Let $F\left(X\right)={\sum }_{n\ge 0}{(-1)}^{\epsilon ₙ}{X}^{-\lambda ₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and ${\lambda }_{n+1}/\lambda ₙ>2$. It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that $Q_{\omega }\left(X\right)$ is a polynomial if and only if ω ∈ ℤ. In all the other cases $Q_{\omega }\left(X\right)$ is an infinite formal power series; we discuss its algebraic...

Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent $p_{rn+i}/q_{rn+i}$ (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping...

Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form $p_{rn+i}/q_{rn+i}$ (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.

Let d ≥ 2 be a square-free integer and for all n ≥ 0, let $l\left({\left(\surd d\right)}^{2n+1}\right)$ be the length of the continued fraction expansion of ${\left(\surd d\right)}^{2n+1}$. If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that $C₁{\left(\surd d\right)}^{2n+1}\ge l\left({\left(\surd d\right)}^{2n+1}\right)\ge C₂{\left(\surd d\right)}^{2n+1}$ for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].