Fourier, Gegenbauer, and Jacobi Expansions Related to a Fundamental Solution of the Polyharmonic Equation in Euclidean Space

This talk focuses on the Fourier and Gegenbauer analysis of a fundamental solution for the polyharmonic equation in rotationally invariant and hyperspherical coordinate systems which parametrize points in d-dimensional Euclidean space. The analysis of a fundamental solution for the polyharmonic equation leads to a generalized Heine's identity for complex Fourier series of binomials, as well as an even more general complex expansion over Gegenbauer polynomials whose coefficients are given in terms of associated Legendre functions. We show how these expansions can be used in conjunction with the addition theorem for hyperspherical harmonics to generate an infinite sequence of multi-summation addition theorems, discuss the prospect of determining an even more general expansion over Jacobi polynomials, and conclude by showing how many known expressions for generating functions of Jacobi polynomials can be written even more simply, in terms of associated Legendre and elementary functions.

Speaker Bio: Dr Howard Cohl obtained a B.S. in Astronomy and Astrophysics from Indiana University, a M.S. and Ph.D. in Physics from Louisiana State University, and a Ph.D. in Mathematics from the University of Auckland in New Zealand. He has worked as a research scientist at various research institutions including the National Solar Observatory in Sunspot, New Mexico; Naval Oceanographic Office Major Shared Resource Center in Stennis Space Center, Mississippi; Lawrence Livermore National Laboratory in Livermore, California; and the School of Physics, University of Exeter in Exeter, United Kingdom. Howard started in December 2010, as a National Research Council Postdoctoral Research Associate in the Applied and Computational Mathematics Division here in ITL at NIST. Dr Cohl is currently interested in the special functions associated with fundamental solutions for linear partial differential equations on Riemannian manifolds.

Presentation Slides: PPT

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