On the length of rational continued fractions over $_q(X)$

S. Driss

Displaying similar documents to “On the length of rational continued fractions over $_{q} (X)$ ”

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.

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...

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 $0 \leq max {a, b} \leq c$ .

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.

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.

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

Alain Lasjaunias (2008)

Acta Arithmetica

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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 $a$ .

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...

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.

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 $L_{p} (ℝ)$ .

A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

Diego Marques, Elaine Silva (2017)

Communications in Mathematics

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In this note, we prove that there is no transcendental entire function $f (z) \in ℚ [[z]]$ such that $f (ℚ) \subseteq ℚ$ and $den f (p / q) = F (q)$ , for all sufficiently large $q$ , where $F (z) \in ℤ [z]$ .

On the unit group of a semisimple group algebra $𝔽_{q} S L (2, ℤ_{5})$

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of $𝔽_{q} S L (2, ℤ_{5})$ , where $𝔽_{q}$ is a finite field with $q = p^{k}$ elements for prime $p > 5,$ and $S L (2, ℤ_{5})$ denotes the special linear group of $2 \times 2$ matrices having determinant $1$ over the cyclic group $ℤ_{5}$ .

On sums and products in a field

Guang-Liang Zhou, Zhi-Wei Sun (2022)

Czechoslovak Mathematical Journal

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We study sums and products in a field. Let $F$ be a field with $ch (F) \neq 2$ , where $ch (F)$ is the characteristic of $F$ . For any integer $k \geq 4$ , we show that any $x \in F$ can be written as $a_{1} + \dots + a_{k}$ with $a_{1}, \dots, a_{k} \in F$ and $a_{1} \dots a_{k} = 1$ , and that for any $α \in F ∖ {0}$ we can write every $x \in F$ as $a_{1} \dots a_{k}$ with $a_{1}, \dots, a_{k} \in F$ and $a_{1} + \dots + a_{k} = α$ . We also prove that for any $x \in F$ and $k \in {2, 3, \dots}$ there are $a_{1}, \dots, a_{2 k} \in F$ such that $a_{1} + \dots + a_{2 k} = x = a_{1} \dots a_{2 k}$ .

Repdigits in generalized Pell sequences

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

Archivum Mathematicum

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For an integer $k \geq 2$ , let ${(n)}_{n}$ be the $k -$ generalized Pell sequence which starts with $0, ..., 0, 1$ ( $k$ terms) and each term afterwards is given by the linear recurrence $n = 2 n - 1 + n - 2 + \dots + n - k$ . In this paper, we find all $k$ -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${(P_{n}^{(2)})}_{n}$ . ...

Approximation properties of β-expansions

Simon Baker (2015)

Acta Arithmetica

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Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${(ϵ_{i})}_{i = 1}^{\infty} \in {0, 1}^{ℕ}$ a β-expansion for x if $x = \sum_{i = 1}^{\infty} ϵ_{i} β^{- i}$ . We call a finite sequence ${(ϵ_{i})}_{i = 1}^{n} \in {0, 1}^{n}$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $Ψ : ℕ \to ℝ_{\geq 0}$ , we introduce the following subset of ℝ: $W_{β} (Ψ) : = ⋂_{m = 1}^{\infty} ⋃_{n = m}^{\infty} ⋃_{{(ϵ_{i})}_{i = 1}^{n} \in {0, 1}^{n}} [\sum_{i = 1}^{n} (ϵ_{i}) / (β^{i}), \sum_{i = 1}^{n} (ϵ_{i}) / (β^{i}) + Ψ (n)]$ In other words, $W_{β} (Ψ)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0 \leq x - \sum_{i = 1}^{n} (ϵ_{i}) / (β^{i}) \leq Ψ (n)$ . When $\sum_{n = 1}^{\infty} 2^{n} Ψ (n) < \infty$ , the Borel-Cantelli lemma tells us that the Lebesgue measure...

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Displaying similar documents to “On the length of rational continued fractions over q ( X ) ”

Displaying similar documents to “On the length of rational continued fractions over $_{q} (X)$ ”