Abstract
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.
Highlights
We have previously obtained antiderivatives and integral representations for the associated Legendre and Ferrers functions of the second kind with degree and order equal to within a sign while using analysis for fundamental solutions of the Laplace equationon Riemannian manifolds of constant curvature
Our integral representations are consistent with the known special values for the associated Legendre and Ferrers functions of the first kind when the order is equal to the negative degree
We produce a lemma for the Gauss hypergeometric function, which will be useful in our analysis of antiderivatives for associated Legendre functions of the second kind below
Summary
We have previously obtained antiderivatives and integral representations for the associated Legendre and Ferrers functions of the second kind with degree and order equal to within a sign while using analysis for fundamental solutions of the Laplace equationon Riemannian manifolds of constant curvature. We derive an antiderivative and an integral representation for the Ferrers function of the second kind with order equal to the negative degree in ([1] Theorem 1), using the d-dimensional hypersphere with d = 2, 3, 4,. Associate Legendre and Ferrers functions appear in any place where harmonic analysis needs to be performed on the surface of a sphere or on an oblate or prolate spheroid These are analogs of the 1/r potential in Euclidean space for Riemannian spaces of constant curvature.
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