Asymptotic Analysis of q-Recursive Sequences

TL;DR

This paper analyzes the asymptotic behavior of q-recursive sequences, showing they are q-regular, and provides detailed asymptotic results for specific sequences like Stern's sequence and generalized Pascal's triangle.

Contribution

It establishes that all q-recursive sequences are q-regular and offers a method to compute their q-linear representations efficiently, with detailed asymptotic analysis for key sequences.

Findings

q-recursive sequences are q-regular

Explicit asymptotic formulas for specific sequences

Efficient computation of q-linear representations

Abstract

For an integer $q\ge2$, 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.