Multi-integral representations for associated Legendre and Ferrers functions

TL;DR

This paper develops new multi-integral and multi-derivative formulas for associated Legendre and Ferrers functions, generalizing classical formulas and revealing new properties and special values of these functions.

Contribution

It introduces novel multi-integral representations for associated Legendre and Ferrers functions, extending classical formulas and identifying new zero and symmetry properties.

Findings

Derived new multi-integral formulas for Legendre and Ferrers functions.

Computed special values and integral representations for specific parameters.

Discovered parameter conditions where the second kind function is identically zero.

Abstract

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind including parameter values for which this function is identically zero.