Theoretical Study of The Generalized Laguerre Polynomials

Abstract

The Laguerre polynomials play an important role in several diverse fields of physics and in many given physical problems a numerical analytical evaluation of the Laguerre polynomials is required. This paper aims at presenting Laguerre polynomials in two variables which provide an unification and a generalization of the classical Laguerre polynomials and their various generalizations introduced in the literatures from time to time. In particular, generating functions, integral representations, differential equation, recurrence relations, Laplace transform, expansions and some applications of this polynomials are established. We also derive explicit representation for this Laguerre polynomials defined through generalized hypergeometric functions, which naturally yield numerous other potentially useful (numerical and analytical) properties of the Laguerre polynomials. Finally, some of these results are employed to derived explicit relationships for the wavefunctions of the Morse and Pöschl-Teller potentials, which are frequently used to describe molecular systems.