Ferrers functions of arbitrary degree and order and related functions

2021/10/26 by P. Malits, Malits, P.
Mathematics · Physics and Astronomy · #33C45(Primary) #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2110.13564

openalex publication_date 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Asymptotic Expansion of Generalized Hypergeometric Functions

Related

• Mutually inverse series relating Ferrers and associated Legendre functions and generating functions pertaining to them
• On the relation between Gegenbauer polynomials and the Ferrers function of the first kind
• Generating Functions for Lacunary Legendre and Legendre-like Polynomials
• Fourier series representation of Ferrers function P
• On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)