{VERSION 2 3 "DEC ALPHA UNIX" "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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 0 1 0 0 0 0 1 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 -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 "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 0 256 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 41 "Ore_algebra, a Package fo r Skew Operators" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 53 "by Frederic Chyzak, Algorithms Project , INRIA, France" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 17 "" 0 "" {TEXT -1 24 "Frederic.Chyzak@inria.fr" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 " Several algorithms for integration and summation have a natural descri ption in terms of linear differential and difference operators, which \+ in turn are well described by skew (or Ore) polynomials. This was the starting point for the " }{HYPERLNK 17 "Ore_algebra" 2 "Ore_algebra" "" }{TEXT -1 9 " package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "with(Ore_algebra);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74%-Ore_to_D ESolG%-Ore_to_RESolG%,Ore_to_diffG%-Ore_to_shiftG%-annihilatorsG%)appl yoprG%-diff_algebraG%-poly_algebraG%/qshift_algebraG%/rand_skew_polyG% .shift_algebraG%-skew_algebraG%*skew_elimG%+skew_gcdexG%*skew_pdivG%+s kew_powerG%*skew_premG%-skew_productG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "To work with an Ore algebra, we first have to declare it . The package creates a table which implements and remembers the\nope rations in this algebra. Here is the example of the algebra of linear differential operators in the differential operator " }{XPPEDIT 18 0 "Dx" "I#DxG6\"" }{TEXT -1 33 " with (rational) coefficients in " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=diff_algebra([Dx,x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%,Ore_algebraG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "(This is the name of the table.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Although the usual product in Maple is commutative, we us e " }{XPPEDIT 18 0 "`*`" "I\"*G6\"" }{TEXT -1 0 "" }{TEXT -1 60 " to d enote skew products with the convention that powers of " }{XPPEDIT 18 0 "Dx" "I#DxG6\"" }{TEXT -1 0 "" }{TEXT -1 18 " are on the right." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Multiplication of operators is obt ained by the function skew_product." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P1:=x*Dx^2-1: P2:=Dx-x: P1_P2:=skew_product(P1,P2,A); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&P1_P2G,*%\"xG\"\"\"*&,&F&!\"#! \"\"F'F'%#DxGF'F'*&F&\"\"#F,F.F+*&F&F'F,\"\"$F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Remember that in skew algebras of linear operators, \+ factorizations are seldom unique." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "DEtools[DFactor](P1_P2,[Dx,x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"xG\"\"\",&*$%#DxG\"\"#F&*$F%!\"\"F,F&,&F)F&F%F, " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "(Univariate) Ore polynomial \+ rings have a Euclidean algorithm. This is implemented in the followin g routines: " }{HYPERLNK 17 "skew_pdiv" 2 "Groebner[skew_pdiv]" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "skew_prem" 2 "Groebner[skew_prem]" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "skew_gcdex" 2 "Groebner[skew_gcdex]" " " }{TEXT -1 5 " and " }{HYPERLNK 17 "skew_elim" 2 "Groebner[skew_elim] " "" }{TEXT -1 54 ". An extension to the multivariate case is availab le." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "As a first application, w e show how the package can be used to derive a proof of the Reed-Dawso n identity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "U[n]=Sum(bin omial(n,k)*binomial(2*k,k)*(-2)^(n-k),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"UG6#%\"nG-%$SumG6$*(-%)binomialG6$F'%\"kG\"\"\"-F- 6$,$F/\"\"#F/F0)!\"#,&F'F0F/!