An explicit Formula for Bernoulli Polynomials with a $q$ Parameter in Terms of $r$-Whitney Numbers

TL;DR

This paper introduces a new explicit formula for Bernoulli polynomials with a parameter q, expressed via r-Whitney numbers, and explores their algebraic and combinatorial properties.

Contribution

It provides the first explicit formula connecting Bernoulli polynomials with a q parameter to r-Whitney numbers of the second kind, along with derived identities.

Findings

Derived algebraic properties of Bernoulli polynomials with q parameter.

Established combinatorial identities involving r-Whitney numbers.

Connected Bernoulli and Cauchy polynomials through these identities.

Abstract

We define the Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of the second kind. Some algebraic properties and combinatorial identities of these polynomials are given. Also, we obtain several relations between the Cauchy and Bernoulli polynomials with a $q$ parameter in terms of $r$-Whitney numbers of both kinds.