On some fractional generalizations of the Laguerre polynomials and the Kummer function

This paper refers to some generalizations of the classical Laguerre polynomials. By means of the Riemann–Liouville operator of fractional calculus and Rodrigues’ type representation formula of fractional order, the Laguerre functions are derived and some of their properties are given and compared with the corresponding properties of the classical Laguerre polynomials. Further generalizations of the Laguerre functions are introduced as a solution of a fractional version of the classical Laguerre differential equation. Likewise, a generalization of the Kummer function is introduced as a solution of a fractional version of the Kummer differential equation. The Laguerre polynomials and functions are presented as special cases of the generalized Laguerre and Kummer functions. The relation between the Laguerre polynomials and the Kummer function is extended to their fractional counterparts.