Large-degree asymptotic expansions for the Jacobi and allied functions

2024/11/17 by Gergő Nemes, Nemes, Gergő
Mathematics · #33C05 #33C45 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2411.10942

openalex publication_date 2024/11/17 · openalex created_date 2024/11/21 · openalex updated_date 2026/07/28

Abstract

Simple asymptotic expansions for the Jacobi functions P_ν^{(α, β)}(z) and Q_ν^{(α, β)}(z) for large degree ν, with fixed parameters α and β, 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 Q_ν^{(α, β)}(x) and Q_ν^{(α, β)}(x).

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