Asymptotic expansions of the Appell’s function $F_1$

The first Appell’s hypergeometric function ${F_1}\left ( a, b, c, d; x, y \right )$ is considered for large values of its variables $x$ and/or $y$. An integral representation of ${F_1}\left ( a, b, c, d; x, y \right )$ is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of $1/\left ( 1 - x \right )$ and/or $1/\left ( 1 - y \right )$. For certain values of the parameters $a, b, c$ and $d$, two of these expansions also involve logarithmic terms in the asymptotic variables $1 - x$ and/or $1 - y$. Coefficients of these expansions are given in terms of the Gauss hypergeometric function $_2{F_1}\left ( \alpha , \beta , \gamma ; x \right )$ and its derivative with respect to the parameter $\alpha$. All of the expansions are accompanied by error bounds for the remainder at any order of the approximation. These error bounds are obtained from the error test and, as numerical experiments show, they are considerably accurate.