Lacunary formal power series and the Stern–Brocot sequence

Abstract

Let F (X) = n≥0 (−1) εn X −λn be a real lacunary formal power series, where ε n = 0, 1 and λ n+1 /λ n > 2. It is known that the denominators Q n (X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Q n (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 ω (X) is a polynomial if and only if ω ∈ Z. In all the other cases Q ω (X) is an infinite formal power series, the algebraic properties of which we discuss in the special case λ n = 2 n+1 − 1.Keywords: Stern-Brocot sequence, continued fractions of formal power series, automatic sequences, algebraicity of formal power series.

Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

“…There are polynomial analogs of Stern's diatomic sequence (as in the work of Dilcher and Stolarsky [13,14], of Coons [9], of Dilcher and Ericksen [12], of Klavžar, Milutinović and Petr [30], of Ulas [50,51], of Vargas [52], of Bundschuh [5], of Bundschun and Väänänen [6] and of Allouche and Mendès France [1]). What are the polynomial analogs for TRIP-Stern sequences?…”

Note on the Stern-Brocot sequence, some relatives, and their generating power series

“…In their Remark 1.3, Allouche and Mendès France [1] defined a map C associating with two sequences a = (a n ) n≥0 and b = (b n ) n≥0 the sequence sequence), which is identical with the sequence (a n+1 ) n≥0 considered in [2, (1)]. The authors of [1] recalled that the ±1 Thue-Morse sequence t = (t n ) n≥0 is given by t 0 = 1 and, for all n ≥ 0, by t 2n = t n and t 2n+1 = −t n .…”

“…The authors of [1] recalled that the ±1 Thue-Morse sequence t = (t n ) n≥0 is given by t 0 = 1 and, for all n ≥ 0, by t 2n = t n and t 2n+1 = −t n . Then they introduced the sequences α = (α n ) n≥0 , β = (β n ) n≥0 , γ = (γ n ) n≥0 by α := C(t, 1), β := C(1, t), γ := C(t, t) to which σ := C(1, 1) is added, σ for Stern, and noted that these four sequences satisfy the recurrences…”