Abstract
Let m 1 , ⋯ , m r be nonnegative integers, and set [...]
Highlights
Laboratoire de Mathematiques et Modélisation d’Évry (UMR 8071), Université d’Évry Val d’Essonne, Department of Mathematics, Institute of Pure and Applied Mathematics, Chonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si 54896, Korea
Let q be a positive integer, we denote by χ a Dirichlet character modulo q, and L(s, χ) a Dirichlet
Studies of Bernoulli numbers and polynomials have been performed in many areas
Summary
We define the Bernoulli polynomials Bn ( x ) through the generating function:. For a Dirichlet character χ modulo q, the generalized Bernoulli numbers: Bn,χ ∈ Q(χ(1), χ(2), . Nielsen gave the following important result for the product of two Bernoulli polynomials. Nörlund [3] gave formulas for the integral of the product of two Bernoulli polynomials. Carlitz [7] studied the integrals of the product of three and four Bernoulli polynomials. There exist real numbers ak (m1 , ..., md ), 0 ≤ k ≤ Md , such that: Md. As the first goal of this paper, we prove explicit formulas for the coefficients ak (m1 , ..., md ) with 0 ≤ k ≤ Md. In the second part of this paper, we establish relationships between these coefficients and mean values of Dirichlet L-series at negative integers
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