Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus

TL;DR

This paper provides an asymptotic analysis of the summatory functions of q-regular sequences, revealing periodic fluctuations linked to eigenvalues, with applications to transducer outputs and Pascal's rhombus.

Contribution

It introduces a novel asymptotic framework for q-regular sequences, extending pseudo Tauberian methods to handle convergence issues and expressing fluctuations via Dirichlet residues.

Findings

Periodic fluctuations are characterized by eigenvalues exceeding the joint spectral radius.

Fourier coefficients of fluctuations are expressed through residues of Dirichlet generating functions.

Two detailed examples illustrate the application to transducer sums and Pascal's rhombus entries.

Abstract

The summatory function of a $q$-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations for eigenvalues of absolute value larger than the joint spectral radius of the matrices of a linear representation of the sequence. The Fourier coefficients of the fluctuations are expressed in terms of residues of the corresponding Dirichlet generating function. A known pseudo Tauberian argument is extended in order to overcome convergence problems in Mellin--Perron summation. Two examples are discussed in more detail: The case of sequences defined as the sum of outputs written by a transducer when reading a $q$ary expansion of the input and the number of odd entries in the rows of Pascal's rhombus.