Some nonlinear functions of Bernoulli and Euler umbrae

TL;DR

This paper explores the probabilistic interpretations of nonlinear moments related to Bernoulli and Euler umbrae, extending previous results to Bernoulli polynomials and providing new insights into their properties.

Contribution

It introduces the probabilistic counterparts of nonlinear moments of Bernoulli umbrae and extends these results to Bernoulli polynomials, advancing understanding of their stochastic properties.

Findings

Probabilistic interpretation of nonlinear moments of Bernoulli umbrae

Extension of results to Bernoulli polynomials

New relationships between Bernoulli and Euler umbrae

Abstract

In a recent paper, Yi-Ping Yu has given some interesting nonlinear moments of the Bernoulli umbra; the aim of this paper is to show the probabilistic counterpart of these results and to extend them to Bernoulli polynomials.