Expansion in Terms of Appropriate Functions and Transmutations

Abstract

This work presents and summarizes the main steps of the work of Fitouhi et al. on the expansions in series of appropriate functions, namely the Bessel functions of the first kind for second-order differential Bessel perturbed operators. By changing functions or variables we can reduce the operators associated with certain polynomials and special functions to the operators considered like the Jacobi polynomials and the Whittaker functions. Taking into account that the principal part of these operators is closely related to the function of Bessel and that these latter verify recursive relations, we show that their eigenfunctions can be developed in series of Bessel functions which induce two integral representations of Mehler and Sonine type.