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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Published in:	Algorithmica Vol. 84; no. 9; pp. 2480 - 2532
Main Authors:	Heuberger, Clemens, Krenn, Daniel, Lipnik, Gabriel F.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2022 Springer Nature B.V
Subjects:	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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Abstract	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.
AbstractList	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. For an integer , a -recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of . In this article, -recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every -recursive sequence is -regular in the sense of Allouche and Shallit and that a -linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for -recursive sequences are then obtained based on a general result on the asymptotic analysis of -regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions ofStern's diatomic sequence,the number of non-zero elements in some generalized Pascal's triangle andthe number of unbordered factors in the Thue-Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms. For an integer $$q\ge 2$$ 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. 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 ofStern’s diatomic sequence,the number of non-zero elements in some generalized Pascal’s triangle andthe number of unbordered factors in the Thue–Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms. For an integer $$q\ge 2$$ 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 ofStern’s diatomic sequence,the number of non-zero elements in some generalized Pascal’s triangle andthe number of unbordered factors in the Thue–Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms. 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 ofStern's diatomic sequence,the number of non-zero elements in some generalized Pascal's triangle andthe number of unbordered factors in the Thue-Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.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 ofStern's diatomic sequence,the number of non-zero elements in some generalized Pascal's triangle andthe number of unbordered factors in the Thue-Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.
Author	Lipnik, Gabriel F. Heuberger, Clemens Krenn, Daniel
Author_xml	– sequence: 1 givenname: Clemens orcidid: 0000-0003-0082-7334 surname: Heuberger fullname: Heuberger, Clemens organization: Alpen-Adria-Universität Klagenfurt – sequence: 2 givenname: Daniel orcidid: 0000-0001-8076-8535 surname: Krenn fullname: Krenn, Daniel email: math@danielkrenn.at organization: Paris Lodron University of Salzburg – sequence: 3 givenname: Gabriel F. orcidid: 0000-0002-4362-429X surname: Lipnik fullname: Lipnik, Gabriel F. organization: Graz University of Technology
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Cites_doi	10.1017/CBO9780511550447 10.1017/CBO9780511566097 10.1016/j.aam.2016.04.006 10.4169/000298910X496714 10.2307/2304500 10.1142/S0129054112400448 10.1016/j.disc.2017.01.003 10.1016/S0304-3975(03)00090-2 10.1016/0304-3975(92)90001-V 10.1051/ita:2008006 10.1142/S0129054113400182 10.1007/s00453-019-00631-3 10.1145/3127585 10.1016/j.disc.2011.07.029 10.1016/j.laa.2012.10.013 10.1142/S0129054114400267 10.1016/j.ejc.2004.04.009 10.1017/CBO9780511546563 10.1016/j.aam.2004.11.004 10.1007/978-3-540-95980-9 10.1016/S1385-7258(60)50046-1 10.1515/crll.1858.55.193 10.1017/CBO9781107326019 10.37236/6581 10.1080/00150517.2003.12428593 10.4230/LIPIcs.AofA.2018.27 10.1007/978-3-642-37064-9_27 10.1007/978-3-319-69152-7_2
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Keywords	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
Language	English
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Snippet	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... For an integer $$q\ge 2$$ q ≥ 2 , a q -recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q . In this... For an integer , a -recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of . In this article, -recursive... 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... 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,... For an integer $$q\ge 2$$ q≥2, a q-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q. In this article,...
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SubjectTerms	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
Title	Asymptotic Analysis of q-Recursive Sequences
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