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Lacunary formal power series and the Stern-Brocot sequence

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Allouche J.-P., Mendès France M.

Acta Arithmetica

2013

tom 159

nr 1

47-61

Let $F(X) = ∑_{n ≥ 0} (-1)^{εₙ} X^{-λₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and $λ_{n+1}/λₙ > 2$. 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 $Q_{ω}(X)$ is a polynomial if and only if ω ∈ ℤ. In all the other cases $Q_{ω}(X)$ is an infinite formal power series; we discuss its algebraic properties in the special case $λₙ = 2^{n+1} - 1$.

Ograniczanie wyników

1 Acta Arithmetica

1 Allouche J.-P.

1 Mendès France M.

1 2013

Wyniki wyszukiwania

Lacunary formal power series and the Stern-Brocot sequence