On periodic continued fractions over \(\mathbb F_q((X^{-1}))\)
The authors consider the problem of periodic continued fractions over \(\mathbb F_q((X^{-1}))\) and derive some nice results on it. Let \(\mathbb F_q\) be a field with \(q\) elements of characteristic \(p\) and \(\mathbb F_q((X^{-1}))\) be the field of formal power series over \(\mathbb F_q\). Let \(f\) be a quadratic formal power series of continued fraction expansion \([b_0; b_1, \dots, b_s, \overline{a_1,\dots, a_t}]\), we denote by \(t = \text{Per}(f)\) the period length of the partial quotients of \(f\). The aim of this paper is to study the continued fraction expansion of \(Af\) where \(A\) is a polynomial \(\in \mathbb F_q[X]\). They also study the asymptotic behavior of the functions \(S(N,n)\) and \(R(N)\), where \[ S(N,n) =\sup_{\deg A=N}\sup_{f\in\Lambda_n}\text{Per}(Af)\quad\text{and}\quad R(N) =\sup _{n\geq 1}\frac{S(N,n)}{n}, \] where \(\Lambda_n\) is the set of quadratic formal power series of period \(n\) in \(\mathbb F_q((X^{-1}))\).
scientific article; zbMATH DE number 5910196