Bernoulli多项式级数：基于函数的解析可和性

中文摘要 AI 辅助

实函数与复函数的解析可和性于2016年被引入。在该主题中，伯努利数和伯努利多项式被用于在开域$D$上定义给定函数（具有幂级数展开）的解析和项。本文研究了解析和项函数的导数与积分，并证明了这些导数与相同的Bernoulli多项式级数$\u003cspan class=\"math-inline\"\u003e$\sum_{n=0}^{\infty}c_nB_n(z)$\u003c/span\u003e相差一个单位前移。因此，解析可和性主题为研究Bernoulli多项式级数提供了一个平台。通过这种方式，我们获得了关于Bernoulli多项式级数的许多新结果，例如一些相关的级数收敛检验以及Bernoulli多项式和所述级数的上界。例如，我们观察到：如果数值级数$\u003cspan class=\"math-inline\"\u003e$\sum_{n=0}^{\infty}\frac{n!}{\pi^n}c_n$\u003c/span\u003e绝对收敛，则$\u003cspan class=\"math-inline\"\u003e$\sum_{n=0}^{\infty}c_nB_n(z)$\u003c/span\u003e在$\mathbb{C}$上绝对收敛。此外，我们给出了该主题的一些应用和各种例子，例如不等式$$|\sum_{n=1}^{\infty}\frac{\pi^n}{n!n^p}B_n(z)|\leq 2e^{\pi|z|}\zeta(p)$$对每个固定的实数$p>1$和所有$z\in \mathbb{C}$成立。

英文摘要

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 $\sum_{n=0}^{\infty}\frac{n!}{π^n}c_n$ is absolutely convergent. Also, we present some applications and various examples of the topic such as the inequality $$|\sum_{n=1}^{\infty}\frac{π^n}{n!n^p}B_n(z)|\leq 2e^{π|z|}ζ(p)$$ held for every fixed real number $p>1$ and all $z\in \mathbb{C}$.

发表机构

• Islamic Azad University（伊斯兰阿扎德大学）
• Zand Institute of Higher Education（赞德高等教育学院）

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