Matrix Continued Fractions

A matrix continued fraction is defined and used for the approximation of a function known as a power series in 1/zwith matrix coefficientsp×q, or equivalently by a matrix of functions holomorphic at infinity. It is a generalization of P-fractions, and the sequence of convergents converges to the given function. These convergents have as denominators a matrix, the columns of which are orthogonal with respect to the linear matrix functional associated to . The case where the algorithm breaks off is characterized in terms of .     

Hurwitz and Tasoev Continued Fractions

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e^1/a (a 1), ae^1/a and (1/a)e^1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.     

Hurwitz and Tasoev Continued Fractions

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e^1/a (a 1), ae^1/a and (1/a)e^1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.     

Additive Partitions and Continued Fractions

A set S of positive integers is avoidable if there exists a partition of the positive integers into two disjoint sets such that no two distinct integers from the same set sum to an element of S. Much previous work has focused on proving the avoidability of very special sets of integers. We vastly broaden the class of avoidable sets by establishing a previously unnoticed connection with the elementary theory of continued fractions.     

连分式的不等式

本文研究二元连分式G(m,λ)的单调性质,得到一些新的连分式不等式．     

Interpolational Integral Continued Fractions

For nonlinear functionals defined on the space of piecewise-continuous functions, we construct an interpolational integral continued fraction on continual piecewise-continuous nodes and establish conditions for the existence and uniqueness of interpolants of this type.     

Continued Fractions and Unique Additive Partitions

A partition of the positive integers into sets A and B avoids a set S N if no two distinct elements in the same part have a sum in S. If the partition is unique, S is uniquely avoidable. For any irrational > 1, Chow and Long constructed a partition which avoids the numerators of all convergents of the continued fraction for , and conjectured that the set S which this partition avoids is uniquely avoidable. We prove that the set of numerators of convergents is uniquely avoidable if and only if the continued fraction for has infinitely many partial quotients equal to 1. We also construct the set S and show that it is always uniquely avoidable.     

Quasi-Elliptic Integrals and Periodic Continued Fractions

 In this report we detail the following story. Several centuries ago, Abel noticed that the well-known elementary integral

Convergence Acceleration of Some Continued Fractions

The tails of a continued fraction satisfy a bilinear recurrent equation. Transforming iteratively these tails (in a special manner) as well as these equations one may obtain finally, for a given fraction, a new, so-called diagonal continued fraction (DF) having the same value. For many important classes of continued fractions the DF has a calculable analytical form and converges qualitatively faster. Using the same method one may transform some hypergeometrical series directly into fast convergent DFs.     

Singularity of Some Random Continued Fractions

We prove singularity of some distributions of random continued fractions that correspond to iterated function systems with overlap and a parabolic point. These arose while studying the conductance of Galton-Watson trees.     

Convergence of Associated Continued Fractions Revised

This is an expository article which contains alternative proofs of many theorems concerning convergence of a continued fraction to a holomorphic function. The continued fractions which are studied are continued fractions of the form

Limiting Behavior of Random Continued Fractions

Let K(a _n /b _n ) be a continued fraction with elements (a _n ,b _n ) picked randomly and independently from $(\mathbb{C}\setminus\{0\})\times\mathbb{C}$ according to some probability distribution μ. We find sufficient conditions on μ for K(a _n /b _n ) to converge with probability 1 or to be restrained with probability 1. More generally, we also consider μ-random sequences {τ _n } of independent Möbius transformations and find sufficient conditions for $\{\tau_{1}\circ\tau_{2}\circ\cdots\circ\tau_{n}\}_{n=1}^{\infty}$ to converge or be restrained with probability 1. The analysis is based on an important paper by Furstenberg.     

Approximation Properties of Two-Dimensional Continued Fractions

By using the difference formula for approximations of two-dimensional continued fractions, the method of fundamental inequalities, the Stieltjes–Vitali theorem, and generalizations of divided and inverse differences, we estimate the accuracy of approximations of two-dimensional continued fractions with complex elements by their convergents and obtain estimates for the real and imaginary parts of remainders of two-dimensional continued fractions. We also prove an analog of the van Vleck theorem and construct an interpolation formula of the Newton–Thiele type.     

On Limit Theorems for Continued Fractions

It is shown that for sums of functionals of digits in continued fraction expansions the Kolmogorov-Feller weak laws of large numbers and the Khinchine-Lévy-Feller-Raikov characterization of the domain of attraction of the normal law hold.      

Haas Molnar Continued Fractions and Metric Diophantine Approximation

Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi’s backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number x, if (p _n /q _n )_n≥1 denotes its sequence of regular continued fraction convergents, set θ _n (x) = q _n ² |x ? p _n /q _n |, n = 1, 2.... The metric behaviour of the Cesàro averages of the sequence (θ _n (x))_n≥1 has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence (θ _n (x))_n≥1 for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of \(({\theta _{{k_n}}}(x))\)_n≥1 for certain sequences (k _n )_n≥1, initiated by the second named author, to Haas–Molnar maps.     

On the Extremal Theory of Continued Fractions

Letting \(x=[a_1(x), a_2(x), \ldots ]\) denote the continued fraction expansion of an irrational number \(x\in (0, 1)\), Khinchin proved that \(S_n(x)=\sum \nolimits _{k=1}^n a_k(x) \sim \frac{1}{\log 2}n\log n\) in measure, but not for almost every \(x\). Diamond and Vaaler showed that, removing the largest term from \(S_n(x)\), the previous asymptotics will hold almost everywhere, this shows the crucial influence of the extreme terms of \(S_n (x)\) on the sum. In this paper we determine, for \(d_n\rightarrow \infty \) and \(d_n/n\rightarrow 0\), the precise asymptotics of the sum of the \(d_n\) largest terms of \(S_n(x)\) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.     

Large and Moderate Deviation Principles for Engel Continued Fractions

Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.     

Continued Fractions for Rogers–Szegö Polynomials

We evaluate different Hankel determinants of Rogers–Szegö polynomials, and deduce from it continued fraction expansions for the generating function of RS polynomials. We also give an explicit expression of the orthogonal polynomials associated to moments equal to RS polynomials, and a decomposition of the Hankel form with RS polynomials as coefficients.