On non-holonomicity, transcendence and $p$-adic valuations

TL;DR

This paper investigates the algebraic and transcendental nature of power series with coefficients based on p-adic valuations, revealing their non-holonomicity and transcendence for many rational and algebraic irrationals, with applications to automatic sequences.

Contribution

It demonstrates that certain p-adic valuation-based power series are non-holonomic and transcendental, providing new insights into their algebraic properties and applications to automatic sequences.

Findings

Power series with coefficients ${ u}_q(n)$ are non-holonomic and not algebraic.

Infinitely many rational and algebraic irrational numbers yield transcendental series values.

Applications to p-automatic sequences, including the period-doubling sequence.

Abstract

Let ${\nu}_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients ${\nu}_q(n)$, respectively ${\nu}_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many rational numbers and infinitely many algebraic irrational numbers for which the values of these series are transcendental. We apply these results to some $p$-automatic sequences, one of them being the period-doubling sequence.