Algebraic continued fractions in $_q((T^{-1}))$ and recurrent sequences in $_q$

Alain Lasjaunias

Displaying similar documents to “Algebraic continued fractions in $_{q} ((T^{- 1}))$ and recurrent sequences in $_{q}$ ”

Automatic continued fractions are transcendental or quadratic

Yann Bugeaud (2013)

Annales scientifiques de l'École Normale Supérieure

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We establish new combinatorial transcendence criteria for continued fraction expansions. Let $α = [0; a_{1}, a_{2}, ...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ${(a_{ℓ})}_{ℓ \geq 1}$ of $α$ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

On the length of rational continued fractions over $_{q} (X)$

S. Driss (2015)

Discussiones Mathematicae - General Algebra and Applications

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Let $_{q}$ be a finite field and $A (Y) \in_{q} (X, Y)$ . The aim of this paper is to prove that the length of the continued fraction expansion of $A (P); P \in_{q} [X]$ , is bounded.

Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, Michel Mendès France (2013)

Acta Arithmetica

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Let $F (X) = \sum_{n \geq 0} {(- 1)}^{ε ₙ} X^{- λ ₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and $λ_{n + 1} / λ ₙ > 2$ . 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...

Perfect unary forms over real quadratic fields

Dan Yasaki (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $F = ℚ (\sqrt{d})$ be a real quadratic field with ring of integers $𝒪$ . In this paper we analyze the number $h_{d}$ of ${GL}_{1} (𝒪)$ -orbits of homothety classes of perfect unary forms over $F$ as a function of $d$ . We compute $h_{d}$ exactly for square-free $d \leq 200000$ . By relating perfect forms to continued fractions, we give bounds on $h_{d}$ and address some questions raised by Watanabe, Yano, and Hayashi.

Multidimensional Gauss reduction theory for conjugacy classes of $SL (n, ℤ)$

Oleg Karpenkov (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we describe the set of conjugacy classes in the group $SL (n, ℤ)$ . We expand geometric Gauss Reduction Theory that solves the problem for $SL (2, ℤ)$ to the multidimensional case, where $ς$ -reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in $GL (n, ℤ)$ in terms of multidimensional Klein-Voronoi continued fractions.

Fermat $k$ -Fibonacci and $k$ -Lucas numbers

Jhon J. Bravo, Jose L. Herrera (2020)

Mathematica Bohemica

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Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all $k$ -Fibonacci and $k$ -Lucas numbers which are Fermat numbers. Some more general results are given.

Estimates of $L_{p}$ norms for sums of positive functions

Ilgiz Kayumov (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We present new inequalities of $L_{p}$ norms for sums of positive functions. These inequalities are useful for investigation of convergence of simple partial fractions in $L_{p} (ℝ)$ .

5-dissections and sign patterns of Ramanujan's parameter and its companion

Shane Chern, Dazhao Tang (2021)

Czechoslovak Mathematical Journal

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In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction $R (q)$ and its reciprocal. We obtain the 5-dissections for functions $R (q) R {(q^{2})}^{2}$ and $R {(q)}^{2} / R (q^{2})$ , which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

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In this paper, we find all Pell and Pell-Lucas numbers written in the form $- 2^{a} - 3^{b} + 5^{c}$ , in nonnegative integers $a$ , $b$ , $c$ , with $0 \leq max {a, b} \leq c$ .

The growth speed of digits in infinite iterated function systems

Chun-Yun Cao, Bao-Wei Wang, Jun Wu (2013)

Studia Mathematica

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Let ${f ₙ}_{n \geq 1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${a ₙ (x)}_{n \geq 1}$ of integers, called the digit sequence of x, such that $x = l i m_{n \to \infty} f_{a ₁ (x)} \circ \dots \circ f_{a ₙ (x)} (1)$ . We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set $x \in Λ : a ₙ (x) \in B (\forall n \geq 1), l i m_{n \to \infty} a ₙ (x) = \infty$ for any infinite subset B ⊂ ℕ, a question posed by...

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $ℚ (α)$ and $ℚ (β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $ℚ (β) \to ℚ (α)$ . The algorithm is particularly efficient if there is only one isomorphism.

An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

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Let $α > 1$ be irrational. Several authors studied the numbers $ℓ^{m} (α) = inf {| y | : y \in Λ_{m}, y \neq 0},$ where $m$ is a positive integer and $Λ_{m}$ denotes the set of all real numbers of the form $y = ϵ_{0} α^{n} + ϵ_{1} α^{n - 1} + \dots + ϵ_{n - 1} α + ϵ_{n}$ with restricted integer coefficients $| ϵ_{i} | \leq m$ . The value of $ℓ^{1} (α)$ was determined for many particular Pisot numbers and $ℓ^{m} (α)$ for the golden number. In this paper the value of $ℓ^{m} (α)$ is determined for irrational numbers $α$ , satisfying $α^{2} = a α \pm 1$ with a positive integer $a$ .

Governing fields for the 2-classgroup of $Q (s q r t - q_{1} q_{2} p)$ and a related reciprocity law

Patrick Morton (1990)

Acta Arithmetica

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$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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Displaying similar documents to “Algebraic continued fractions in q ( ( T - 1 ) ) and recurrent sequences in q ”

Displaying similar documents to “Algebraic continued fractions in $_{q} ((T^{- 1}))$ and recurrent sequences in $_{q}$ ”