Generalized elementary functions

We use the theory of generalized linear differential equations to introduce new definitions of the exponential, hyperbolic and trigonometric functions. We derive some basic properties of these generalized functions, and show that the time scale elementary functions with Lebesgue integrable arguments represent a special case of our definitions.     

Continued fractions,the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.     

Some completely monotonic functions involving polygamma functions and an application

By using the first Binet's formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained. As direct consequences, two inequalities are proved. As an application, the best lower and upper bounds of the nth harmonic number are established.     

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.     

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of

     

Vector Stieltjes Continued Fraction and Vector QD Algorithm

The definition, in previous studies, of vector Stieltjes continued fractions in connexion with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. A new version of the vector QD algorithm allows to extend to the vector case the result which was known for one scalar function. Beside this connexion, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. It gives also some effective computations of coefficients linked to vector orthogonality.     

On some properties of digamma and polygamma functions

In this note we present some new and structural inequalities for digamma, polygamma and inverse polygamma functions. We also extend, generalize and refine some known inequalities for these important functions.     

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．     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

Absolutely (completely) monotonic functions and Jordan-type inequalities

Detailed analysis shows that a function f admits the double Jordan-type inequality if and only if f is analytic and even. Associated with f is the function g with f(x)=g(x²). In this short note, based on this association, and using properties of absolutely and/or (completely) monotonic functions, we propose a concise method to derive the inequality from the coefficients in the Taylor’s series of f. The results include some existing ones as special cases.     

Monotonicity of some functions involving the gamma and psi functions

Let , where is Euler's gamma function. We determine conditions for the numbers so that the function is strongly completely monotonic on . Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of and functions. Some limiting and particular cases are also considered.

     

On the functions whose inverse is jet-absolutely monotonic

The infinitely differentiable real valued functions whose inverse have all the Taylor coefficients nonnegative at the fixed point of the axis are considered. An application to the theory of the inverse pair of the Radon measures is given.     

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

Continued fractions of operator-valued analytic functions

The subject of this work is the convergence of infinite continued fractions whose coefficients are analytic functions which take their values in the space of bounded linear operators on a complex Hilbert space. By appropriately combining a prior result of Fair with Vitali's theorem, we show that convergence under the uniform operator topology occurs on certain angular regions of the complex plane whenever the operator coefficients are commutative, invertible, and satisfy certain conditions on their numerical ranges.     

Inequalities on generalized trigonometric and hyperbolic functions

In this paper, we prove some inequalities involving the generalized trigonometric and hyperbolic functions.     

On the structure of homothetic functions

Summary Homothetic function is a term which refers to some extension of the concept of a homogeneous function. We study different hierarchies of generalized homogeneous functions. The main result is a general classification of those functions.     

一个二元矩阵插值连分式的展开式

本文借助于文[1]定义的一种实用的矩阵广义逆,构造了一个二元Stieltjes型矩阵值插值连分式的展开式,它的截断分式可以定义二元矩阵值插值函数.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Continued fractions with three limit points

On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if A_n/B_n denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A_n/B_n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.