Asymptotic Analysis of Regular Sequences

Abstract

In this article, q-regular sequences in the sense of Allouche and Shallit are analysed asymptotically. It is shown that the summatory function of a regular sequence can asymptotically be decomposed as a finite sum of periodic fluctuations multiplied by a scaling factor. Each of these terms corresponds to an eigenvalue of the sum of matrices of a linear representation of the sequence; only the eigenvalues of absolute value larger than the joint spectral radius of the matrices contribute terms which grow faster than the error term.The paper has a particular focus on the Fourier coefficients of the periodic fluctuations: They are expressed as residues of the corresponding Dirichlet generating function. This makes it possible to compute them in an efficient way. The asymptotic analysis deals with Mellin-Perron summations and uses two arguments to overcome convergence issues, namely Hölder regularity of the fluctuations together with a pseudo-Tauberian argument.Apart from the very general result, three examples are discussed in more detail:• sequences defined as the sum of outputs written by a transducer when reading a q-ary expansion of the input; • the amount of esthetic numbers in the first N natural numbers; and • the number of odd entries in the rows of Pascal's rhombus.For these examples, very precise asymptotic formulae are presented. In the latter two examples, prior to this analysis only rough estimates were known. The extended abstract [31] (with appendices containing proofs available as arXiv:1802.03266) imposes a restriction on the asymptotic growth. The extended abstract [30] (with appendices containing proofs available as arXiv:1808.00842) lifts this restriction by completely getting rid of the corresponding technical condition. This article now contains the full (majorly restructured) proof covering all cases. It is shorter and simpler. We now use a generating functions approach which also gives additional insights. For example, the cancellations in the proof in [31] seem to be a kind of magic at that point, but with the new approach, it is now clear and no surprise anymore that they have to appear. Besides, the examples investigated in [31,30] are now presented with full details. A new part on computational aspects of the computation of Fourier coefficients is added. Reading strategies for various interests are now outlined in Part I so that readers find their ways through this article.The authors thank Sara Kropf for her comments on an early version of this paper.

The periodic complexity function of the Thue-Morse word, the Rudin-Shapiro word, and the period-doubling word

“…Dumas [1] obtained a description of the asymptotics of the summatory function of q-regular sequences. Heuberger and Krenn [2] have also recently given a similar description of these asymptotics.…”

“…Note that we have defined the linear representation of x(n) in terms of the most-significantdigit first representation of n. It can also be defined using the least-significant-digit first representation of n (as it is in [2]). One can easily convert from one representation to the other by taking the transpose of v, M 0 , .…”

Asymptotic Analysis of q-Recursive Sequences

“…SageMath [ 37 ] provides an implementation of this minimization algorithm. The statement of Theorem A for and is mentioned by the first two authors of this article in [ 20 , Remark 5.1]. …”

“…The statement of Theorem A for and is mentioned by the first two authors of this article in [ 20 , Remark 5.1]. …”

“…There exist explicit results for many particular regular sequences and also some quite general results. Dumas [ 13 ] as well as the first two authors of this paper together with Prodinger [ 20 , 22 ] studied the asymptotics of q -regular sequences in general. The two works [ 20 , 22 ] will also be one of the main ingredients for obtaining the asymptotic behavior of q -recursive sequences.…”