Specialisation and Reduction of Continued Fractions of Formal Power Series

Poorten, Alfred J. van der

doi:10.1007/s11139-005-0827-3

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The Sequence P N Modulo1

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2013

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“…This implies that F (X) ≡ Φ(X) mod 2. Hence P n (X) ≡ π n (X) mod 2 and Q n (X) ≡ κ n (X) mod 2: to be sure that the convergents of the reduction modulo 2 of F are equal to the reduction modulo 2 of the convergents of F (X), the reader can look at, e.g., [34]. Thus P n (X) ≡ π n (X) = κ n−1 (X) ≡ Q n−1 (X) mod 2.…”

Section: The Sequence P N Modulomentioning

confidence: 99%

Lacunary formal power series and the Stern–Brocot sequence

Allouche¹,

France²

2013

Acta Arith.

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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.

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Section: The Sequence P N Modulomentioning

confidence: 99%