International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 73750, 5 pages
doi:10.1155/2007/73750
Research Article
On a Class of Integrals Involving a Bessel Function Times Gegenbauer Polynomials
Department of Mathematics and Statistics, University of Cyprus, P. O. Box 20537, Nicosia 1678, Cyprus
Received 18 August 2006; Revised 1 November 2006; Accepted 16 November 2006
Academic Editor: Feng Qi
Copyright © 2007 Stamatis Koumandos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We provide information and explicit formulae for a class of integrals involving Bessel functions and Gegenbauer polynomials. We present a simple proof of an old formula of Gegenbauer. Some interesting special cases and applications of this result are
obtained. In particular, we give a short proof of a recent result of A. A. R. Neves et al. regarding the analytical evaluation of an integral of a Bessel function times associated Legendre functions. These integrals arise in problems of vector diffraction
in electromagnetic theory.