\"\"F0/F/;\"\"!F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bino mial(n,n/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)binomialG6$%\"nG,$F &#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "for even " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "0" "\"\"!" }{TEXT -1 0 "" }{TEXT -1 9 " for odd " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Let us introduce an algebra of linear rec urrence operators in " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 " " } {TEXT -1 4 "and " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 0 "" }{TEXT -1 3 ". " }{XPPEDIT 18 0 "Sn" "I#SnG6\"" }{TEXT -1 0 "" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Sk" "I#SkG6\"" }{TEXT -1 0 "" }{TEXT -1 31 " de note the shift operators in " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 14 " \+ respectively." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A:=shift_a lgebra([Sn,n],[Sk,k]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We comp ute first order recurrences satisfied by the summand (which we denote \+ by " }{XPPEDIT 18 0 "h" "I\"hG6\"" }{TEXT -1 0 "" }{TEXT -1 2 ")." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "h:=binomial(n,k)*binomial(2* k,k)*(-2)^(n-k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*(-%)binomia lG6$%\"nG%\"kG\"\"\"-F'6$,$F*\"\"#F*F+)!\"#,&F)F+F*!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Sn-normal(applyopr(Sn,h,A)/h,expand ed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#SnG\"\"\"*&,&%\"nG!\"#F)F% F%,(F(F%F%F%%\"kG!\"\"F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Sk-normal(applyopr(Sk,h,A)/h,expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#SkG\"\"\"*&,**$%\"kG\"\"#F**&%\"nGF%F)F%!\"#F)F%F,! \"\"F%,(F(F%F)F*F%F%F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G:=map(numer,\{%%,%\}):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "We el iminate " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 0 "" }{TEXT -1 74 " \+ between these operators by computing their greatest common right divis or." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "GCD:=skew_gcdex(op(G ),k,A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$GCDG7',@!#;\"\"\"%#SkG\" \")%\"nG!#C%#SnG!#7*&F-F(F+F(!#9*&F-F(F)F(\"#7*&F)F(F+F(F2*(F)F(F-F(F+ F(\"#9*&F-F(F+\"\"#!\"%*$F+F7!\")*(F)F(F-F(F+F7\"\"%*(F)F(F-F7F+F7F(*& F-F7F)F(F<*&F)F(F+F7F<*(F)F(F-F7F+F(F<,,F:F(F+F8*&,&F+!\"#F8F(F(F)F(! \"\"*&,&%\"kGF7F(F(F(F-F(FE*(,(FHFE!\"$F(F+FEF(F-F(F)F(FE*$F-F7,2F>FE* &F-F7FHF(FK*&F-F7FHF7F7*(F+F(F-F7FHF(FDFLFD*&F-F7F+F(FE*(F-F7F)F(FHF(F D*(F-F7F)F(FHF7FE,,FQF7FL\"\"'*&F-\"\"$FHF(FE*&F-FWF+F(F(*$F-FWF7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series(GCD[1],Sk=1);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+',&%#SkG\"\"\"!\"\"F&,.!\")F&%\"nG!#7 *$F*\"\"#!\"%*&%#SnGF-F*F&\"\"%*$F0F-F1*&F0F-F*F-F&\"\"!,4*&F0F&F*F&\" #9*&F0F&F*F-F1F3F&\"\")F&F2F1F0\"#7F,F1F*F:F/F1\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The factor " }{XPPEDIT 18 0 "Sk-1" ",&%#S kG\"\"\"\"\"\"!\"\"" }{TEXT -1 0 "" }{TEXT -1 68 " corresponds to taki ng a finite difference. Definite summation for " }{XPPEDIT 18 0 "k" " I\"kG6\"" }{TEXT -1 0 "" }{TEXT -1 82 " over all integers yields a tel escoping series which collapses to 0. We thus get:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(applyopr(coeff(%,Sk-1,0),u[n,k],A),k= -infinity..infinity)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$, &*&,(!\")\"\"\"%\"nG!#7*$F,\"\"#!\"%F+&%\"uG6$F,%\"kGF+F+*&,(F,\"\"%F7 F+F.F+F+&F26$,&F,F+F/F+F4F+F+/F4;,$%)infinityG!\"\"F>\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Therefore, the sum " }{XPPEDIT 18 0 "U[n] " "&%\"UG6#%\"nG" }{TEXT -1 0 "" }{TEXT -1 10 " satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "collect(primpart(coeff(%%,Sk-1,0),S n),Sn,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&%\"nG\"\"\"\" \"#F'F'%#SnGF(F'F&!\"%F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "applyopr(%,U[n],A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"nG !