Large-degree asymptotic expansions for the Jacobi and related functions

TL;DR

This paper derives simple asymptotic expansions with explicit error bounds for Jacobi functions of large degree, filling a notable gap in the literature and extending results to related functions.

Contribution

It provides the first comprehensive derivation of inverse factorial asymptotic expansions for Jacobi functions with fixed parameters, including explicit error bounds.

Findings

Derived simple inverse factorial expansions for Jacobi functions

Provided explicit, computable error bounds for the expansions

Extended results to related functions $ extsf{Q}_ u^{(oldsymbol{eta})}$ and $ extsf{Q}_ u^{(oldsymbol{eta})}$

Abstract

Simple asymptotic expansions for the Jacobi functions $P_\nu^{(\alpha, \beta)}(z)$ and $Q_\nu^{(\alpha, \beta)}(z)$ for large degree $\nu$, with fixed parameters $\alpha$ and $\beta$, are surprisingly rare in the literature, with only a few special cases covered. This paper addresses this notable gap by deriving simple (inverse) factorial expansions for these functions, complemented by explicit and computable error bounds. Additionally, we provide analogous results for the associated functions $\mathsf{Q}_\nu^{(\alpha, \beta)}(x)$ and $\mathrm{Q}_\nu^{(\alpha, \beta)}(x)$.