New identities for Legendre associated functions of integral order and degree - Schach, Stephen Ronald (Department of Mathematics and Applied Mathematics, 2016)
In the solution of the boundary value problems of mathematical physics in a separable 3-dimensional coordinate system, the shape of the boundary of the space may be such that the Green's function of the second order differential operator can be expanded as an infinite series of orthogonal functions. In many coordinate systems (such as the spherical, spheroidal and some cyclidal systems) these expansions are given in terms of Legendre associated functions of integral order and degree. Starting with Dougall's identities for Legendre associated functions of non-integral degree, new identities for infinite series of Legendre associated functions of integral degree are derived. Uniform convergence of each new identity is investigated in detail. The direct applicability of these identities is demonstrated by using them to verify theorems satisfied by the Dirichlet Green's function of the infinite half-space and of the interior of the prolate hemispheroid. The results and techniques are then generalized, and a sufficient condition found under which a generalized orthogonal function which satisfies Dougall's identity will also satisfy the new identity. This theorem is applied to the Legendre associated function, the generalized Legendre associated function and to the Jacobi function.