Some quadratic irrationals with explicit continued fraction and Engel series expansions

Abstract

A remarkable class of quadratic irrational elements having both explicit Engel series and continued fraction expansions in the field of Laurent series, mimicking the case of real numbers discovered by Sierpiński and later extended by Tamura, is constructed. Linear integer-valued polynomials which can be applied to construct such class are determined. Corresponding results in the case of real numbers are mentioned.