Hankel determinants of the Thue-Morse sequence

Allouche, Jean‐Paul; Peyrière, Jacques; Zhou, Wen; Wen, Zhi-Ying

doi:10.5802/aif.1609

Cited by 47 publications

(77 citation statements)

References 9 publications

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“…We make use of a result of Allouche, Peyrière, Wen, and Wen [5] on the non-vanishing of Hankel determinants of the Thue-Morse sequence to get many Padé approximants to the generating function of the Thue-Morse sequence. By means of these rational fractions we construct infinitely many good rational approximations to the Thue-Morse-Mahler number ξ t,b , which form a sufficiently dense sequence to conclude that the irrationality exponent of ξ t,b cannot exceed 2.…”

Section: Resultsmentioning

confidence: 99%

On the rational approximation to the Thue–Morse–Mahler numbers

Bugeaud¹

2011

Ann. inst. Fourier

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Let p be a prime number and ξ an irrational p-adic number. Its multiplicative irrationality exponent µ × (ξ) is the supremum of the real numbers µ × for which the inequalityhas infinitely many solutions in nonzero integers a, b. We show that µ × (ξ) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of ξ. We establish that µ × (ξ t,p ) = 3, where ξ t,p is the p-adic number 1 − p − p 2 + p 3 − p 4 + . . ., whose sequence of digits is given by the Thue-Morse sequence over {−1, 1}.

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Section: Resultsmentioning

confidence: 99%

On the rational approximation to the Thue–Morse–Mahler numbers

Bugeaud¹

2011

Ann. inst. Fourier

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“…And the block sequence {u(m)u(m + 1) · · · u(m + 2n)} m≥0 is k-automatic. Hence, the determinant sequence {|u m n |} m≥0 is k-automatic, please see [4]. There are some results about the automaticity of the Hankel determinant sequences.…”

Section: Definition 2 Given An Integer Sequencementioning

confidence: 99%

“…There are some results about the automaticity of the Hankel determinant sequences. Allouche, Peyrière, Wen and Wen first studied the Hankel determinant of the Thue-Morse sequence t in [4]. They proved that the sequences {|t m n |(mod2)} n≥0 are 2-automatic.…”

Section: Definition 2 Given An Integer Sequencementioning

confidence: 99%

On the regularity of the Hankel determinant sequence of the characteristic sequence of powers of 2

Guo

2019

Advances in Applied Mathematics

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For any sequences u = {u(n)} n≥0 , v = {v(n)} n≥0 , we define uv := {u(n)v(n)} n≥0 and u + v := {u(n) + v(n)} n≥0 . Let f i (x) (0 ≤ i < k) be sequence polynomials whose coefficients are integer sequences. We say an integer sequence u = {u(n)} n≥0 is a polynomial generated sequence ifIn this paper, we study the polynomial generated sequences. Assume k ≥ 2 and f i (x) = a i x+b i (0 ≤ i < k). If a i are k-automatic and b i are k-regular for 0 ≤ i < k, then we prove that the corresponding polynomial generated sequences are k-regular. As a application, we prove that the Hankel determinant sequence {det(p(i+j)) n−1 i,j=0 } n≥0 is 2-regular, where {p(n)} n≥0 = 0110100010000 · · · is the characteristic sequence of powers 2. Moreover, we give a answer of Cigler's conjecture about the Hankel determinants.

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“…One of the key ingredients of that paper is the result from [2] about non-vanishing of Hankel determinants of f T M (x) (they will be properly defined and discussed in Section 3). Later this approach was further developed and generalised to cover many other Mahler functions, see for example [7,8,14].…”

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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Hankel determinants of the Thue-Morse sequence

Cited by 47 publications

References 9 publications

On the rational approximation to the Thue–Morse–Mahler numbers

On the rational approximation to the Thue–Morse–Mahler numbers

On the regularity of the Hankel determinant sequence of the characteristic sequence of powers of 2

Continued fractions of certain Mahler functions

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