Inverse relations and the products of Bernoulli polynomials
type: Item
instance of: scholarly article
title: Inverse relations and the products of Bernoulli polynomials
zbMATH Open document ID: 1238.11020
author: Amy M. Fu
author: Iris F. Zhang
published in: Graphs and Combinatorics
publication date: 2010-10-07T00:00:00Z
review text: In this article, the authors obtain two inverse pairs of identities involving the Bernoulli polynomials and the Bernoulli polynomials of the second kind. More precisely, the following are given: Theorem 1. For \(y=x_1+x_2+\cdots +x_N\) and \(n\geq N\), \[ \begin{multlined} \sum_{k_1+\cdots+k_N=n, k_1,\dots,k_N\geq 0}{n \choose {k_1,\dots,k_n}}B_{k_1}(x_1)\cdots B_{k_N}(x_N)=(-1)^{N-1}N{n \choose N}\\ \times \sum_{j=0}^{N-1}(-1)^j\frac{B_{n-j}(y)}{n-j}\sum_{k=0}^j{{N+k-j-1} \choose k}s(N,N-j+k)y^k,\end{multlined} \] and for \(n\geq N+1\), \[ \begin{multlined} (n-N)\sum_{k=0}^{N}{N \choose k}(-y)^k \frac{B_{n-k}(y)}{n-k}=(-1)^N\sum_{k=0}^N{{n-N+k} \choose k}^{-1}S(N+1,k+1)\\ \times \sum_{l_0+\cdots + l_k=n-N+k,l_0,\dots,l_k\geq 0}{{n-N+k} \choose {l_0,\dots,l_k}}B_{l_0}(y-\sum_{i=1}^k x_i)\prod_{i=1}^kB_{l_i}(x_i),\end{multlined} \] where \(s(n,k)\) and \(S(n,k)\) are the Stirling numbers of the first kind and the second kind, respectively. Also similar identities for the Bernoulli polynomials of the second kind are given as Theorem 2.
reviewed by: Kaori Ota
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 5797147
zbMATH Keywords: Bernoulli polynomials
zbMATH Keywords: Bernoulli polynomials of the second kind
zbMATH Keywords: Stirling number of the first kind and the second kind
MaRDI profile type: Publication
full work available at URL: s00373-010-0947-4
OpenAlex ID: W1987863055
cites work: Convolution identities and lacunary recurrences for Bernoulli numbers
cites work: Sums of products of generalized Bernoulli polynomials.
cites work: Sums of products of Bernoulli numbers
cites work: Q4320535
cites work: Sums of Products of Bernoulli numbers of the second kind
DOI: 10.1007/S00373-010-0947-4
description: scientific article; zbMATH DE number 5797147