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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 49 "/home/m262f99/KOEPF/works
heetsV.4/extendedalg.mws" }{MPLTEXT 1 0 0 "" }}{PARA 258 "" 0 "" 
{TEXT 257 78 "Math 262a, Fall 1999, Glenn Tesler\nKoepf's \"Extended G
osper\" algorithm\n11/1/99" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "read `hsum.mpl`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZCopyright~19
98~~Wolfram~Koepf,~Konrad-Zuse-Zentrum~BerlinG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "gosper(k*k!,k);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%*factorialG6#%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "a1 := (k/2)*(k/2)!;\ngosper(a1,k);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#a1G,$*&%\"kG\"\"\"-%*factorialG6#,$F'#F(\"\"#F(F-" }}{PARA 8 
"" 1 "" {TEXT -1 43 "Error, (in gosper) algorithm not applicable" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "s1 := extended_gosper(a1,k);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G,&-%*factorialG6#,$%\"kG#\"
\"\"\"\"#F,-F'6#,&F*F+F+F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "
Note: the above is not a hypergeometric term, but it is an antidiffere
nce of (k/2)*(k/2)! with a simple, closed formula.  Let's check it:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify((subs(k=k+1,s1)-s1) - a1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 19 "So we conclude that" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 
"Sum(a1,k=a..(b-1)) = subs(k=b,s1)-subs(k=a,s1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$,$*&%\"kG\"\"\"-%*factorialG6#,$F)#F*\"\"#F*F
//F);%\"aG,&%\"bGF*!\"\"F*,*-F,6#,$F5F/F*-F,6#,&F5F/F/F*F*-F,6#,$F3F/F
6-F,6#,&F3F/F/F*F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "A variatio
n: instead of computing s(k) s.t. s(k+1)-s(k)=a(k), we supply an integ
er j and compute s(k) s.t. s(k+j)-s(k)=a(k).  This is called a j-fold \+
antidifference.\n    extended_gosper(f(k),k,j);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "extended_gosper((k/2)*(k/2)!,k,2);" }}{PARA 11 "" 1 "
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