Asymptotics of Humbert functions Y1 and Y2 for y → ∞ with arbitrary parameters

Establish general asymptotic expansions for the Humbert functions Y1[a, b; c, c'; x, y] and Y2[a; c, c'; x, y] as y → ∞ for arbitrary complex parameters a, b, c, c', without restrictive assumptions, so that the leading and subsequent terms are rigorously characterized across the full parameter ranges.

Background

The paper develops several asymptotic results for Humbert functions Y1 and Y2 under conditions where one variable is small or both variables are large, and also provides Mellin–Barnes representations and Laplace-type analyses. However, the authors note a gap in obtaining asymptotics as y becomes large when parameters are unrestricted.

They attribute the difficulty to the lack of effective expansions for the Gauss hypergeometric function 2F1 and the Kummer function 1F1 in regimes where parameters themselves become large. Filling this gap would enable broader use of Mellin–Barnes techniques to treat multiple hypergeometric functions with one large argument.

References

Remarkably, we could not obtain the asymptotics of Y1x, y and Y2[x, y] as y -> oo with arbitrary values of parameters, because we lack of useful expansions of 2F1 and 1F1, which are expanded in terms of large parameters. If such effective expansions of pFq were known, one might use the Mellin-Barnes integral technique to establish the asymptotics of multiple hypergeometric functions for one large argument.

— Asymptotics of the Humbert functions $Ψ_1$ and $Ψ_2$ (2501.07281 - Hang et al., 13 Jan 2025) in Section 7, Concluding remarks