Large-degree asymptotic expansions for the Jacobi and related functions

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 ${\mathrm{Q}}_{\nu }^{(\alpha ,\beta )}(x)$.

40 pages, 5 figures, accepted for publication in Mathematics of Computation