Asymptotic Analysis of q-Recursive Sequences
Abstract
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 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.
Cite
@article{arxiv.2105.04334,
title = {Asymptotic Analysis of q-Recursive Sequences},
author = {Clemens Heuberger and Daniel Krenn and Gabriel F. Lipnik},
journal= {arXiv preprint arXiv:2105.04334},
year = {2024}
}