Sums of products of two \(q\)-Bernoulli numbers
type: Item
instance of: scholarly article
title: Sums of products of two \(q\)-Bernoulli numbers
zbMATH Open document ID: 0916.11014
author: Junya Satoh
published in: Journal of Number Theory
publication date: 1999-07-07T00:00:00Z
review text: Let \(q\in \mathbb C\), \(| q| <1\). The sequence of \(q\)-Bernoulli numbers \(\beta _n\) (\(n\geq 0\)) is defined by the relations \[ \beta _0=1,\quad \sum \limits_{j=0}^nq^{j+1}\binom nj\beta_j- \beta_n=\begin{cases} 1, &\text{for \(n=1\)}\\ 0, &\text{for \(n>1\)}.\end{cases} \] Let \(G_q\) be the generating function of the sequence \(\{ \beta_n\}\). It is proved that \[ G_q*\varphi (G_q)=G_q-(G_q+qG_q')*\varphi \] where \(*\) is a \(q\)-analogue of the product of formal power series [\textit{J. Satoh}, Nagoya Math. J. 127, 129-143 (1992; Zbl 0761.11023)], while \(\varphi\) is the differential operator \(I+(q-1)\frac{d}{dt}\).
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 1271088
zbMATH Keywords: \(q\)-Bernoulli number
zbMATH Keywords: generating function
zbMATH Keywords: \(*\)-product
reviewed by: Anatoly N. Kochubei
MaRDI profile type: Publication
full work available at URL: jnth.1998.2331
OpenAlex ID: W2314203635
cites work: \(q\)-analogue of Riemann's \(\zeta\)-function and \(q\)-Euler numbers
cites work: A construction of q-analogue of Dedekind sums
cites work: Q4306516
cites work: Sums of products of Bernoulli numbers
DOI: 10.1006/JNTH.1998.2331
description: scientific article; zbMATH DE number 1271088