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DNA Nanotubes as a Versatile Tool to Study Semiflexible Polymers

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Asymptotic Analysis of Regular Sequences.

Clemens Heuberger¹, Daniel Krenn¹

¹Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt am Wörthersee, Austria.

|February 29, 2020

Summary

This summary is machine-generated.

This study analyzes q-regular sequences, showing their summatory functions decompose into periodic fluctuations. These fluctuations are linked to eigenvalues, with precise asymptotic formulas derived for key examples like esthetic numbers and Pascal

Keywords:

Esthetic numbers Mellin–Perron summation Pascal’s rhombus Regular sequence Summatory function Tauberian theorem Transducer

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Area of Science:

Number Theory
Combinatorics
Theoretical Computer Science

Background:

q-regular sequences are fundamental in various mathematical fields.
Previous analyses often provided only rough estimates for related quantities.
Understanding the asymptotic behavior of these sequences is crucial for advanced mathematical research.

Purpose of the Study:

To asymptotically analyze q-regular sequences.
To decompose the summatory function of regular sequences into periodic fluctuations.
To provide efficient methods for computing Fourier coefficients of these fluctuations.

Main Methods:

Asymptotic analysis using Mellin-Perron summations.
Employing Hölder regularity and a pseudo-Tauberian argument to address convergence.
Linear representation of sequences and eigenvalue analysis.
Residue calculus on Dirichlet generating functions.

Main Results:

Demonstrated asymptotic decomposition of summatory functions into scaled periodic fluctuations.
Identified eigenvalues of linear representations as key contributors to growth rates.
Developed efficient computation of Fourier coefficients via Dirichlet generating functions.
Derived precise asymptotic formulas for specific sequences (transducer outputs, esthetic numbers, Pascal's rhombus entries).

Conclusions:

The study provides a general framework for the asymptotic analysis of q-regular sequences.
The derived methods offer significant improvements over previous estimation techniques.
The results have direct applications in number theory and combinatorics, offering new insights into classical problems.