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I try to calculate the following hypergeometric function:

HypergeometricPFQ[{n+1, n+1, n+3/2}, {2n+3}, -1]

for several positive natural values $n=0,1,2,\ldots$, but Mathematica give as result Infintity, which is logical because this hypergeometric function of course diverges for any value, but Maple is capable of giving values. Is it possible to get the same results in Mathematica?

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  • $\begingroup$ What did you try so far? Please provide your Mathematica code. Where did you find Mathematica function HyperGeometric3F1? $\endgroup$ Mar 30, 2021 at 9:29
  • $\begingroup$ In Mathematica syntax: HypergeometricPFQ[{n+1, n+1, n+3/2}, {2n+3}, -1] $\endgroup$
    – Roman
    Mar 30, 2021 at 9:31
  • $\begingroup$ XE Maple say that $$\, _3F_1\left(1,1,\frac{3}{2};3;-1\right)=0.7391205037880855294637891218743701306027945181746889830708299919761323612223322$$ $\endgroup$
    – capea
    Mar 30, 2021 at 9:39

1 Answer 1

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$$\, _3F_1\left(n+1,n+1,n+\frac{3}{2};2 n+3;-1\right)=\\\sum _{k=0}^{\infty } \frac{\left((n+1)_k\right){}^2 \left(n+\frac{3}{2}\right)_k (-1)^k}{(2 n+3)_k k!}=\\\sum _{k=0}^{\infty } \frac{(-1)^k 2^{1-2 k} (1+n) \Gamma (1+k+n) \Gamma (2+2 k+2 n)}{\Gamma (1+k) \Gamma (1+n) \Gamma (3+k+2 n)}=\\\sum _{k=0}^{\infty } \frac{(-1)^k 2^{1-2 k} (1+n) \Gamma (1+k+n) \int_0^{\infty } t^{1+2 k+2 n} \exp (-t) \, dt}{\Gamma (1+k) \Gamma (1+n) \Gamma (3+k+2 n)}=\\\int_0^{\infty } \left(\sum _{k=0}^{\infty } \frac{(-1)^k 2^{1-2 k} (1+n) \Gamma (1+k+n) t^{1+2 k+2 n}}{\Gamma (1+k) \Gamma (1+n) \Gamma (3+k+2 n)}\right) \, dt=\\\int_0^{\infty } 2 e^{-t} (1+n) t^{1+2 n} \, _1\tilde{F}_1\left(1+n;3+2 n;-\frac{t^2}{4}\right) \, dt=\\\frac{4^{n+1} (n+1) G_{2,3}^{3,1}\left(1\left| \begin{array}{c} 1,2 n+3 \\ n+1,n+1,n+\frac{3}{2} \\ \end{array} \right.\right)}{\sqrt{\pi } \Gamma (n+1)}$$

f[n_]:= (4^(1 + n) (1 + n) MeijerG[{{1}, {3 + 2 n}}, {{1 + n, 1 + n, 3/2 + n}, {}}, 1])/(Sqrt[\[Pi]] Gamma[1 + n])

N[f[0], 100]
(*0.7391205037880855294637891218743701306027945181746889830708299919761323612223322420425589635789669557 + 0.*10^-101 I*)

ListLinePlot[Table[{n, f[n]}, {n, 0, 10, 1/5}], PlotRange -> All](*A nice Plot*)

In Maple you can use convert function to convert Hypergeometric function to MeijerG function: convert(hypergeom([n + 1, n + 1, n + 3/2], [2*n + 3], -1), MeijerG)

then you get:

4*MeijerG([[1], [2*n + 3]], [[n + 1, n + 1, n + 3/2], []], 1)*(n + 1)*2^(2*n)/(sqrt(Pi)*GAMMA(n + 1))

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