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 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.