Fractional calculus and generalized Rodrigues formula

The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers. The two generalizations of the Rodrigues formula of the Laguerre polynomials L-alpha(beta)(x) = (X-beta/n!)e(x)D(x)e(-x)x(n+beta), and L-alpha(beta)(x) = ((x(-beta)e(x))/Gamma(1 + alpha))D(alpha)e(-x)x(x+beta), are defined in [Math. Sci. Res. Hot-line 1 (10) (1997) 7; Appl. Math. Comput. 106 (1) (1999) 51] and some of their properties are proved. Here we define the new special function L-alpha(beta)(y, a; x) based on a generalization of the Rodrigues formula, then we study some of its properties, some recurrence relations and prove that the set of functions {L-alpha(beta) (y, a; x), alpha is an element of R} is continuous as a function of alpha is an element of R. The continuation as alpha, y --> n and a = 1 to the Rodrigues formula of the Laguerre polynomials L-alpha(beta)(x) are proved. Also the confluent hypergeometric representation will be given. (C) 2002 Elsevier Inc. All rights reserved.