Fractional calculus and generalized Rodrigues formula

摘要

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 Lαβ(x)=(x−β/n!)exDαe−xxn+β, and Lαβ(x)=((x−βex)/Γ(1+α))Dαe−xxα+β, 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 Lαβ(γ,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 {Lαβ(γ,a;x),α∈R} is continuous as a function of α∈R. The continuation as α,γ→n and a=1 to the Rodrigues formula of the Laguerre polynomials Lnβ(x) are proved. Also the confluent hypergeometric representation will be given.