Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$

Abstract

The main purpose of this paper is to show some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level m. Our approach is based on the use of Fourier expansions for the periodic generalized Bernoulli functions of level m, as well as quadrature formulae of Euler-Maclaurin type. Some illustrative examples involving such relations are also given.2010 Mathematics Subject Classification. 65D32, 41A55, 65B15, 33F05. Key words and phrases. Bernoulli polynomials; generalized Bernoulli polynomials of level m; Euler-Maclaurin quadrature formulae; quadrature formula; Riemann zeta function.(1) Supported in part by Decanato de Investigación y Desarrollo, Universidad Simón Bolívar, Venezuela, grant DID-USB (S1-IC-CB-004-17).

Mixed-Type Hypergeometric Bernoulli–Gegenbauer Polynomials

“…(0) for all n ≥ 0. The hypergeometric Bernoulli polynomials also are called generalized Bernoulli polynomials of level m [5,6]. It is clear that if m = 1 in (3), then we obtain the definition of the classical Bernoulli polynomials B n (x) and classical Bernoulli numbers, respectively, i.e.,…”

“…The following results summarize some properties of the hypergeometric Bernoulli polynomials (cf. [5,6,11,12,15]).…”

“…For a comprehensive treatment of the diverse aspects, including summation formulas and applications, interested readers can refer to recent works [1,2]. Inspired by recent articles [3][4][5][6][7] where authors explore analytic and numerical aspects of hypergeometric Bernoulli polynomials, hypergeometric Euler polynomials, generalized mixed-type Bernoulli-Gegenbauer polynomials, and Lagrange-based hypergeometric Bernoulli polynomials, this article focuses on the algebraic and differential properties of the polynomials…”