Connection Coefficients for Higher-order Bernoulli and Euler Polynomials: A Random Walk Approach

TL;DR

This paper explores how random walk models, specifically reflected Brownian motion and Bessel processes, can be used to derive identities involving higher-order Bernoulli and Euler polynomials through hitting times.

Contribution

It introduces a novel approach using stochastic processes to obtain connection coefficients for higher-order Bernoulli and Euler polynomials.

Findings

Derived identities involving higher-order Bernoulli and Euler polynomials from hitting times.

Established connections between random walk models and polynomial identities.

Provided new tools for analyzing special polynomials via stochastic processes.

Abstract

We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a $1$-dimensional linear reflected Brownian motion and of a $3$-dimensional Bessel process. Considering the successive hitting times of two, three, and four fixed levels by these random walks yields non-trivial identities that involve higher-order Bernoulli and Euler polynomials.