\"%F'\"\"\"F(&%\"UG6#F&F(F(*&,&F&F(\"\"#F(F(&F*6#F-F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Now, the announced result follows from in itial conditions:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "n=1" "/%\"nG\"\"\"" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "U[1]:=add(subs(n=1,h),k=0..1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"UG6#\"\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "U[n]=0" "/&%\"UG6#%\"nG\"\"!" } {TEXT -1 9 " for odd " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "n=0" "/%\"n G\"\"!" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "U [0]:=add(subs(n=0,h),k=0..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" UG6#\"\"!\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " } {XPPEDIT 18 0 "V(p)=U[2*p]" "/-%\"VG6#%\"pG&%\"UG6#*&\"\"#\"\"\"F&F," }{TEXT -1 12 " is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "2*(p+1)*V(p+1)-4*(2*p+1)*V(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*&,&%\"pG\"\"\"F'F'F'-%\"VG6#F%F'\"\"#*&,&F&F+F'F'F'-F)6#F&F'!\"%" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "LREtools[hypergeomsols](%, V(p),\{V(0)=1\},output=basis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,&% #_CG6#\"\"\"F')\"\"#%\"pGF)-%&GAMMAG6#,&F*F'#F'F)F'F'-F,6#,&F*F'F'F'! \"\"%#PiG#F3F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "which is " } {XPPEDIT 18 0 "binomial(2*p,p)" "-%)binomialG6$*&\"\"#\"\"\"%\"pGF'F( " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "As a more ad vanced application, we derive contiguity relations for Gauss hypergeom etric function. This function is known to Maple as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=hypergeom([a,b],[c],z);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fG-%*hypergeomG6%7$%\"aG%\"bG7#%\"cG%\"zG" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "It is " }{XPPEDIT 18 0 "Sum(u[n]* z^n,n=0..infinity)" "-%$SumG6$*&&%\"uG6#%\"nG\"\"\")%\"zGF)F*/F);\"\"! %)infinityG" }{TEXT -1 4 " for" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "u:=pochhammer(a,n)*pochhammer(b,n)/pochhammer(c,n)/n!:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "When considering the summand, we i ntroduce the following algebra." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A:=shift_algebra([Sn,n],[Sa,a],comm=\{b,c\}):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The bivariate sequence " }{XPPEDIT 18 0 " u" "I\"uG6\"" }{TEXT -1 0 "" }{TEXT -1 38 " vanishes at both following operators:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "G:=\{(c+n)*( n+1)*Sn-(a+n)*(b+n),a*Sa-(a+n)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "normal(map(applyopr,G,u,A),expanded);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "(T here are general algorithms to find such operators.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "From the previous first order recurrences, we d erive relations on " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 0 "" } {TEXT -1 45 " in the mixed differential difference algebra" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A:=skew_algebra(diff=[Dz,z],shift=[ Sa,a],comm=\{b,c\}):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The equat ions are obtained by multiplication of the recurrences by " }{XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 0 "" }{TEXT -1 46 ", followed by su mmation over all non-negative " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 42 ". Formally, this corresponds to changing " } {XPPEDIT 18 0 "Sn" "I#SnG6\"" }{TEXT -1 0 "" }{TEXT -1 6 " into " } {XPPEDIT 18 0 "1/z" "*&\"\"\"\"\"\"%\"zG!\"\"" }{TEXT -1 0 "" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 0 "" }{TEXT -1 6 " into " }{XPPEDIT 18 0 "z*Dz" "*&%\"zG\"\"\"%#DzGF$" }{TEXT -1 0 "" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "map(proc( p)\n\011collect(add(add(coeff(coeff(p,Sn,i),n,j)*\n\011 skew_produc t(skew_power(z*Dz,j,A),1/z^i,A),\n\011 j=0..degree(coeff(p,Sn,i),n) ),\n\011 i=0..degree(p,Sn)),Dz,factor)\n end,G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,&*&%\"zG\"\"\"%#DzGF'!