An analytic approach to special numbers and polynomials

TL;DR

This paper introduces an analytic approach to special numbers and polynomials using derivative polynomials, offering new integral representations for Bernoulli numbers and polynomials, and serves as a review with novel elements.

Contribution

It presents a simplified analytic method based on derivative polynomials and introduces new integral representations for Bernoulli numbers and polynomials.

Findings

New integral representations for Bernoulli numbers

New integral representations for Bernoulli polynomials

A review of analytic approaches to special numbers and polynomials

Abstract

The purpose of this article is to present, in a simple way, an analytic approach to special numbers and polynomials. The approach is based on the derivative polynomials. The paper is, to some extent, a review article, although it contains some new elements. In particular, it seems that some integral representations for Bernoulli numbers and Bernoulli polynomials can be seen as new.