Related papers: Fourier transform of a Bessel function multiplied …
We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty \frac{J_\alpha(x j_{m,\nu})J_\beta(y j_{m,\nu})} {j_{m,\nu}^{2n+\alpha+\beta-2\nu+2} J_{\nu+1}(j_{m,\nu})^2}, \] where $0<x$, $0<y$, $x+y<2$, $n$ is an integer,…
Using Bauer's expansion and properties of spherical Bessel and Legender functions, we deduce a new transform and briefly indicate its use.
The article discusses the fractional powers of the Bessel operator and their numerical implementation. An extensive literature is devoted to the study of fractional powers of the Laplace operator and their applications. Such degrees are…
A few approaches are derived to calculate three-particle integrals which include spherical Bessel functions of the first and second kind, i.e., the $j_{\ell}(V r)$ and $n_{\ell}(V r)$ functions. Such integrals are important in applications…
As for the Fourier transforms of positive and integrable functions supported in the unit interval, we make a list of improvements for P\'olya's results on the distribution of their positive zeros and give new sufficient conditions under…
In this paper we give the q-analogue of the higher-order Bessel operators studied by M. I. Klyuchantsev [12] and A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [3]. Our objective is twofold. First, using the q-Jackson integral and…
A new representation for a regular solution of the perturbed Bessel equation of the form $Lu=-u"+\left( \frac{l(l+1)}{x^2}+q(x)\right)u=\omega^2u$ is obtained. The solution is represented as a Neumann series of Bessel functions uniformly…
The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(sl_2)$. A linear combination of its generators gives…
In this paper we study the variation diminishing kernel as a part of the $q$-calculus. We introduce the $q$-Macdonald function a newborne in the family of the $q$-special functions which play a central role in this study.
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…
Closed-form evaluations of certain integrals of $J_{0}(\xi)$, the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from…
We study the non homogeneous quadratic Bessel zeta function $\zeta_{RB}(s,\nu,a)$ defined as the sum of the square of the positive zeros of the Bessel function $J_\nu(z)$ plus a positive constant. In particular, we give explicit formulas…
We improve on Fourier transforms (FT) between imaginary time $\tau$ and imaginary frequency $\omega_n$ used in certain quantum cluster approaches using the Hirsch-Fye method. The asymptotic behavior of the electron Green's function can be…
The Bessel-Neumann expansion (of integer order) of a function $g:\mathbb{C}\rightarrow\mathbb{C}$ corresponds to representing $g$ as a linear combination of basis functions $\phi_0,\phi_1,\ldots$, i.e., $g(z)=\sum_{\ell = 0}^\infty w_\ell…
The best bounds of the form $B(\alpha,\beta,\gamma,x)=(\alpha+\sqrt{\beta^2+\gamma^2 x^2})/x$ for ratios of modified Bessel functions are characterized: if $\alpha$, $\beta$ and $\gamma$ are chosen in such a way that…
We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions…
In this paper, we consider the Clenshaw-Curtis-Filon method for the highly oscillatory Bessel transform $\int_0^1x^\alpha (1-x)^\beta f(x) J_{\nu}(\omega x)dx$, where $f$ is a smooth function on $[0, 1]$, and $\nu\geq0.$ The method is based…
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized…
Given $\alpha > -1$, consider the second order differential operator in $(0,\infty)$, $$L_\alpha f \equiv (x^2 \frac{d^2}{dx^2} + (2\alpha+3)x \frac{d}{dx} + x^2 + (\alpha+1)^2)(f), $$ which appears in the theory of Bessel functions. The…
It is known that the Bessel--Fourier coefficients $f_m$ of a function $f$ such that $\sqrt{x}f(x)$ is integrable over $[0,1]$ satisfy $f_m/\sqrt{m}\to 0$. We show a partial converse, namely that for $0\leq \alpha<1/2$ and any non-negative…