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Calculating the numbers of representations and the Garsia entropy in linear numeration systems

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Abstract

Given a ≥ b, let G 0 = 1, G 1 = a + 1, and G n+2 = aG n+1 + bG n for n ≥ 0. For each choice of a and b, we have a linear recurrence that defines a numeration system. Every positive integer n may be written as the sum of the G n , with alphabet A = {0,1, . . . , a}, in one or more different ways. Let R (a,b)(n) be the function that counts the number of distinct representations of an integer as a sum of the G n . We extend results of Berstel, Kocábová, Masáková, and Pelantová, and Edson and Zamboni and give two distinct methods for calculating R (a,b)(n). One formula involves products of 2 × 2 matrices and the other sums of binomial coefficients modulo 2. For the main result, we consider the limiting measure μ β of a convergent infinite convolution of measures (Bernoulli convolutions), where β is the dominating root of the characteristic equation of the recurrence above. We study the Garsia entropy of these measures and calculate explicitly the limiting entropy associated with μ β . This result extends those of Alexander and Zagier, and Grabner, Kirschenhofer, and Tichy. We then see that all these results can be generalized further to confluent numeration systems.

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Author notes

  1. Marcia Edson

    Present address: Institute of Analysis and Computational Number Theory (Math A), Graz University of Technology, Steyrergasse 30/II, 8010, Graz, Austria

Authors and Affiliations

  1. Department of Mathematics and Statistics, Murray State University, Murray, KY, 42071, USA

    Marcia Edson

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  1. Marcia Edson
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Correspondence to Marcia Edson.

Additional information

Communicated by K. Schmidt.

M. Edson is supported by the Austrian Science Foundation FWF, project S9605, part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

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Edson, M. Calculating the numbers of representations and the Garsia entropy in linear numeration systems. Monatsh Math 169, 161–185 (2013). https://doi.org/10.1007/s00605-012-0397-6

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  • DOI: https://doi.org/10.1007/s00605-012-0397-6

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