# Large-parameter asymptotic expansions for the Legendre and allied functions

## TL;DR

This paper develops new simple asymptotic expansions for Legendre functions and related functions at large parameters, providing sharp error bounds and finite representations in special cases, filling a significant gap in mathematical literature.

## Contribution

It introduces novel inverse factorial asymptotic expansions for Legendre and allied functions, with explicit error bounds and finite forms in special cases, many of which are new.

## Key findings

- Derived simple asymptotic expansions for Legendre functions at large parameters
- Provided sharp bounds on the error terms of these expansions
- Included new finite representations for special cases and bounds for hypergeometric series

## Abstract

Surprisingly, apart from some special cases, simple asymptotic expansions for the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for large degree $\nu$ or large order $\mu$ are not available in the literature. The main purpose of the present paper is to fill this gap by deriving simple (inverse) factorial expansions for these functions and provide sharp and realistic bounds on their error terms. Analogous results for the Ferrers functions and the closely related Gegenbauer function are also included. In the cases that $\nu$ is an integer or $2\mu$ is an odd integer, many of these new expansions terminate and provide finite representations in terms of simple functions. Most of these representations appear to be new. It is well known that the hypergeometric series can be regarded as a large-$c$ asymptotic expansion for the hypergeometric function $F(a,b;c;z)$. We also derive computable bounds for the remainder term of this expansion. To our best knowledge, no such estimates have been given in the literature prior to this paper.