Transcendental $p$-adic continued fractions

TL;DR

This paper introduces a new transcendence criterion for Ruban $p$-adic continued fractions, providing explicit examples and showing that a classical $p$-adic Lagrange theorem does not hold in this context.

Contribution

It establishes a novel transcendence criterion for Ruban $p$-adic continued fractions and demonstrates the failure of the $p$-adic Lagrange theorem analogy.

Findings

Explicit transcendental Ruban continued fractions with bounded partial quotients.

A new transcendence criterion for $p$-adic continued fractions.

The $p$-adic Lagrange theorem analogy does not hold for Ruban continued fractions.

Abstract

We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial quotients. This is $p$-adic analogy of Baker's result. We also prove that $p$-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.