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.     

On the transcendence and algebraic independence of certain continued fractions

The transcendence of continued fractions =[a ₀;a ₁,a ₂,...] is proved under growth conditions involving the denominatorsq _n of the convergents and shifted partial quotientsa _n+k . Extending this idea, conditions for the algebraic independence of several continued fractions are given. The proofs use the approximation properties of continued fractions in combination with the Thue-Siegel-Roth Theorem or a criterion for algebraic independence of Bundschuh.     

Continued fraction of e2 with confluent hypergeometric functions

The tanh-type, tan-type, and e-type Hurwitz continued fractions have been generalized by the author. In this paper, we study a generalized form of e²-type Hurwitz continued fractions by using confluent hypergeometric functions. We also obtain a similar type of Tasoev continued fractions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 513–531, October–December, 2006.     

Hurwitz continued fractions with confluent hypergeometric functions

Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions ₀ F1(;c;z). In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions. This research was supported in part by the Grant-in-Aid for Scientific research (C) (No. 18540006), the Japan Society for the Promotion of Science.     

Simple continued fractions and cutting sequences

Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ?², the modular group Γ = PSL(2, ?) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ? and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T ₀-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ?, the Farey tessellation and the simple continued fraction expansion of w ∈ ? _∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.     

On Gauss-Kuz’min statistics for finite continued fractions

The article is devoted to finite continued fractions for numbers a/b when integer points (a, b) are taken from a dilative region. Properties similar to the Gauss-Kuz’min statistics are proved for these continued fractions. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 195–208, 2005.     

Expansion of a ratio of hypergeometric functions of two variables in branching continued fractions

We obtain two-dimensional analogs of the continued fractions of Gauss which are an expansion of the ratio of the Appel hypergeometric functions. It is proved that these fractions are those corresponding to the formal power series expansion of the given ratio. Convergence criteria are established for the branching continued fractions under consideration.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 40–44.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

The Borel—Bernstein Theorem for multidimensional continued fractions

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms. Supported by a grant from the CNP _q -Brazil, 301456/80, and FINEP/CNP _q /MCT 41.96.0923.00 (PRONEX).     

Repeated modifications of limitk-periodic continued fractions

Summary The advantages of using modified approximants for continued fractions, can be enhanced by repeating the modification process. IfK(a _n /b _n) is limitk-periodic, a natural choice for the modifying factors is ak-periodic sequence of right or wrong tails of the correspondingk-periodic continued fraction, if it exists. If the modified approximants thus obtained are ordinary approximants of a new limitk-periodic continued fraction, we repeat the process, if possible. Some examples where this process is applied to obtain a convergence acceleration are also given.     

Red numbers and quadratic field

A natural number is said red if the period of the continued fraction of its square root has odd length. For any quadratic field \mathbbQ(?D)\mathbb{Q}(\sqrt{D}), we show how the parity of the periods length of the continued fractions of its irrationalities depends on the redness of their discriminant.     

A priori estimates for truncation error of continued fractionsK(1/b n )

A priori estimates are obtained for the truncation error of continued fractions of the formK(1/b _n ), with complex elementsb _n . The method employed is based on the calculation of bounds for successive diameters of a sequence of nested disks, where then-th approximant of the continued fraction is contained in then-th disk. Numerical examples are given to illustrate useful procedures and typical error estimates for continued fraction expansions of the complex logarithm and the ratio of consecutive Bessel functions.This research was supported by the National Science Foundation under Grant No. GP-9009 and by the United States Air Force through the Air Force Office of Scientific Research under Grant No. AFOSR-70-1888.     

Some twin regions of convergence for branched continued fractions

We study twin regions of convergence for branched continued fractions and establish an estimate of the rate of convergence; we construct a counterexample showing that the natural formulation of Thron's convergence criterion for continued fractions does not extend to branched continued fractions. Translated fromMatematichni Metodi ta Fiziko-Makhanichni Polya, Vol. 39, No. 2, 1996, pp. 62–64.     

Extremal values of continuants and transcendence of certain continued fractions

We prove a criterion for the transcendence of continued fractions whose partial quotients are contained in a finite set {b₁,…,b_r} of positive integers such that the density of occurrences of b_i in the sequence of partial quotients exists for 1ir. As an application we study continued fractions [0,a₁,a₂,a₃,…] with a_n=1+([nθ]modd) where θ is irrational and d2 is a positive integer.     

Lebesgue measure and Hausdorff dimension of special sets of real numbers from (0,1)

We estimate the Hausdorff dimension and the Lebesgue measure of sets of continued fractions of the type a=[a ₁,a ₂,…] where a _n belongs to a set S _n ⊂ℕ for every n∈ℕ. An upper bound for the Hausdorff dimension of the set of numbers with continued fraction expansions which fulfill some properties of asymptotic densities is also included.     

Continued fractions with multiple limits

For integers m2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity.We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k2. The k limits are evaluated in terms of ratios of certain q-series.Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m2.     

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

Singular values of the Rogers-Ramanujan continued fraction

Let z∊ C be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e ^{2π iz} is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals whenever z is imaginary quadratic. 2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40     

An analog of the Vorpits'kii convergence criterion for branched continued fractions of special form

We study branched continued fractions of a special form with inequivalent variables. We establish a multidimensional analog of the Vorpits'kii convergence criterion for continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 35–38.