Algebraic independence of the generating functions of Stern’s sequence and of its twist

Bundschuh, Peter; Väänänen, Keijo

doi:10.5802/jtnb.824

Cited by 13 publications

(10 citation statements)

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“…Recently, Bundschuh and Väänänen [BV13] proved that µ(S(1/b)) ≤ 2.929 and µ(T (1/b)) ≤ 3.555 for all integers b ≥ 2. Our next result gives the exact irrationality exponent of the Stern number and also that of the twisted Stern number, and it will be proved in Section 7.…”

Section: Resultsmentioning

confidence: 99%

Hankel Determinants, Padé Approximations, and Irrationality Exponents

Bugeaud

Han

Wen

et al. 2015

Int Math Res Notices

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The irrationality exponent of an irrational number ξ, which measures the approximation rate of ξ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of ξ. Results obtained so far are rather fragmentary and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to 2. Our classes contain the Thue-Morse-Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.

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Section: Resultsmentioning

confidence: 99%

Hankel Determinants, Padé Approximations, and Irrationality Exponents

Bugeaud

Han

Wen

et al. 2015

Int Math Res Notices

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“…We now apply Theorem 6 with F (z) = S(z) and G(z) = S(z 4 ). The use of (7) gives d = 4, r = 1 and F (z 4 ) = G(z), G(z 4 ) = −zF (z) + (1 + z + z 2 )G(z).…”

Section: Proof Of Theorems 1-5mentioning

confidence: 99%

On linear independence measures of the values of Mahler functions

Väänänen

Wen²

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.

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“…Note also that in [4] we proved the algebraic independence of Γ σ (z) and its twist. The sole transcendence over C(z) of each of the functions Γ σ , Γ α , Γ β will now be generalized in a different direction.…”

Section: And Iterationmentioning

confidence: 99%