On integral representations and asymptotics of some hypergeometric functions in two variables

The leading asymptotic behaviour of the Humbert functions ${\mathrm{\Phi }}_2$, ${\mathrm{\Phi }}_3$, ${\mathrm{\Xi }}_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these functions are given. These are re-expressed as inverse Laplace transformations and the asymptotics is then found from a Tauberian theorem. Some integrals of the Humbert functions are also analysed.