Asymptotic expansions of the Lauricella hypergeometric function FD

Abstract

The Lauricella hypergeometric function F D r ( a, b 1,…, b r ; c; x 1,…, x r ) with r∈ N , is considered for large values of one variable: x 1, or two variables: x 1 and x 2. 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− x 1 or 1− x 1 and 1− x 2. For certain values of the parameters a, b i and c, two of these expansions also involve logarithmic terms in the asymptotic variable(s). For large x 1, coefficients of these expansions are given in terms of the Lauricella hypergeometric function F D r−1 ( a, b 2,…, b r ; c; x 2,…, x r ) and its derivative with respect to the parameter a, whereas for large x 1 and x 2 those coefficients are given in terms of F D r−2 ( a, b 3,…, b r ; c; x 3,…, x r ) 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.