Papers › Asymptotic Analysis of q-Recursive Sequences
Asymptotic Analysis of q-Recursive Sequences
Clemens Heuberger, Daniel Krenn, Gabriel F. Lipnik
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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 formul\ae{} without error terms.
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