* Bill Gosper <billgosper@gmail.com> [Aug 05. 2015 07:43]:
http://dlmf.nist.gov/15.8 It seems to need another term on the right. --rwg [...]
Works for me (fixing parameters because of Pari's limitations with symbolic computations): -------------------------------------------- \\ http://dlmf.nist.gov/15.8 \\ eq. 15.8.23 /* \[\mathop{F\/}\nolimits\!\left({a,1-a\atop c};z\right)=\left(\sqrt{1-z^{-1}}-1% \right)^{1-a}\left(\sqrt{1-z^{-1}}+1\right)^{a-2c+1}\left(1-z^{-1}\right)^{c-1% }\mathop{F\/}\nolimits\!\left({c-a,c-\tfrac{1}{2}\atop 2c-1};\frac{4\sqrt{1-z^% {-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}\right),\] */ \r hypergeom default(echo,1); N=15; z = 1/2/'z^2+O('z^N); c=1 a=1/2 L=hypergeom([a,1-a], [c], z, N) S=sqrt(1-1/z) R = (S-1)^(1-a) * (S+1)^(a-2*c+1) * (1-1/z)^(c-1) R *= hypergeom([c-a, c-1/2], [2*c-1], 4*S/(S+1)^2 , N) L - R quit; -------------------------------------------- The expression 'L - R' near end gives: 93990019574025/147573952589676412928*z^-30 + 25145962430625/18446744073709551616*z^-28 + 1690195005625/576460752303423488*z^-26 + 457028729521/72057594037927936*z^-24 + 7775536041/562949953421312*z^-22 + 2133423721/70368744177664*z^-20 + 147744025/2199023255552*z^-18 + 41409225/274877906944*z^-16 + 184041/536870912*z^-14 + O(z^-13)