Asymptotic Analysis of q-Recursive Sequences

Abstract

For an integer $q\ge 2$, 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 of Stern's diatomic sequence, the number of non-zero elements in some generalized Pascal's triangle and the number of unbordered factors in the Thue--Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.