Combinatorics on words and generating Dirichlet series of automatic sequences

TL;DR

This paper explores Dirichlet series associated with automatic sequences and restricted digit sets, extending previous results and connecting to well-known integer sequences and automatic sequence theory.

Contribution

It unifies and extends prior work on Dirichlet series of sequences defined by digit restrictions and automaticity, introducing new methods and results in this area.

Findings

Unified and extended previous results on Dirichlet series of digit-restricted sequences.

Connected sequences to known integer sequences and automatic sequence classes.

Identified a specific non-$b$-regular sequence within this framework.

Abstract

Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base $b$; i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words over the alphabet $\{0, 1, \dots, b-1\}$ that do not begin with a $0$. We show how to unify and extend results proved by Nathanson in 2021 and by K\"ohler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous $b$-automatic sequences or $b$-regular sequences. We also consider a specific sequence that is not $b$-regular.