Asymptotic expansions of the Lauricella hypergeometric function FD

The Lauricella hypergeometric function FDr(a,b1,…,br;c;x1,…,xr) with r∈N, is considered for large values of one variable: x1, or two variables: x1 and x2. An integral representation of this function 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 the asymptotic variable(s) 1−x1 or 1−x1 and 1−x2. For certain values of the parameters a, bi and c, two of these expansions also involve logarithmic terms in the asymptotic variable(s). For large x1, coefficients of these expansions are given in terms of the Lauricella hypergeometric function FDr−1(a,b2,…,br;c;x2,…,xr) and its derivative with respect to the parameter a, whereas for large x1 and x2 those coefficients are given in terms of FDr−2(a,b3,…,br;c;x3,…,xr) and its derivative. All the expansions are accompanied by error bounds for the remainder at any order of the approximation. Numerical experiments show that these bounds are considerably accurate.