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A joint central limit theorem for the sum-of-digits function, and asymptotic divisibility of Catalan-like sequences


Authors: Michael Drmota and Christian Krattenthaler
Journal: Proc. Amer. Math. Soc. 147 (2019), 4123-4133
MSC (2010): Primary 11K65; Secondary 05A15, 11A63, 11N60, 60F05
DOI: https://doi.org/10.1090/proc/14349
Published electronically: July 8, 2019
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Abstract: We prove a central limit theorem for the joint distribution of $ s_q(A_jn)$, $ 1\le j \le d$, where $ s_q$ denotes the sum-of-digits function in base $ q$ and the $ A_j$'s are positive integers relatively prime to $ q$. We do this in fact within the framework of quasi-additive functions. As an application, we show that most elements of ``Catalan-like'' sequences--by which we mean integer sequences defined by products/quotients of factorials--are divisible by any given positive integer.


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Additional Information

Michael Drmota
Affiliation: Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria

Christian Krattenthaler
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

DOI: https://doi.org/10.1090/proc/14349
Keywords: Sum-of-digits function, quasi-additive functions, central limit theorem, Catalan numbers, central binomial coefficients
Received by editor(s): March 30, 2018
Received by editor(s) in revised form: August 8, 2018
Published electronically: July 8, 2019
Additional Notes: This research was partially supported by the Austrian Science Foundation FWF, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”, project F50-N15.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2019 American Mathematical Society