Continued Fraction Research Articles

TRANSCENDENTAL RUBAN p-ADIC CONTINUED FRACTIONS

Abstract We establish explicit constructions of Mahler’s p-adic $U_{m}$ -numbers by using Ruban p-adic continued fraction expansions of algebraic irrational p-adic numbers of degree m.

On the Analytic Continuation of Appell’s Hypergeometric Function F2 to Some Symmetric Domains in the Space C2

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of Appell’s hypergeometric function F2 with real and complex parameters, using their branched continued fraction expansions whose elements are polynomials in the space C2. To do this, we used a technique that extends the domain of convergence of the branched continued fraction, which is already known for a small domain, to a larger domain, as well as the PC method to prove that it is also the domain of analytical continuation. A few examples are provided at the end to illustrate this.

Some lacunarity properties of partial quotients of real numbers

Some lacunarity properties of partial quotients of real numbers

Recovering orthogonality from quasi‐type kernel polynomials using specific spectral transformations

In this work, the concept of quasi‐type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The process of recovering orthogonality for the linear combination of a quasi‐type kernel polynomial with another orthogonal polynomial, which is identified by involving linear spectral transformation, is provided. This process involves an expression of ratio of iterated kernel polynomials. This leads to considering the limiting case of ratio of kernel polynomials involving continued fractions. Special cases of such ratios in terms of certain continued fractions are exhibited. Moreover, examples are discussed for obtaining the orthogonality of quasi‐type kernel polynomials of order 1.

A renormalization scheme for semi-regular continued fractions

AbstractIn this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, $$\hat{T}_{slow}$$ T ^ slow and $$\hat{T}_{fast}$$ T ^ fast : these maps are defined for $$(x,y)\in [0,1]$$ ( x , y ) ∈ [ 0 , 1 ] , where x is the number for which a semi- regular continued fraction representation is developed by $$\hat{T}_{slow}$$ T ^ slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map $$\hat{T}_{fast}$$ T ^ fast is a “sped up" version of the map $$\hat{T}_{slow}$$ T ^ slow , and we show that $$\hat{T}_{fast}$$ T ^ fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, $$\hat{T}_{slow}$$ T ^ slow preserves no such measure, but does preserve an infinite, $$\sigma $$ σ -finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y under rotation by x modulo one. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada’s $$\alpha $$ α -continued fractions.

Transcendence and Normality of Complex Numbers via Hurwitz Continued Fractions

Abstract We study the topological, dynamical, and descriptive set-theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number that is not a Gaussian rational. The resulting space of sequences of Gaussian integers $\Omega $ is not closed. Using an iterative procedure, we show that $\Omega $ contains a natural subset whose closure $\overline{\textsf{R}}$ encodes continued fraction expansions of complex numbers that are not Gaussian rationals. We prove that $(\overline{\textsf{R}}, \sigma )$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

Efficacy of Continued Fraction Expansion technique in the approximation of fractional order systems

Efficacy of Continued Fraction Expansion technique in the approximation of fractional order systems

On Ramanujan’s cubic continued fraction

On Ramanujan’s cubic continued fraction

Research on the convergence of some types of functional branched continued fractions

An analysis of research on the problem of convergence of various types of functional branched continued fractions has been carried out. Branched continued fractions with $N$ branching branches and branched continued fractions with independent variables are considered. The definition and, in our opinion, characteristic criteria of convergence of multidimensional generalizations of C-, S-, g-, J-fractions are given, both for branched continued fractions of the general form with $N$ branching branches and branched continued fractions with independent variables. Such multidimensional generalizations of continued fractions arise, in particular, in the development of various classes of hypergeometric functions of several variables, in particular, the functions of Appel, Lauricella, Horn, etc.

On the transcendance of quasi-periodic Rosen continued fractions

AbstractIn this paper, we consider the two Hecke groups $$G_{4}$$ G 4 and $$G_{6}$$ G 6 and we use the Schmidt Subspace Theorem to establish the transcendence of some quasi-periodic Rosen continued fractions in order to get the exact analogues of the results established with the regular continued fractions.

Diophantine approximation by rational numbers of certain parity types

AbstractFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

Analysis on the derivative of the discriminant polynomial for the critical almost Mathieu operator

This article first introduces the discriminant polynomial and the central spectral band of the critical almost Mathieu operator. The Hardy–Littlewood method and the symbolic coding of continued fractions are introduced and applied to estimate the first-order and the second-order derivatives of the discriminant polynomial. Finally, these results are applied to obtain a lower bound for the measure of the central spectral band.

