Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials with applications to series involving zeros of Bessel functions

TL;DR

This paper introduces new classes of polynomials based on Dunkl operators, generalizing classical Bernoulli and Apostol-Euler polynomials, and applies them to evaluate series involving Bessel function zeros.

Contribution

The paper develops Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials and demonstrates their use in summing series related to Bessel function zeros.

Findings

New polynomial families generalizing classical polynomials

Explicit formulas for series involving Bessel zeros

Applications to summation of special function series

Abstract

We introduce Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials as generalizations of Bernoulli and Apostol-Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to sum a bunch of series involving the zeros of Bessel functions.