Random Walk Models for Nontrivial Identities of Bernoulli and Euler Polynomials

TL;DR

This paper derives new identities for Bernoulli and Euler polynomials using loop decompositions of hitting times in reflected Brownian motion and Bessel processes, linking stochastic processes with special polynomials.

Contribution

It introduces loop identities for hitting times and connects them to Bernoulli and Euler polynomials, providing a novel probabilistic approach to these classical polynomials.

Findings

Derived loop identities for hitting times of stochastic processes.

Expressed Bernoulli and Euler polynomials via higher-order Euler polynomials.

Established combinatorial and inductive proofs for the identities.

Abstract

We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.