Laguerre polynomials of arbitrary (fractional) orders

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. In this paper we define the Laguerre polynomials Łαβ(x) of arbitrary order α∈R (the set of all real numbers), and prove that this definition generalizes and interpolates the well known definition Lnβ(x),n=1,2,… of Laguerre polynomials, and prove that {Łαβ(x),α∈R} is continuous as a function of α,α∈R. Also the confluent hypergeometric representation of these polynomials will be given.