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

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.