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.     

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.     

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.     

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.     

Convergence criteria for branched continued fractions with nonnegative components

For branched continued fractions with nonnegative components and a fixed or variable number of branchings we establish necessary and sufficient conditions for their approximants to be well-defined. We study necessary and sufficient conditions for convergence that are multivariable analogs of the known Seidel-Stern and Stern criteria for continued fractions with positive elements. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 7–13.     

The element sets and value sets of two-dimensional continued fractions

For two-dimensional continued fractions we prove the existence and uniqueness of an optimal sequence of value sets corresponding to an arbitrarily given sequence of element sets. We compute the element set for a given sequence of disk value sets and as a corollary, give the element sets and value sets that are used in convergence criteria for two-dimensional continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 55–61.     

On the expansion of some functions in a two-dimensional <Emphasis Type="Italic">g</Emphasis>-fraction with independent variables

We expand some functions in a two-dimensional g -fraction with independent variables and show the efficiency of approximations of the obtained expansion by branched continued fractions.     

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)     

On the convergence of periodic integral continued fractions with variable limits of integration

We establish estimates for the “tails” of periodic integral continued fractions with variable upper limits of integration. We prove a theorem on the uniform and absolute convergence of such fractions, and we obtain an estimate of their rate of convergence. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 28–34     

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.     

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).     

On the convergence of branched continued fractions

We give a survey of research on the theory of convergence of branched continued fractions. Translated fromMatematychni Metody ta Fizyko-Mechanichni Polya, Vol. 41, No. 1, 1998, pp. 117–126.     

Nota sulle frazioni continuep-adiche

Summary In this note it is proved that for p-adic continued fractions a result analogous to Galois theorem for ordinary continued fractions holds. Moreover some results concerning the p- adic continued fraction expansions of the square roots of the integers are obtained.     

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.     

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.     

Convergence of two-dimensional continued fractions with complex elements

We prove an analog of Vorpits'kii's theorem for two-dimensional continued fractions, applying the formulas for computing their absolute error. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 75–83.     

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.     

Two types of explicit continued fractions

Summary Two types of explicit continued fractions are presented. The continued fractions of the first type include those discovered by Shallit in 1979 and 1982, which were later generalized by Pethő. They are further extended here using Peth\H o's method. The continued fractions of the second type include those whose partial denominators form an arithmetic progression as expounded by Lehmer in 1973. We give here another derivation based on a modification of Komatsu's method and derive its generalization. Similar results are also established for continued fractions in the field of formal series over a finite base field.     

Hypergeometric series and continued fractions

Ramanujan’s results on continued fractions are simple consequences of three-term relations between hypergeometric series. Theirq-analogues lead to many of the continued fractions given in the ‘Lost’ notebook in particular the famous one considered by Andrews and others.