Fractional operators and multi-integral representations for associated Legendre functions

Abstract

In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for the associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps. These are of potential use in a number of physical problems. We show here how their results can be derived simply from more general relations involving non-integer changes in the order obtained using the fractional group operator methods developed earlier for SO(2,1), E(2,1) and its conformal extension, and SO(3). We also present general integral relations for fractional changes of the degrees of the functions and related multi-derivative and multi-integral representations.