Related papers: An extended Cauchy integral
We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents…
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging…
It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new…
During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exist a rigorous and an approximate extensions. The rigorous extension introduces the…
Inversion of function sinc(x) is studied. New series and integral representations of branches of inverse function are obtained using Fourier analysis.
The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the…
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…
The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying theory of regular variation, without appeal to the general…
We extend Painlev\'e's determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general 'multiple-valued' Cauchy's problems. We study $C^0-$continuability (near singularities) of…
We introduce poly-Cauchy permutations that are enumerated by the poly-Cauchy numbers. We provide combinatorial proofs for several identities involving poly-Cauchy numbers and some of their generalizations. The aim of this work is to…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even…
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_p\Psi_q(z)$ of one variable. Some consequences of…
We prove an analogue of the Cauchy integral theorem for hyperholomorphic functions given in three-dimensional domains with non piece-smooth boundaries and taking values in an arbitrary finite-dimensional commutative associative Banach…
In this paper, new integral representations for the Bessel $J$ and $I$ functions were presented and their results were used to derive an expression for the Modified Bessel $K$ function.
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…
Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the…
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
A translation of Kummer`s paper On certain definite integrals and infinite series