Derivative[1, 0][SphericalBesselJ][-n - 1/2, z] == Sqrt[Pi/2] ((-1)^(n - 1)/Sqrt[z]) Sum[((n - k - 1)!/k!) (z/2)^(2 k - n), {k, 0, n - 1}] + (((-1)^n Pi)/2) SphericalBesselY[n - 1/2, z] + ((n!/2) Sum[(1/((n - k) k!)) SphericalBesselJ[k - 1/2, z] (z/2)^k, {k, 0, n - 1}])/(-(z/2))^n + Sqrt[Pi/2] (1/(Sqrt[z] n!)) (-(z/2))^n Sum[(1/j) HypergeometricPFQ[{j}, {1 + j, 1 + n}, -(z^2/4)], {j, 1, n}] /; Element[n, Integers] && n >= 0

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2007-05-02