Transcendental continued β-fraction with quadratic Pisot basis over Fq((x−1))

Abstract

Let Fq be a finite field and Fq((x−1)) is the field of formal power series with coefficients in Fq. Let β ∈ Fq((x−1)) be a quadratic Pisot series with deg(β) = 2. We
establish a transcendence criterion depending on the continued β-fraction of one element
of Fq((x−1)).