Dimension-theoretical results for a family of generalized continued fractions

2018 ◽  
Vol 66 (2) ◽  
pp. 115-122
Author(s):  
Jörg Neunhäuserer
Keyword(s):  
Continued Fractions ◽  
Theoretical Results ◽  
Generalized Continued Fractions

Markov chains and generalized continued fractions

1992 ◽  
Vol 29 (04) ◽  
pp. 838-849 ◽  
Author(s):  
Thomas Hanschke
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Wide Class ◽  
Stochastic Models ◽  
Continuous Time ◽  
Continued Fractions ◽  
Invariant Measures ◽  
Continuous Time Markov Chains ◽  
Generalized Continued Fractions

This paper deals with a class of discrete-time Markov chains for which the invariant measures can be expressed in terms of generalized continued fractions. The representation covers a wide class of stochastic models and is well suited for numerical applications. The results obtained can easily be extended to continuous-time Markov chains.

scholarly journals On irrationality exponents of generalized continued fractions

2015 ◽  
Vol 151 ◽  
pp. 18-35 ◽  
Author(s):  
Jaroslav Hančl ◽  
Kalle Leppälä ◽  
Tapani Matala-aho ◽  
Topi Törmä
Keyword(s):  
Continued Fractions ◽  
Generalized Continued Fractions

Metrical property for GCFϵ expansion with the parameter function ϵ(k) = c(k + 1)

2015 ◽  
Vol 11 (07) ◽  
pp. 2065-2072 ◽  
Author(s):  
Ting Zhong ◽  
Quanwu Mu ◽  
Luming Shen
Keyword(s):  
Continued Fractions ◽  
Partial Quotient ◽  
The Real ◽  
Metric Properties ◽  
Metrical Property ◽  
Parameter Function ◽  
Generalized Continued Fractions

This paper is concerned with the metric properties of the generalized continued fractions (GCFϵ) with the parameter function ϵ(kn), where kn is the nth partial quotient of the GCFϵ expansion. When -1 < ϵ(kn) ≤ 1, Zhong [Metrical properties for a class of continued fractions with increasing digits, J. Number Theory128 (2008) 1506–1515] obtained the following metrical properties: [Formula: see text] which are entirely unrelated to the choice of ϵ(kn) ∈ (-1, 1]. Here we deal with the case of ϵ(k) = c(k + 1) with constant c ∈ (0, ∞). It is proved that: [Formula: see text] which change with the real c ∈ (0, ∞). Note that [Formula: see text] as c → 0, it indicates that when c → 0, the GCFϵ has the same metrical property as the case of -1 < ϵ(kn) ≤ 1.

INFINITE-DIMENSIONAL GENERALIZED CONTINUED FRACTIONS, DISTRIBUTION OF QUADRATIC RESIDUES AND NON-RESIDUES, AND ERGODIC THEORY

2002 ◽  
Vol 05 (04) ◽  
pp. 555-570
Author(s):  
L. D. PUSTYL'NIKOV
Keyword(s):  
Prime Number ◽  
Ergodic Theory ◽  
Continued Fractions ◽  
Classical Problem ◽  
Infinite Dimensional ◽  
Ergodic Properties ◽  
Quadratic Residues ◽  
Generalized Continued Fractions

A new theory of generalized continued fractions for infinite-dimensional vectors with integer components is constructed. The results of this theory are applied to the classical problem on the distribution of quadratic residues and non-residues modulo a prime number and are based on the study of ergodic properties of some infinite-dimensional transformations.

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