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Asymptotic Analysis of q-Recursive Sequences
Clemens Heuberger1, Daniel Krenn2, Gabriel F Lipnik3
1Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria.
Algorithmica
|August 17, 2022
Summary
This study analyzes q-recursive sequences, demonstrating they are q-regular. The research provides a method to compute their q-linear representation and details their asymptotic behavior, including precise formulas for specific sequences.
Area of Science:
- Number Theory
- Discrete Mathematics
- Theoretical Computer Science
Background:
- q-recursive sequences are defined by recurrence relations based on indices modulo powers of q.
- Understanding the asymptotic behavior of these sequences is crucial for various mathematical and computational applications.
Purpose of the Study:
- To study q-recursive sequences and analyze the asymptotic behavior of their summatory functions.
- To establish the connection between q-recursive and q-regular sequences.
- To derive precise asymptotic formulas for specific mathematical sequences.
Main Methods:
- Defining q-recursive sequences using recurrence relations modulo powers of q.
- Proving that every q-recursive sequence is q-regular (Allouche & Shallit).
- Utilizing a general result on the asymptotic analysis of q-regular sequences.
Main Results:
- Every q-recursive sequence is shown to be q-regular.
- A q-linear representation of q-recursive sequences can be easily computed.
- Detailed asymptotic results for q-recursive sequences are derived.
- Precise formulas without error terms are obtained for Stern's diatomic sequence and the number of non-zero elements in generalized Pascal's triangles.
Conclusions:
- The study establishes q-recursive sequences as a subset of q-regular sequences.
- The methods provide a framework for analyzing the asymptotic behavior of a broad class of sequences.
- The findings offer exact results for specific, notable sequences in number theory and combinatorics.
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