Ferrers functions of arbitrary degree and order and related functions

TL;DR

This paper introduces new integral and series representations, relations, and asymptotic expansions for Ferrers functions of arbitrary degree and order, enhancing understanding and computational methods for these special functions.

Contribution

It provides novel integral, series, and differential relations, along with uniform asymptotic expansions for Ferrers functions, and establishes new addition theorems and relations to Gegenbauer polynomials.

Findings

Derived new integral and series representations for Ferrers functions.

Established addition theorems for Ferrers functions with hyperbolic argument.

Connected Ferrers functions to Gegenbauer polynomials as special cases.

Abstract

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for Ferrers functions of the argument tanh(a+b) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.