Abstract In our recent work SIGMA 20, 074 (2024), the leading behaviour of the Humbert function Ψ 1 a, b ; c, c ′; x, y when x → ∞ and y → +∞ has been derived in a direct and simple manner. In this paper, we obtain the complete asymptotics of Ψ 1 in the general case x, y → ∞ along a new path. Indeed, our proof is based on a sharp estimate on F 2 2 a, b − n ; c, d − n ; z ₂F₂a, b-n;c, d-n;z, which is valid uniformly for n ∈ Z ≥ 0 n Z ₀ and large z.
Hang et al. (Fri,) studied this question.