COMPLETE CLASSIFICATION OF QUADRATIC IRRATIONALS WITH PERIOD TWO

This article presents a comprehensive investigation into the classification of quadratic irrationals with period two in their continued fraction representations. Building upon foundational results in Number Theory, particularly in the context of continued fractions and Pell's equation, the study reveals intricate relationships between quadratic irrationals and their periodic structures. The main object of study is √N and properties of its continued fractions. While it is well-known that continued fractions of √N is periodic with periodic part being palindrome, the distribution of the lengths of the periodic parts are far from being complete. Our main goal will be to focus on the period two case and provide a complete characterization. The research's proved theorems clarify the conditions under which the period length is exactly two and give an insight into the underlying algebraic features. Additionally, it delves deeper by offering numerical analysis and illustrations demonstrating the distribution of period lengths among quadratic irrationals. This research opens up new paths for future studies on quadratic irrationals and how they're shown as continued fractions.

Quasinormal Modes and Stability Analysis of Rotating Hairy Black Holes in Asymptotically Flat Spacetime

Abstract Kerr black holes, characterized solely by their mass and angular momentum, are considered the predominant type in the universe. Recent studies, however, suggest that black holes may possess additional parameters, known as “hair”. This paper investigates the quasinormal modes (QNMs) of rotating hairy black holes in asymptotically flat spacetimes using the continued fraction method to analyze their stability. Our results demonstrate that the introduction of hair parameters modifies the QNM frequencies, indicating a nuanced stability profile that reduces to the classical Kerr solution under specific conditions. These findings provide new insights into the dynamic properties of hairy black holes and their potential astrophysical implications.

Scalar Quasi-Normal Modes of a loop quantum black hole

We compute the Quasi-Normal Mode (QNM) frequencies for scalar perturbations for modified Schwarzschild black holes in Loop Quantum Gravity. We study the singularity-free polymerized metric characterized by two parameters encoding loop quantum effects: the minimal area gap a 0 and the polymeric deformation parameter P. We perform numerical computations using Leaver's continued fraction method and compare our results to other semi-analytical methods and existing literature. We study the effects on the QNM spectrum of variation of both deformation parameters and systematically compare to the standard Schwarzschild case. In particular we find that the scalar fundamental mode is modified from the third decimal for values of P in accordance with the most recent astrophysical constraints. We also show that qualitative differences arise for highly damped modes: on the one hand, a new crossing of the imaginary axis occurs for high values of a 0 and, on the other hand, increasing P produces a positive shift of the real part and an increase of the spacing in imaginary part between modes.

Uniform Diophantine approximation and run-length function in continued fractions

Abstract We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set $$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$ for a class of quadratic irrational numbers $y\in [0,1)$ . These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.

Lattice paths and the Prouhet-Thue-Morse sequence

Using Flajolet's combinatorial theory of continued fractions and axial symmetry of lattice paths, we show that the enumeration of certain lattice paths modulo two yields the ubiquitous Prouhet-Thue-Morse sequence. This answers an open problem of Berstel et al.

Continued fractions for cycle-alternating permutations

AbstractA permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index i is either a cycle valley ($$\sigma ^{-1}(i)>i<\sigma (i)$$ σ - 1 ( i ) > i < σ ( i ) ) or a cycle peak ($$\sigma ^{-1}(i)<i>\sigma (i)$$ σ - 1 ( i ) < i > σ ( i ) ). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes–Rogers and Jacobi–Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs.

The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

Improvement of lactose digestion by highland barley (Hordeum vulgare var. coeleste L.) β-glucan: Activation of lactase under simulated gastric/small intestinal digestive conditions

β-Galactosidase (lactase) plays a crucial role as a dietary supplement in managing lactose intolerance. Here, the catalytic activity of lactase was successfully activated for the first time through complexation with water-extractable β-glucans from highland barley (WHBG). Under simulated gastric/small intestinal digestive conditions, WHBG and lactase spontaneously formed complexes, resulting in a remarkable increase in catalytic activity up to 172.6 %. Structural analyses revealed that the incorporation of WHBG caused partial unfolding of lactase, thereby exposing its hydrophobic regions with active sites, and the electrostatic and hydrophobic interactions between the two played pivotal roles. Meanwhile, according to microstructure and particle size analyses, the dissociation of aggregates and the re-distribution of lactase molecules were also observed. Consequently, the enzyme-substrate contact was promoted, and the hydrolysis efficiency of complexed lactase in the digestion of lactose in milk was superior to that of native lactase. Notably, among WHBG30/50/70 obtained by continuous fractionation of WHBG with 30 %/50 %/70 % ethanol, WHBG70 exhibited the lowest molecular weights and size, and the highest negative ζ-potential, potentially contributing to its superior activation abilities on lactase. These findings challenge the traditional view of polysaccharides as enzyme inhibitors and highlight their potential for diverse applications.