Fermat’s principle applied to a flat metric in the plane yields the phase of a Bessel function in the periodic domain for a constant index of refraction. Gravitational forces cause the index of refraction to vary and lead to a modified phase of the Bessel function. A distinction is made between the forces that cause acceleration: the gravitational force affects the optical properties of the med...

In this study, our main objective is to determine some monotonic and log-concavity properties of generalized $k$-Bessel function by using its Hadamard product representation earlier results on power series. addition, the relationships between Bessel-type special functions basic functions, we present specific examples related trigonometric hyperbolic functions.

This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre second-order differential equations...

The theory of c-frames and c-Bessel mappings are the generalizationsof the theory of frames and Bessel sequences. In this paper, weobtain several equivalent conditions for dual of c-Bessel mappings.We show that for a c-Bessel mapping $f$, a retrievalformula with respect to a c-Bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-Besselmapping which  wea...

This paper studies an extended Bessel function of the form [Formula: see text] Representation formulations for [Formula: see text] are derived in terms of the parameters a, b, and p. An important consequence is the derivation of an [Formula: see text]-order differential equation satisfied by the function [Formula: see text]. Interesting functional inequalities are established, particularly for ...

in this paper, we first show that the tensor product of a finite number of standard g-frames (resp. fusion frames, frames) is a standard g-frame (resp. fusion frame, frame) for the tensor product of hilbert $c^ast-$ modules and vice versa, then we consider tensor products of g-bessel multipliers, bessel multipliers and bessel fusion multipliers in hilbert $c^ast-$modules. moreover, we obtain so...

in this paper we have solved analytically the schrödinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square ...

The Bessel-Muirhead hypergeometric system (or 0 F 1-system) in two variables (and three variables) is solved using symmetric series, with an explicit formula for coefficients , in order to express the K-Bessel function as a linear combination of the J-solutions. Limits of this method and suggestions for generalizations to a higher rank are discussed. 1. Introduction. The Bessel functions (of th...

On page 335 in his lost notebook, Ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary Bessel functions. We provide the first published proof of this result. The identity yields as corollaries representations of weighted divisor sums, in particular, the summatory function for r2(n), the number of representations of the positiv...