Exponential generating functions for the associated Bessel functions

Abstract

Similar to the associated Legendre functions, the differential equation for the associated Bessel functions B l,m (x) is introduced so that its form remains invariant under the transformation l → −l − 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l < 0 and l ≥ 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l − m) but with different values of l and m are independent of each other. So, all the functions B l,m (x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.[1, 2, 3], random graphs and complex networks [4], polymerization kinetics [5], counting problems in combinatorics [6], are some applications of the theory of generating functions. The generating functions are used to obtain expected values (averages), variances, moments and cumulants of distributions, and also, to establish relationships between distributions [7].The exponential generating functions and their numerous generalizations have been alternatively introduced and studied by various methods for the orthogonal polynomials and special functions (see Refs. [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]). The generating functions as continuous functions generally describe the convergence of an infinite summation of given infinite sequences of functions. In this sense, the special polynomials and functions of a given sequence in one variable x can be defined as the coefficients in the expansion of their generating functions. The generating functions include various useful properties and all information that is needed to generate the solutions corresponding to a differential equation or a set of recursion relations between those solutions. Therefore, generating functions are very useful to analyze problems involving summations on the infinite sequences of functions such as coherent states. The application of generating functions for known orthonormal special functions, allows one to derive a compact formula for the coherent states. Generating function corresponding to a given set of special functions is not unique. This manuscript has been devoted to introducing new generating functions for the solutions of the differential equation of associated Bessel functions which can be applied to obtain bound states of some solvable models in the framework of supersymmetric quantum mechanics, such as radial bound states of the hydrogen-like atoms [23]. Then, let us remember that Krall and Frink have first studied the Bessel polynomials in the formalism of hypergeometric functions [24]. Also, some authors have introduced some generating functions for Bessel polynomials [25,26,27]. Moreover, the generating functions associated with the group-theoretic techni...

Landau levels as a limiting case of a model with the morse-like magnetic field

“…From our comparison with the Ref. [28] we also find that the allowed energies of the electron are quantized as a positive quadratic function of both quantum numbers l and n:…”

“…(16) Z is an arbitrary complex variable with the polar form Z = re iϕ so that 0 ≤ r < ∞ and 0 ≤ ϕ < 2π. In addition, the following relations have been used to derive (15) [28,29]:…”

“…The square integrable solutions can be obtained by comparing the last equation with the associated Bessel differential equation of Ref. [28]:…”

“…The square integrability conditions n ≥ 1 and 0 ≤ l ≤ n−1 for the bound-state wavefunctions ψ l,n (x, y) can be deduced from Ref. [28]. They form an orthonormal set with respect to both integer indices l and n, that is, (bar is for the complex conjugation)…”

Approach of the Generating Functions to the Coherent States for Some Quantum Solvable Models

“…It should be noted that these methods have different and distinct results for other physical models, respectively. Following the extension of the above idea of other classical orthogonal polynomials, we have tried it on several cases including the associated Legendre polynomials P m l (x) as well as the associated Laguerre polynomials L m n (x) and the associated Bessel functions B q,β n,m [45]. Because of the relations between the associated Legendre functions P m l (x) and the spherical harmonics Y lm (θ, φ), they correspond to the quantum states of many physical problems and have, also, been studied in the framework of the coherent states theory (see, for example, Refs.…”