An infinite product with bounded partial quotients

Allouche, Jean‐Paul; France, Michel Mendès; Poorten, Alf van der

doi:10.4064/aa-59-2-171-182

Cited by 14 publications

(29 citation statements)

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“…Specifically, in that example the partial quotients of G 3 over Q all have good reduction modulo 3 (as was shown with inappropriate effort in [1]). …”

Section: Definitionmentioning

confidence: 85%

Specialisation and Reduction of Continued Fractions of Formal Power Series

Poorten¹

2005

Ramanujan J

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“…Specifically, in that example the partial quotients of G 3 over Q all have good reduction modulo 3 (as was shown with inappropriate effort in [1]). …”

Section: Definitionmentioning

confidence: 85%

Specialisation and Reduction of Continued Fractions of Formal Power Series

Poorten¹

2005

Ramanujan J

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“…bounded degree) continued fraction expansion; the coefficient sequence of that algebraic series of degree 3 is now known as the Baum-Sweet sequence. In [3], Allouche, Mendés France and van der Poorten showed that functions given by certain infinite products have linear partial quotients. In [22,23], van der Poorten also studied continued fraction expansions for other infinite products.…”

Section: Introductionmentioning

confidence: 99%

“…The Stieltjes continued fraction Stiel r (x) defined by the Rudin-Shapiro sequence r is congruent modulo 4 to an algebraic series in Z[[x]]. Namely, Stiel r (x) ≡ 4 x + 2x 2 + 2x 3 + (3x + 2x3 )φ(x) + x 1 − 4xφ(x).Proof. The Stieltjes continued fraction Stiel r (x) can be obtained by a subsequence of its convergents.…”

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confidence: 99%

Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

Wen

2020

Advances in Applied Mathematics

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We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents P n (x)/Q n (x) modulo 4, we give the formal power series expansions (modulo 4) of these two continued fractions and prove that they are congruent modulo 4 to algebraic series in Z [[x]]. Therefore, the coefficient sequences of the formal power series expansions are 2-automatic. Write Q n (x) = i≥0 a n,i x i . Then (Q n (x)) n≥0 defines a two-dimensional coefficient sequence (a n,i ) n,i≥0 . We prove that the coefficient sequences (a n,i mod 4) n≥0 introduced by both (Q n (x)) n≥0 and (P n (x)) n≥0 are 2-automatic for all i ≥ 0. Moreover, the pictures of these two dimensional coefficient sequences modulo 4 present a kind of self-similar phenomenon.

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“…The question of computing the continued fraction of certain Mahler functions (to the best of authors knowledge) goes back to 1991, when Allouche, Mendès France and van der Poorten [1] showed that all partial quotients of the infinite product…”

Section: Introductionmentioning

confidence: 99%

Continued fractions of certain Mahler functions

Badziahin

2019

Acta Arith.

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We investigate the continued fraction expansion of the infinite product g(where the polynomial P (x) satisfies P (0) = 1 and deg(P ) < d. We construct relations between the partial quotients of g(x) which can be used to get recurrent formulae for them. We provide formulae for the cases d = 2 and d = 3. As an application, we prove that for P (x) = 1 + ux where u is an arbitrary rational number except 0 and 1, and for any integer b with |b| > 1 such that g(b) = 0 the irrationality exponent of g(b) equals two. In the case d = 3 we provide a partial analogue of the last result with several collections of polynomials P (x) giving the irrationality exponent of g(b) strictly bigger than two.

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An infinite product with bounded partial quotients

Cited by 14 publications

References 0 publications

Specialisation and Reduction of Continued Fractions of Formal Power Series

Specialisation and Reduction of Continued Fractions of Formal Power Series

Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

Continued fractions of certain Mahler functions

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