Asymptotic expansions for double Shintani zeta-functions of several variables

This is a summarized version of the forthcoming paper [19]. Let m and n be any positive integers. We write e(x)=e^2π√-1, and use the vectorial notation x=(x₁,...,x_m) for any complex x and x_i(i=1,...,m). The main object of this paper is the Shintani zeta-function φ̃_n(s,a,λ;z) defined by (1.4) below, where s_j(j=1,...,n) are complex variables, a_i and λ_i(i=1,2) real parameters with a_i>0, and z _j complex parameters with |argz_j|<π(j=1,...,n). We shall first present a complete asymptotic expansion of φ̃ _n(s,a,λ;z) in the ascending order of z_n as z _n→0 (Theorem 1), and that in the descending order of z _n as z_n→∞ (Theorem 2), both through the sectorial region |argz_n-θ₀|<π/2 for any angle θ₀ with |θ₀|<π/2, while other z _j's move within the same sector upon satisfying the conditions z _j≈z_n(j=1,...,n-1). It is significant that the Lauricella hypergeometric functions (defined by (2.1) below) appear in each term of the asymptotic series on the right sides of (2.7) and (2.10). Our main formulae (2.6) (with (2.7) and (2.8)) and (2.9) (with (2.10) and (2.11)) further yield several functional properties of φ̃_n(s,a,λ;z) (Corollaries 1-3).