Random Walk Models for Nontrivial Identities of Bernoulli and Euler Polynomials

2021/12/16 by Lin Jiu, Jiu, Lin, Italo Simonelli +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #05A19 #11B68 #60G50 #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #Complex Systems and Time Series Analysis #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2112.08716

openalex publication_date 2021/12/16 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

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.

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