Orthogonal Polynomials in Analytical Method of solving Differential Efquations Describing Dynamics of Multilevel Systems

Abstract

An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete analogues. All the obtained solutions are expressed by way of special or elementary functions.

Relative Fisher information of discrete classical orthogonal polynomials

“…The knowledge of the algebraic and spectral properties of the orthogonal polynomials in a discrete variable [1,13,28,32,40] as well as the elucidation of their universal structure [42] have been issues of permanent interest since the early years of the last century up until now, not only because of its mathematical interest [3,4,6,13,24,27,28,32,34,36,40,42] but also because of the increasing number of applications of these functions in so many scientific and technological fields [9, 22-24, 35-39, 49, 52]. In particular, the classical or hypergeometric discrete orthogonal polynomials do not only play a relevant role in the theory of difference analogues of special functions and other branches of mathematics [2,10,22,24,36,40,49], but also for mathematical modelling of a great deal of simple [8,9,35,37,39,49,50,52] and complex [12,14,23,23,38,48] systems, as well as for the compression of information for signal processing [29,41,42].…”