Bernoulli polynomials series via analytic summability of functions

Abstract

Analytic summability of real and complex functions was introduced in 2016. In the topic, the Bernoulli numbers and polynomials were used for defining analytic summand of a given function with a power series on an open domain $D$. In this paper, we study derivatives and integrals of the analytic summand functions and show that the derivatives are the same Bernoulli polynomials series $\sum_{n=0}^{\infty}c_nB_n(z)$ up to a unit forward shift. Therefore, the topic of the analytic summability is a platform for studying Bernoulli polynomials series. In the way, we obtain many new results for the Bernoulli polynomials series such as some related series convergent tests and upper bounds for the Bernoulli polynomials and the mentioned series. For instance, we observe that $\sum_{n=0}^{\infty}c_nB_n(z)$ is absolutely convergent on $\mathbb{C}$, if the numerical series $\sum_{n=0}^{\infty}\frac{n!}{\pi^n}c_n$ is absolutely convergent. Also, we present some applications and various examples of the topic such as the inequality $$|\sum_{n=1}^{\infty}\frac{\pi^n}{n!n^p}B_n(z)|\leq 2e^{\pi|z|}\zeta(p)$$ held for every fixed real number $p>1$ and all $z\in \mathbb{C}$.