AppellF1[n + 1/2, 1/2, 1, n + 3/2, Subscript[z, 1], Subscript[z, 2]] == ((2 n + 1)/Sqrt[Subscript[z, 1]]) (ArcTanh[(Sqrt[Subscript[z, 1]] Sqrt[-1 + Subscript[z, 2]/Subscript[z, 1]])/ Sqrt[1 - Subscript[z, 1]]]/(Subscript[z, 2]^n Sqrt[-1 + Subscript[z, 2]/Subscript[z, 1]]) - Sum[(-1)^j Binomial[n, 1 + j] ((-2 Subscript[z, 1] + Subscript[z, 2])/ Subscript[z, 2])^(1 + j) Sum[(Binomial[1 + j, i] Subscript[z, 1]^i (Sum[(1/q!) ((Binomial[-1 + i, 2 q] Pochhammer[1/2, q] ((2 Subscript[z, 1] - Subscript[z, 2])/Subscript[z, 1])^(-1 + i - 2 q) Subscript[z, 2]^(2 q) (2 ArcSin[Sqrt[Subscript[z, 1]]] + Sqrt[1 - Subscript[z, 1]] Sqrt[Subscript[z, 1]] Sum[((-1 + p)! (1 - 2 Subscript[z, 1])^(-1 + 2 p))/ Pochhammer[1/2, p], {p, 1, q}]))/Subscript[z, 1]^(2 q)), {q, 0, Floor[(1/2) (-1 + i)]}] + Sum[2^(1 + 2 q) Binomial[-1 + i, 1 + 2 q] q! (1 - Subscript[z, 1])^(1/2 + q) Subscript[z, 1]^(-(1/2) - q) ((2 Subscript[z, 1] - Subscript[z, 2])/Subscript[z, 1])^(-2 + i - 2 q) Subscript[z, 2]^(1 + 2 q) Sum[(1 - 2 Subscript[z, 1])^(2 p)/(2^(2 p) (1 - Subscript[z, 1])^ p Subscript[z, 1]^p (p! Pochhammer[3/2, -p + q])), {p, 0, q}], {q, 0, -1 + Floor[i/2]}]))/ (-2 Subscript[z, 1] + Subscript[z, 2])^i, {i, 0, 1 + j}], {j, 0, -1 + n}]/(2^n Subscript[z, 1]^n))

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2007-05-02