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 $e(x) = e2π−1,$ and use the vectorial notation $x = (x1,…,xm)$ for any complex x and $xi(i = 1,…,m).$ The main object of this paper is the Shintani zeta‐function $φ̃n(s,a,λ;z)$ defined by (1.4) below, where $sj(j = 1,…,n)$ are complex variables, $ai$ and $λi(i = 1,2)$ real parameters with $ai>0,$ and $zj$ complex parameters with $|argzj|<π(j = 1,…,n).$ We shall first present a complete asymptotic expansion of $φ̃n(s,a,λ;z)$ in the ascending order of $zn$ as $zn→0$ (Theorem 1), and that in the descending order of $zn$ as $zn→∞$ (Theorem 2), both through the sectorial region $|argzn−θ0|<π/2$ for any angle $θ0$ with $|θ0|<π/2,$ while other $zj$’s move within the same sector upon satisfying the conditions $zj≈zn (j = 1,…,n−1).$ 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 $φ̃n(s,a,λ;z)$ (Corollaries 1–3).