Fractional operators and multi-integral representations for associated Legendre functions

TL;DR

This paper introduces fractional operator methods to derive multi-integral and multi-derivative relations for associated Legendre functions, extending previous results to non-integer order changes with applications in physics.

Contribution

It provides a unified fractional operator framework to generate new integral and derivative relations for Legendre functions, generalizing prior integer-step results.

Findings

Derived new integral relations for fractional order changes

Unified approach using fractional group operators

Extended relations to degrees of Legendre functions

Abstract

In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for 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.