Formulas of special polynomials involving Bernoulli polynomials derived from matrix equations and Laplace transform

TL;DR

This paper develops new formulas for special polynomials, especially Bernoulli polynomials, using matrix equations and Laplace transforms, revealing connections to the Hurwitz zeta function and infinite series.

Contribution

It introduces a linear transformation on polynomial rings, derives matrix equations involving Bernoulli and Bell polynomials, and applies Laplace transforms to generate novel formulas.

Findings

New formulas involving Bernoulli polynomials and matrix equations

Derivation of formulas connecting Bernoulli polynomials with Hurwitz zeta function

Application of Laplace transform to generate infinite series formulas

Abstract

The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some special cases of this matrix, matrix equations including inverse matrices, the Bell polynomials will be given. With the help of these equations, new formulas containing different polynomials, especially the Bernoulli polynomials, will be given. Finally, by applying the Laplace transform to the generating function for the Bernoulli polynomials, we derive some novel formulas involving the Hurwitz zeta function and infinite series.