Asymptotic Analysis of q-Recursive Sequences
Abstract
For an integer , 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 ofFor the first two sequences, our analysis even leads to precise formulæ without error terms.
•
Stern’s diatomic sequence,
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the number of non-zero elements in some generalized Pascal’s triangle and
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the number of unbordered factors in the Thue–Morse sequence.
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Publication History
Published: 01 September 2022
Accepted: 22 January 2022
Received: 11 May 2021
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