The authors apply a certain novel technique based on the combined use of operational methods and of some special multivariable and multiindex polynomials to derive several families of generating functions involving the products of Bessel and related functions. The possibility of extending this technique to the derivation of generating functions of hybrid nature (involving, for example, the prod...

We provide information and explicit formulae for a class of integrals involving Bessel functions and Gegenbauer polynomials. We present a simple proof of an old formula of Gegenbauer. Some interesting special cases and applications of this result are obtained. In particular, we give a short proof of a recent result of A. A. R. Neves et al. regarding the analytical evaluation of an integral of a...

An increasing attention has been concentrated on nondiffracting Bessel beams, due to their novel properties and prospective applications. In order to study their properties entirely, including the transverse modes, the polarization states and the flow of energy, vector analyses are necessary. In this paper, based on auxiliary functions of Hertzian vector potential, nondiffracting Bessel beams a...

For a delta operator aD − bD we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case of D − 1 2D 2 we show that the corresponding Bessel–Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions νt, t > 0, for which {yn(t)}, the Bessel polynomials at t, is the moment ...

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view a...

In this work, we extend the analytic treatment of Bessel functions of large order and/or argument. We examine uniform asymptotic Bessel function expansions and show their accuracy and range of validity. Such situations arise in a variety of applications, in particular the Fourier transform of the gravitational wave signal from a pulsar. The uniform expansion we consider here is found to be vali...

We study the effect of Two-Photon Absorption (TPA) nonlinear losses on Gaussian pulses, with power that exceeds the critical power for self-focusing, propagating in bulk kerr media. Experiments performed in fused silica and silicon highlight a spontaneous reshaping of the input pulse into a pulsed Bessel beam. A filament is formed in which sub-diffractive propagation is sustained by the Bessel-...

The fuzzy disc is a matrix approximation of the functions on a disc which preserves rotational symmetry. In this paper we introduce a basis for the algebra of functions on the fuzzy disc in terms of the eigenfunctions of a properly defined fuzzy Laplacian. In the commutative limit they tend to the eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of the first kind, thu...

In this note our aim is to present some Jordan-type inequalities for generalized Bessel functions in order to extend some recent results concerning generalized and sharp versions of the well-known Jordan’s inequality. Acknowledgements: Research partially supported by the Institute of Mathematics, University of Debrecen, Hungary. The author is grateful to Prof. Lokenath Debnath for a copy of pap...

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge...