Asymptotics of the Humbert Function ${\mathrm{\Psi }}_1$ for Two Large Arguments

Recently, Wald and Henkel (2018) derived the leading-order estimate of the Humbert functions ${\mathrm{\Phi }}_2$, ${\mathrm{\Phi }}_3$ and ${\mathrm{\Xi }}_2$ for two large arguments, but their technique cannot handle the Humbert function ${\mathrm{\Psi }}_1$. In this paper, we establish the leading asymptotic behavior of the Humbert function ${\mathrm{\Psi }}_1$ 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 ${}_p{F}_q$ $(p⩽q)$ for large parameters, which are not contained in NIST handbook.