Ferrers functions of arbitrary degree and order and related functions
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.
Keywords
Cite
@article{arxiv.2110.13564,
title = {Ferrers functions of arbitrary degree and order and related functions},
author = {P. Malits},
journal= {arXiv preprint arXiv:2110.13564},
year = {2022}
}
Comments
39 pages. 1) Inessential details are omitted in the proof of Theorem 13. 2) The statement of Theorem 26 is slightly changed. This change has enabled me to write a new generating function for Ferrers functions as a corollary. 3) The statement of Theorem 29 (Theorem 28 in the previous version) is corrected. 4) Typos are corrected