D[JacobiZeta[z, m], {z, n}] == KroneckerDelta[n] JacobiZeta[z, m] + KroneckerDelta[n - 1] (Sqrt[1 - m Sin[z]^2] - (EllipticE[m]/EllipticK[m]) (1/Sqrt[1 - m Sin[z]^2])) - ((2 I^(n - 1) Pochhammer[-(1/2), n])/ (n - 1)!) Sum[(((-1)^q Binomial[n - 1, q])/(1 - m Sin[z]^2)^q) (Sqrt[1 - m Sin[z]^2]/(1 - 2 q) + (EllipticE[m]/EllipticK[m]) ((1 - 2 n)/((1 + 2 q) Sqrt[1 - m Sin[z]^2]))) Sum[Binomial[q, j] m^j (2 - m)^(q - j) 2^(n - j - q - 1) Sum[Binomial[j, i] (2 i - j)^(n - 1) E^(2 (2 i - j) I z), {i, 0, j}], {j, 0, q}], {q, 1, n - 1}] /; Element[n, Integers] && n >= 0

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2007-05-02