{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "terminal" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 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 "Co urier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 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 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 36 "/home/m262f99/A=B/Maple/s um2int1.mws" }{MPLTEXT 1 0 0 "" }}{PARA 256 "" 0 "" {TEXT 257 88 "Math 262a, Fall 1999, Glenn Tesler\nCreative Telescoping for multisums/int egrals\n10/27/99" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 375 "read D OUBLE_SUM_SINGLE_INTEGRAL:\nP:=1:\nF:=rf(al+1,n)*rf(be+1,n)/n!/rf(al+b e+1,n)/t^(n+1)*\n(1+t)^(-al-be-1)*rf(al+be+1,2*(m+n1))/n1!/m!/rf(al+1, n1)/rf(be+1,m)*\n(t/(4*(1+t)^2))^(m+n1)*\n(1+x)^m*(1+y)^m*(1-x)^n1*(1- y)^n1;\nORDER:=2:\nresh:=[al,be,x,y,n]:\nS1:=1/(t+1):\nS2:=(n1+al)*n1: \nS3:=m*(m+be):\ndisorcon:=continuous:\ncer:=findope(P,F,x,t,n1,m,ORDE R,resh,D_x,S1,S2,S3,disorcon);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%X THIS~IS~VERSION~1.1~OF~THE~MAPLE~PACKAGE~multi1c2d~FOR~G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#%`oINTEGRAND/SUMMANDS~DEPENDING~OF~ONE~CONTINUOU S(INTEGRATION)~VARIABLE~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%FAND~TWO ~DISCRETE(SUMMATION)~VARIABLESG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\" ~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%NTHE~PREOGRAM~multi1c2d~BASED~O N~THE~PAPER~BY:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%boWILF~AND~ZEILBE RGER:\"AN~ALGORITHMIC~PROOF~THEORY~FOR~(ORDINARY~AND~'q')G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%_oMULTISUMS/INTEGRAL~HYPERGEOMETRIC~IDENTIT IES(Invent.~Math.~108(1992)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-pp.5 75-633).G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#%_oThe~primilary~version~of~the~program~was~written~b y~Doron~ZeilbergerG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\o~Please~rep ort~all~bugs~and~comments~to:~zeilberg@math.temple.eduG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%in~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~or~akal u@math.temple.eduG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%A~For~a~list~of~procedures,~type:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%?~?muti1c2d~or~~help(muti1c2d)~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%K~For~help~with~a~specific~procedure,~type:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%K?procedure~name~~or~~help(procedure~name)~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?~Copy~write~1998,~Akalu~TeferaG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FG*B-%&GAMMAG6#,(%#alG\"\"\"F+F+%\"nGF+F +-F'6#,(%#beGF+F+F+F,F+F+-%*factorialG6#F,!\"\"-F'6#,*F*F+F0F+F+F+F,F+ F4)%\"tG,&F,F+F+F+F4),&F+F+F9F+,(F*F4F0F4F4F+F+-F'6#,,F*F+F0F+F+F+%\"m G\"\"#%#n1GFBF+-F26#FCF4-F26#FAF4-F'6#,(F*F+F+F+FCF+F4-F'6#,(F0F+F+F+F AF+F4),$*&F9F+F%$cerG6$,(*&%\"nG\"\"\",*%#alGF)%#beGF)F)F)F(F)F)F)*&,,*&%\"xGF)F+F )!\"\"*&F0F)F,F)F1F0!\"#F+F1F,F)F)%$D_xGF)F)*(,&F0F)F1F)F),&F)F)F0F)F) F4\"\"#F17%,$*(%\"tGF),4F+F)F,F)F)F)F(F)*&F " 0 "" {MPLTEXT 1 0 117 "ope \+ := cer[1]:\nR1 := cer[2][1]: R2 := cer[2][2]: R3 := cer[2][3]:\nwritep ap(P,F,x,y1,k1,k2,D_x,ope,R1,R2,R3,disorcon);" }}{PARA 6 "" 1 "" {TEXT -1 54 "Theorem:\n\nLet G( y1 , k1 , k2 , x ) be " }}{PARA 12 "" 1 "" {XPPMATH 20 "6$*B-%&GAMMAG6#,(%#alG\"\"\"F)F)%\"n GF)F)-F%6#,(%#beGF)F)F)F*F)F)-%*factorialG6#F*!\"\"-F%6#,*F(F)F.F)F)F) F*F)F2)%\"tG,&F*F)F)F)F2),&F)F)F7F),(F(F2F.F2F2F)F)-F%6#,,F(F)F.F)F)F) %\"mG\"\"#%#n1GF@F)-F06#FAF2-F06#F?F2-F%6#,(F(F)F)F)FAF)F2-F%6#,(F.F)F )F)F?F)F2),$*&F7F)F:!\"##F)\"\"%,&F?F)FAF)F)),&F)F)%\"xGF)F?F)),&F)F)% \"yGF)F?F)),&F)F)FUF2FAF)),&F)F)FXF2FAF)%!G" }}{PARA 6 "" 1 "" {TEXT -1 170 "\nand a( x ) be its integral w.r.t to y1 ,sum w.r.t. \+ k2 k1 .\n\nLet D_x be differentiation w.r.t. x .\n\nThe f unction a( x ) satisfies the recurrence" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&,(*&%\"nG\"\"\",*%#alGF(%#beGF(F(F(F'F(F(F(*&,,*&%\" xGF(F*F(!\"\"*&F/F(F+F(F0F/!\"#F*F0F+F(F(%$D_xGF(F(*(,&F/F(F0F(F(,&F(F (F/F(F(F3\"\"#F0F(-%\"aG6#F/F(%#0.G" }}{PARA 6 "" 1 "" {TEXT -1 39 "\n Proof: It is routinely verifiable that" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,(*&%\"nG\"\"\",*%#alGF'%#beGF'F'F'F&F'F'F'*&,,*&%\"xGF'F)F'! \"\"*&F.F'F*F'F/F.!\"#F)F/F*F'F'%$D_xGF'F'*(,&F.F'F/F'F',&F'F'F.F'F'F2 \"\"#F/F'-%\"GG6&%#y1G%#k1G%#k2GF.F'" }}{PARA 6 "" 1 "" {TEXT -1 14 " \n= D_ y1 (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"tG\"\"\",4%# alGF&%#beGF&F&F&%\"nGF&*&F%F&F*F&F&*&F%F&%#n1GF&!\"\"*&F%F&%\"mGF&F.F0 F&F-F&F&,&F&F&F%F&F.-%\"GG6&%#y1G%#k1G%#k2G%\"xGF&F." }}{PARA 6 "" 1 " " {TEXT -1 29 " )\n\n+(E_ k1 -I)(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"xG\"\"\"!\"\"F'F(,&%#n1GF'%#alGF'F'F*F'-%\"GG6 &%#y1G%#k1G%#k2GF&F'\"\"#" }}{PARA 6 "" 1 "" {TEXT -1 17 "\n+(E_ k2 \+ -I)(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&\"\"\"F&%\"xGF&!\"\"% \"mGF&,&F)F&%#beGF&F&-%\"GG6&%#y1G%#k1G%#k2GF'F&!\"#" }}{PARA 6 "" 1 " " {TEXT -1 100 " ),\n\nand the result follows by integrating \+ w.r.t \ny1 , and summing w.r.t. k1 k2 ." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 }