\"\"*&%\"aGF',&%#SaGF'F)F 'F'F',(*(F&F',&F&F'F)F'F'F(\"\"#F)*&,*%\"cGF'F&F)*&F&F'F+F'F)*&F&F'%\" bGF'F)F'F(F'F'*&F+F'F7F'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The refore, we set:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "P:=z*(1- z)*Dz^2+(c-(a+b+1)*z)*Dz-a*b:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "H:=z*Dz/a+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G:=\{ P,numer(Sa-H)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG<$,(*&%\"aG \"\"\"%#SaGF)F)F(!\"\"*&%\"zGF)%#DzGF)F+,(*(F-F),&F-F+F)F)F)F.\"\"#F)* &,&%\"cGF)*&,(F(F)%\"bGF)F)F)F)F-F)F+F)F.F)F)*&F(F)F8F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The linear differential operator " } {XPPEDIT 18 0 "Sa=z*Dz/a+1" "/%#SaG,&*(%\"zG\"\"\"%#DzGF'%\"aG!\"\"F' \"\"\"F'" }{TEXT -1 64 " is called a step-up operator. It relates the forward shift of " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 0 "" } {TEXT -1 19 " to derivatives of " }{XPPEDIT 18 0 "t" "I\"tG6\"" } {TEXT -1 0 "" }{TEXT -1 26 " by the following equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "hypergeom([a+1,b],[c],z)=z/a*diff(h ypergeom([a,b],[c],z),z)+hypergeom([a,b],[c],z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*hypergeomG6%7$%\"bG,&%\"aG\"\"\"F+F+7#%\"cG%\"zG,&* *F.F+F(F+F-!\"\"-F%6%7$F),&F(F+F+F+7#,&F-F+F+F+F.F+F+-F%6%7$F*F(F,F.F+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The elimination of " } {XPPEDIT 18 0 "Dz" "I#DzG6\"" }{TEXT -1 0 "" }{TEXT -1 94 " between th is step-up operator and the differential equation yields a contiguity \+ relation for " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 0 "" }{TEXT -1 89 ", i.e., a purely recurrence equation. It is obtained by the exten ded skew gcd algorithm:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " C:=collect(skew_elim(P,numer(Sa-H),Dz,A),Sa,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,,*(,&%\"zG\"\"\"!\"\"F)F),&%\"aGF)F)F)F)%#SaG \"\"#F**&,.%\"cGF)*&F(F)F,F)F)F,!\"#F(F)*&F(F)%\"bGF)F*F3F)F)F-F)F)F,F )F1F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "In other words, Gaus s hypergeometric function satisfies the following equation:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "applyopr(C,hypergeom([a,b],[ c],z),A)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "More interestingly , the extended Euclidean algorithm yields a step-down operator for " } {XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 0 "" }{TEXT -1 47 ", i.e., a re lation between an inverse shift of " }{XPPEDIT 18 0 "f" "I\"fG6\"" } {TEXT -1 0 "" }{TEXT -1 67 " and its derivatives. This is obtained by computing an inverse of " }{XPPEDIT 18 0 "H" "I\"HG6\"" }{TEXT -1 0 " " }{TEXT -1 8 " modulo " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 0 "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "GCD:=skew _gcdex(P,numer(H),Dz,A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$GCDG7', (*&%\"cG\"\"\"%\"aGF)!\"\"F*F)*$F*\"\"#F)%\"zG,,F)F)*&,&*$F.F-F+F.F)F) %#DzGF)F+F(F+*&F.F)%\"bGF)F)F*F),(*&F.F)F3F)F+F*F+F+F),2*&F.F-F3F-F+*& F.F)F3F-F)*(F3F)F.F)F*F)F+*(F3F)F.F)F5F)F+F7!\"#F5F+*&F*F)F5F)F+*&F3F) F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "From this result, we ha ve " }{XPPEDIT 18 0 "GCD[2]*P+GCD[3]*numer(H)=GCD[1]" "/,&*&&%$GCDG6# \"\"#\"\"\"%\"PGF)F)*&&F&6#\"\"$F)-%&numerG6#%\"HGF)F)&F&6#\"\"\"" } {TEXT -1 0 "" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "GCD[4]*P+GCD[5]*nume r(H)=0" "/,&*&&%$GCDG6#\"\"%\"\"\"%\"PGF)F)*&&F&6#\"\"&F)-%&numerG6#% \"HGF)F)\"\"!" }{TEXT -1 17 ". In particular," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "B:=collect((a-1)*subs(a=a-1,GCD[3]/GCD[1]),Dz, factor);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "is the step-down oper ator:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "hypergeom([a-1,b] ,[c],z)=z*(1-z)*diff(hypergeom([a,b],[c],z),z)/(a-c)+(c-a-z*b)*hyperge om([a,b],[bc],z)/(c-a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "There is also an algorithm to compute (one-sided) Groebner bases in Ore alg ebras. This will be demonstrated in the presentation of the " } {HYPERLNK 17 "Groebner" 2 "Groebner" "" }{TEXT -1 9 " package." }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 }