On an umbral treatment of Gegenbauer, Legendre and Jacobi polynomials

Abstract

Special polynomials, ascribed to the family of Gegenbauer, Legen- dre, and Jacobi and of their associated forms, can be expressed in an operational way, which allows a high degree of flexibility for the for- mulation of the relevant theory. We develop a point of view based on an umbral type formalism, exploited in the past, to study some aspects of the theory of special functions, in general, and in particular those of Bessel functions. We propose a fairly general analysis, allowing a transparent link between different forms of special polynomials