Fractional operators and special functions. II. Legendre functions

TL;DR

This paper extends the theory of fractional differential operators within Lie group representations to derive new integral and recurrence relations for associated Legendre functions, enhancing understanding of their properties.

Contribution

It introduces fractional generalizations of Lie algebra operators applied to SO(2,1), leading to novel relations and integral representations for Legendre functions.

Findings

Derived new fractional operator relations for Legendre functions.

Established integral representations from two-variable fractional relations.

Connected fractional relations to known integral formulas for Legendre functions.

Abstract

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we apply the fractional generalizations $D^\mu$ of these operators developed in an earlier paper in the context of Lie theory to the group SO(2,1) and its conformal extension. The fractional relations give a variety of interesting relations for the associated Legendre functions. We show that the two-variable fractional operator relations lead directly to integral relations among the Legendre functions and to one- and two-variable integral representations for those functions. Some of the relations reduce to known fractional integrals for the Legendre functions when reduced to one variable. The results enlarge the understanding of many properties of the associated Legendre functions on the basis of the underlying group structure.