On polynomials connected to powers of Bessel functions

TL;DR

This paper explores the series expansion of powers of the modified Bessel function of the first kind, introducing new polynomial recurrences and connections to probabilistic identities and umbral calculus.

Contribution

It presents novel recurrence relations for Bessel-related polynomials using Bell polynomials and Bessel zeta functions, expanding theoretical understanding.

Findings

Derived new recurrence relations for Bessel-related polynomials

Connected polynomial identities to probabilistic Euler identities

Linked the polynomials to umbral formalism on Bessel functions

Abstract

The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional recurrences. Connections to the umbral formalism on Bessel functions introduced by Cholewinski are established.