Theory of two-index Bessel functions and applications to physical problems

Abstract

In this article the theory of two-index Bessel functions is presented. Their generating function, series expansion, and integral representations are discussed. Their usefulness in physical problems is also discussed in the context of analysis of radiation emitted by relativistic electrons in two-frequency undulators. Finally, the theoretical analysis proving addition and multiplication theorems for two-index Bessel functions are completed and their modified forms are introduced.

Representation of Lie algebra T3 and 2-variable 2-parameter Bessel functions

“…Dattoli and his co-workers introduced and discussed various generalizations of BF within purely mathematical and applicative contexts (see for example [2][3][4][5][6][7][8]). Generalized Bessel functions (GBF) have become a powerful tool to investigate the dynamical aspects of physical This is the most convenient form to study the modified forms of the first-kind cylindrical GBF.…”

Symbolic methods for the evaluation of sum rules of Bessel functions

“…We have indeed mentioned laser assisted processes in strong laser fields 20 , but we should also include synchrotron radiation problems regarding the emission of electron radiation in nonconventional undulator magnets [21][22][23] . Within this last framework a significant role has been played by the so called multi-index Bessel functions 24 , whose theory can be greatly simplified by the umbral formalism discussed here. …”

Multiplication theorems for multi-variable and multi-index Bessel functions

“…Dattoli and his co-workers introduced and discussed various generalizations of BF within purely mathematical and applicative contexts (see for example [1][2][3][4][5]7,[9][10][11]), as the success of any generalized special functions depend on its usefulness in applications and on the existence of effective numerical procedures for its computation. Generalized Bessel functions (GBF) have become a powerful tool to investigate the dynamical aspects of physical problems such as electron scattering by an intense linearly polarized laser wave, multi-photon processes and undulator radiation.…”

“…Also, the 2-index 1-parameter Bessel functions (2I1PBF)J m,n (x; τ ) defined by means of the series [7] and specified by the series…”

“…where J m,n (x, y, z) denotes the 2-index 3-variable Bessel functions (2I3VBF) defined by the generating function ( [7], p.3639 (13))…”