Related papers: Integral representations and summations of modifie…
A variety of inverse Sturm-Liouville problems is considered, including the two-spectrum inverse problem, the problem of recovering the potential from the Weyl function, as well as the recovery from the spectral function. In all cases the…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby…
Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…
An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y={\lambda}y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations…
In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving…
In this paper, we derive a Laplace-type integral representations for both the generalized Bessel function and the Dunkl kernel associated with the rank-two root system of type B_2. The derivation of the first one elaborates on the integral…
We construct higher order spectral shift functions, extending the perturbation theory results of M. G. Krein and L. S. Koplienko on representations for the remainders of the first and second order Taylor-type approximations of operator…
Highly oscillatory integrals, such as those involving Bessel functions, are best evaluated analytically as much as possible, as numerical errors can be difficult to control. We investigate indefinite integrals involving monomials in $x$…
We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel…
We have discovered three non-power infinite series representations for Bessel functions of the first kind of integer orders and real arguments. These series contain only elementary functions and are remarkably simple. Each series was…
The $\nu$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{i\nu}(x)$, it is known that it possesses a countably infinite…
In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double…
The modified Bessel function of the first kind, $I_{\nu}(x)$, arises in numerous areas of study, such as physics, signal processing, probability, statistics, etc. As such, there has been much interest in recent years in deducing properties…
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…
Some power series representations of the modified Bessel functions (McDonald functions $K_{\alpha}$) are derived using the relatively little known formalism of fractional derivatives. The resulting summation formulae are believed to be new.
The integral representation of the two-parameter Mittag-Leffler function $E_{\rho,\mu}(z)$ is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from…
By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive…
We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized…
In a recent paper we unified Bessel functions of different orders .Here we extend the unification to other linairely independant solutions to Bessel equation, Neumann's and Hankel's functions