Asymptotics of the Humbert Function Ψ₁ for Two Large Arguments

Abstract

Recently, Wald and Henkel (2018) derived the leading-order estimate of the Humbert functions Φ₂, Φ₃, and Ξ₂ for two large arguments, but their technique cannot handle the Humbert function Ψ₁. In this paper, we establish the leading asymptotic behavior of the Humbert function Ψ₁ for two large arguments. Our proof is based on a connection formula of the Gauss hypergeometric function and Nagel's approach (2004). This approach is also applied to deduce asymptotic expansions of the generalized hypergeometric function ₚ𝐹𝑞 (𝑝 ⩽ 𝑞) for large parameters, which are not contained in the NIST handbook.