Finite generating functions for the sum-of-digits sequence
type: Item
instance of: scholarly article
title: Finite generating functions for the sum-of-digits sequence
zbMATH Open document ID: 1464.11016
author: Christophe Vignat
author: Tanay Wakhare
published in: The Ramanujan Journal
publication date: 2020-01-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1708.06479
review text: Among the numerous properties of the sum of digits of an integer \(n\) in an integer base \(b\), denoted \(s_b(n)\), is its occurrence in nice infinite series. We only give three examples: \[ \sum_{n \geq 1} \frac{s_b(n)}{n(n+1)} = \frac{b}{b-1} \log b \] (due to Shallit for general \(b\)) \[ \sum_{n \geq 1} \frac{s_2(n)}{2n(2n+1)(2n+2)} = - \frac{1}{2} \log \pi + \frac{\gamma}{2} + \frac{1}{2} \log 2 \] (similar to a series of Addison also studied by Behrmann, Van Lint, and Gerst) and \[ \sum_{n \geq 1} \frac{s_2(n)}{2n(2n+1)} = - \frac{1}{2} \log \pi + \frac{\gamma}{2} + \log 2 \] (due to Sondow). In the paper under review the authors give nice generalizations of series above; in particular they also study finite sums of the same type (which, of course, converge to series similar to the previous ones). There is some sort of fascination when one looks at identities between seemingly unrelated finite sums or at unexpected closed formulas for infinite series, even after having read a proof: perhaps something similar to the fascination in front of magic tricks -- even when you know how they work. We end with three remarks: -- The following reference could be added to the bibliography: [\textit{I. Gerst}, Am. Math. Mon. 76, 273--275 (1969; Zbl 0174.09503)]. -- It may be the case that the ideas of the authors give some generalizations of results in the paper: [\textit{J.-P. Allouche}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 40, 69--79 (2013; Zbl 1289.11067)]. -- Note that Reference [16] has appeared: see [\textit{M. Merca} and \textit{M. D. Schmidt}, Contrib. Discrete Math. 14, No. 1, 31--45 (2019; Zbl 1470.11008)].
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11B85
zbMATH DE Number: 7150326
zbMATH Keywords: sum-of-digits
zbMATH Keywords: Hurwitz zeta function
zbMATH Keywords: infinite products
zbMATH Keywords: Lambert transform
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
OpenAlex ID: W2963423000
arXiv ID: 1708.06479
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Wikidata QID: Q128954116
DOI: 10.1007/S11139-018-0065-0
describes a project that uses: Digital Library of Mathematical Functions
description: scientific article