1F.F(F(*&\"#^F(F'F(F(\"#XF+F(-F56#,&F*F(F/F +F(F+**F.F(FOF(,6*(F3F(F'F(F;F(F(*&F3F(F;F(F+*(\"#>F(F'F(F?F(F+*&\"#?F (F?F(F(*(\"#FF(F'F(F*F(F(*&\"#JF(F*F(F+*(FGF(F'F(F1F(F(*(F/F(F'F(FIF(F (*&\"\"*F(F'F(F+\"#:F(F(-F56#F)F(F(*.F'F(,&*&F.F(F*F(F(\"\"&F+F(FOF(,& F1F(F*F(F(,(F1F(F*F+F.F(F(-F56#F*F(F(*(F.F(,B**F.F(F'F(F?F(FIF(F(**\" \")F(F'F(F*F(FIF(F+*(F/F(F'F(FIF(F(**F3F(F'F(F?F(F1F(F(**\"#;F(F'F(F*F (F1F(F+*(FGF(F'F(F1F(F(*(FGF(F'F()F*F3F(F(*&FioF(F`pF(F+*(\"#[F(F'F(F; F(F+*&\"#kF(F;F(F(*(\"$P\"F(F'F(F?F(F(*&\"$#=F(F?F(F+*(\"$k\"F(F'F(F*F (F+*&\"$;#F(F*F(F(*&\"#mF(F'F(F(\"#!*F+F(-F56#,&F*F(F.F+F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")52=W!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "RE3:=sumrecursion(Lambdaterm,j,Lambda(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RE3G,***,(%\"nG\"\"\"%\"kGF)F)!\"\"F),(F( F)F*F+F)F+F)F(F)-%'LambdaG6#,&F(F)\"\"$F+F)F+*(,,*&F1F))F(\"\"#F)F+*( \"\"%F)%\"yGF)F5F)F)*(F6F)F9F)F(F)F+*&F6F)F(F)F)*$)F*F6F)F+F)-F.6#,&F( F)F6F+F),&F(F)F)F+F)F+*(F(F),0*&F1F)F5F)F+*&F8F)F(F)F)*(F8F)F9F)F5F)F) *(\"\"'F)F9F)F(F)F+*&F6F)F9F)F)F)F+F " 0 "" {MPLTEXT 1 0 70 " TIME:=time(): \nRE4:=sumrecursion(Lambdaterm,j,Lambda(k));\ntime()-TIM E;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$RE4G,**,%\"yG\"\"\",&%\"kGF(F (!\"\"F(,(%\"nGF(F*F(F(F+F(,(F-F(\"\"$F(F*F+F(-%'LambdaG6#,&F*F(F/F+F( F+*(,&F*F(\"\"#F+F(,2*(F/F(F'F()F*F6F(F(*&\"\"%F(F9F(F+*(\"\")F(F'F(F* F(F+*&\"#5F(F*F(F(*&F'F()F-F6F(F(*(F6F(F'F(F-F(F(*&\"\"'F(F'F(F(FDF+F( -F16#F5F(F+*(F)F(,2*(F/F(F'F(F9F(F(*&F;F(F9F(F+*(F?F(F'F(F*F(F+*&\"#9F (F*F(F(F@F(*&\"\"*F(F'F(F(\"#7F+*(F6F(F'F(F-F(F(F(-F16#F)F(F(*,F'F(F5F (,&F-F(F*F(F(,(F-F(F*F+F6F(F(-F16#F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&_&G!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "The followi ng yields the hypergeometric representations that are used in the proo f of Theorem 8, i.e. Eq. (22)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "df:=sumtohyper(tauterm-subs(n=n-1,tauterm),j);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#dfG*()%\"yG%\"kG\"\"\",&-%)binomialG6$,(%\"nGF) F(F)F)F),&*&\"\"#F)F(F)F)F)F)F)-F,6$,&F/F)F(F)F0!\"\"F)-%*hypergeomG6% 7%F(,&F(F)F/F6F.7$F0,&F(F)F)F)F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "prefactor:=eval(df,hypergeom=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*prefactorG*&)%\"yG%\"kG\"\"\",&-%)binomialG6$,(%\"nG F)F(F)F)F),&*&\"\"#F)F(F)F)F)F)F)-F,6$,&F/F)F(F)F0!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simpcomb(y^k*binomial(n+k,n-k)/pref actor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 11 "Theorem 9: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "s:=sumtohyper(tauterm+Lambdaterm,j);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"sG,$**\"\"#\"\"\")%\"yG%\"kGF(-%)b inomialG6$,(%\"nGF(F+F(F(F(,&*&F'F(F+F(F(F(F(F(-%*hypergeomG6%7&F+,&F+ F(#F(F'F(,&F+F(F0!\"\",(F0F(F+F(F'F(7%,&F+F(F(F(,&F+F(#\"\"$F'F(,$*&F' F(F+F(F(F*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "d:=sumtoh yper(tauterm-Lambdaterm,j);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"dG* 0,&%\"nG\"\"\"%\"kG!\"\"F(,(F'F(F)F(\"\"#F(F(-%)binomialG6$,(F'F(F)F(F (F(,&*&F,F(F)F(F(F(F(F()%\"yG,&F)F(F(F(F(,&*&F,F(F)F(F(\"\"$F(F*F5F*-% *hypergeomG6%7&,(F'F(F)F(F8F(F5,&F)F(#F8F,F(,(F)F(F'F*F(F(7%,&F)F(F,F( ,&F)F(#\"\"&F,F(,&*&F,F(F)F(F(F,F(F4F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "simpcomb(simplify(subs(k=k+1,s)/d));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }