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 ∑n=0∞cnBn(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 ∑n=0∞cnBn(z) is absolutely convergent on ℂ, if the numerical series ∑n=0∞n!/πncn is absolutely convergent. Also, we present some applications and various examples of the topic such as the inequality |∑n=1∞πn/n!npBn(z)|≤ 2eπ|z|ζ(p) held for every fixed real number p>1 and all z∈ ℂ.
Related
- Critical cusp cancellation and analytic response for a Bernoulli random interval map
- New Formulas of Bernoulli Polynomials of the Second Kind Using Several Approaches
- Generalized Morgan-Voyce Polynomials with Positive and Negative Indices: Expansions and Analytic Properties
- Rhaly and generalized Hilbert operators between spaces of analytic functions
- Arc-analytic functions on locally closed semialgebraic sets
BibTeX
@misc{hooshmand2026bernoulli,
title = {{Bernoulli polynomials series via analytic summability of functions}},
author = {M. H. Hooshmand and S. Mehboodi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05526},
eprint = {2609.05526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05526},
}
RIS
TY - UNPB
TI - Bernoulli polynomials series via analytic summability of functions
AU - M. H. Hooshmand
AU - S. Mehboodi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.05526
UR - https://arxiv.org/abs/2609.05526
AB - 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 ∑_n=0^∞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 ∑_n=0^∞c_nB_n(z) is absolutely convergent on ℂ, if the numerical series ∑_n=0^∞n!/π^nc_n is absolutely convergent. Also, we present some applications and various examples of the topic such as the inequality |∑_n=1^∞π^n/n!n^pB_n(z)|≤ 2e^π|z|ζ(p) held for every fixed real number p>1 and all z∈ ℂ.
ER -