If a machine did it, it is probably transcendental (even p-adically)

Abstract

Abstract Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of p -adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the p -adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski–Bugeaud in the real setting and extends previous works that were limited to specific choices of p -adic floor functions and less general classes of words.