Combinatorics on words and generating Dirichlet series of automatic sequences
CoRR(2024)
摘要
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 on the alphabet {0, 1,
…, b-1} that do not begin with a 0. We show how to unify and extend
results proved by Nathanson in 2021 and by Köhler and Spilker in 2009. En
route, we encounter several sequences from Sloane's On-Line Encyclopedia of
Integer Sequences, as well as some famous q-automatic sequences or
q-regular sequences.
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