Asymptotic Analysis of Regular Sequences

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 H\"older 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 formul\ae{} are presented. In the latter two examples, prior to this analysis only rough estimates were known.

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