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 sequen...

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Veröffentlicht in:Algorithmica Jg. 84; H. 9; S. 2480 - 2532
Hauptverfasser: Heuberger, Clemens, Krenn, Daniel, Lipnik, Gabriel F.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.09.2022
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Asymptotic properties
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Mathematics of Computing
Theory of Computation
68Q45
Pascal’s triangle
Recurrence relation
30B50
68R05
Summatory function
Stern’s diatomic sequence
68R15
Thue–Morse sequence
11B37
Regular sequence
Digital function
11A63
Dirichlet series
Asymptotic analysis
05A16
ISSN:0178-4617, 1432-0541
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Zusammenfassung: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.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-022-00950-y