Finite generating functions for the sum-of-digits sequence
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)].
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