Asymptotic Analysis of q-Recursive Sequences.

For an integer q ≥ 2 , a q-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q. In this article, q-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every q-recursive sequence is q-regular in the sense of Allouche and Shallit and that a q-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for q-recursive sequences are then obtained based on a general result on the asymptotic analysis of q-regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions ofStern's diatomic sequence,the number of non-zero elements in some generalized Pascal's triangle andthe number of unbordered factors in the Thue-Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.

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 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; andthe number of odd entries in the rows of Pascal's rhombus. For these examples, very precise asymptotic formulæ are presented. In the latter two examples, prior to this analysis only rough estimates were known.