Explicit formulas for Bernoulli polynomials of order \(n\)
type: Item
instance of: scholarly article
title: Explicit formulas for Bernoulli polynomials of order \(n\)
zbMATH Open document ID: 0860.11009
author: Junesang Choi
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1996-12-11T00:00:00Z
review text: The author shows that the \(n\)-th order Hurwitz zeta function \(\zeta_n (s,x)\), defined as the meromorphic continuation of the series \(\sum (x+k_1+ \cdots+ k_n)^{-s}\) extended over all nonnegative integers \(k_1, \dots, k_n\), can be expressed in terms of classic (first order) Hurwitz zeta functions as a linear combination of the form \[ \zeta_n(s,x) = \sum^{n-1}_{j=0}p_{n,j} (x)\zeta (s-j,x), \] where the multipliers are polynomials in \(x\) whose coefficients involve Stirling numbers of the first kind. By calculating the residue at \(s=k\) of each member of the last equation, the author obtains an explicit formula expressing Bernoulli polynomials of order \(n\) as a finite double sum involving classic Bernoulli polynomials (of order 1) and Stirling numbers.
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 941303
zbMATH Keywords: higher order Hurwitz zeta function
zbMATH Keywords: Stirling numbers of the first kind
zbMATH Keywords: Bernoulli polynomials
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 941303