scholarly journals Continued fraction expansions for q-tangent and q-cotangent functions

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fraction ◽  
Similar Type ◽  
Recursive Relation ◽  
Continued Fraction Expansion ◽  
International Audience ◽  
Continued Fraction Expansions

International audience For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

The distribution of the largest digit in continued fraction expansions

2009 ◽  
Vol 146 (1) ◽  
pp. 207-212 ◽  
Author(s):  
JUN WU ◽  
JIAN XU
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Continued Fraction Expansions

AbstractLet [a1(x), a2(x), . . .] be the continued fraction expansion of x ∈ [0,1). Write Tn(x)=max{ak(x):1 ≤ k ≤ n}. Philipp [6] proved that Okano [5] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T(x)=1/log k. In this paper we show that, for any α ≥ 0, the set is of Hausdorff dimension 1.

Slow increasing functions and the largest partial quotients in continued fraction expansions

2016 ◽  
Vol 164 (1) ◽  
pp. 1-14 ◽  
Author(s):  
JINHUA CHANG ◽  
HAIBO CHEN
Keyword(s):  
Level Sets ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Positive Function ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Increasing Functions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

scholarly journals On the metrical theory of a non-regular continued fraction expansion

2015 ◽  
Vol 23 (2) ◽  
pp. 147-160
Author(s):  
Dan Lascu ◽  
George Cîrlig
Keyword(s):  
Dynamical System ◽  
Continued Fraction ◽  
Probability Function ◽  
Transition Probability ◽  
Continued Fraction Expansion ◽  
Metrical Theory ◽  
Unilateral Shift ◽  
Symbolic Dynamical System ◽  
Continued Fraction Expansions ◽  
Transition Probability Function

Abstract We introduced a new continued fraction expansions in our previous paper. For these expansions, we show the Brodén-Borel-Lévy type formula. Furthermore, we compute the transition probability function from this and the symbolic dynamical system of the natural number with the unilateral shift.

scholarly journals Mean-value of the Riemann zeta-function and other remarks III

1983 ◽  
Vol Volume 6 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Lower Bounds ◽  
Riemann Zeta Function ◽  
Continued Fraction ◽  
Mean Value ◽  
Continued Fraction Expansion ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Unique Integer ◽  
International Audience ◽  
Riemann Zeta

International audience The results given in these papers continue the theme developed in part I of this series. In Part III we prove $M(\frac{1}{2})>\!\!\!>_k (\log H_0/q_n)^{k^2}$, where $p_m/q_m$ is the $m$th convergent of the continued fraction expansion of $k$, and $n$ is the unique integer such that $q_nq_{n+1}\geq \log\log H_0 > q_nq_{n-1}$. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.

scholarly journals Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

2021 ◽  
Author(s):  
philip olivier
Keyword(s):  
Complex Variable ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Mathematical Tool ◽  
Continued Fraction Expansion ◽  
Continued Fraction Expansions ◽  
Irrational Function

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

Optimal approximation by continued fractions

1991 ◽  
Vol 50 (3) ◽  
pp. 481-504 ◽  
Author(s):  
Wieb Bosma ◽  
Cor Kraaikamp
Keyword(s):  
Continued Fractions ◽  
Best Approximation ◽  
Continued Fraction ◽  
Irrational Number ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
The Best Approximation ◽  
Natural Sense ◽  
The One ◽  
Continued Fraction Expansions

AbstractAmong all possible semiregular continued fraction expansions of an irrational number the one with the best approximation properties, in a well-defined and natural sense, is determined. Some properties of this so called optimal continued fraction expansion are described.

scholarly journals Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

2021 ◽  
Author(s):  
philip olivier
Keyword(s):  
Complex Variable ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Mathematical Tool ◽  
Continued Fraction Expansion ◽  
Continued Fraction Expansions ◽  
Irrational Function

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

scholarly journals CLASSIFICATION AND SYMMETRIES OF A FAMILY OF CONTINUED FRACTIONS WITH BOUNDED PERIOD LENGTH

2012 ◽  
Vol 93 (1-2) ◽  
pp. 53-76
Author(s):  
K. H. F. CHENG ◽  
R. K. GUY ◽  
R. SCHEIDLER ◽  
H. C. WILLIAMS
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Period Length ◽  
Continued Fraction Expansion ◽  
Quadratic Irrational ◽  
Continued Fraction Expansions

AbstractIt is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

On the exceptional sets in Sylvester continued fraction expansion

2015 ◽  
Vol 11 (08) ◽  
pp. 2369-2380
Author(s):  
Zhen-Liang Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Exceptional Sets ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Set Of Points

In this paper, we study some exceptional sets of points whose partial quotients in their Sylvester continued fraction expansions obey some restrictions. More precisely, for α ≥ 1 we prove that the Hausdorff dimension of the set [Formula: see text] is one. In addition, we find that the points whose partial quotients in their Sylvester continued fraction expansions obey some property of divisibility have the same Engel continued fraction expansion and Sylvester continued fraction expansion. And we establish that the set of points whose Engel continued fraction expansion and Sylvester continued fraction expansion coincide is uncountable.

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