An application of generating function for Hermite polynomials

Abstract

An important application of the Hermite polynomials with the weight function absorbed into them is that they form the solutions of the differential equation for a simple harmonic oscillator. Motivated by this fact, our focus for this work is to investigate a class of singular integral equations whose kernels are formed by the generating function for the Hermite polynomials with the weighting function absorbed into them. We propose an efficient method for solving such a class of singular integral equations. In addition, we establish the results of applications involving the integral of Hermite polynomials and their generating functions.