TY - GEN

T1 - Asymptotic expansions for double Shintani zeta-functions of several variables

AU - Katsurada, Masanori

PY - 2011/11/25

Y1 - 2011/11/25

N2 - 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 z j complex parameters with |argzj|<π(j=1,...,n). We shall first present a complete asymptotic expansion of φ̃ n(s,a,λ;z) in the ascending order of zn as z n→0 (Theorem 1), and that in the descending order of z n as zn→∞ (Theorem 2), both through the sectorial region |argzn-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≈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).

AB - 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 z j complex parameters with |argzj|<π(j=1,...,n). We shall first present a complete asymptotic expansion of φ̃ n(s,a,λ;z) in the ascending order of zn as z n→0 (Theorem 1), and that in the descending order of z n as zn→∞ (Theorem 2), both through the sectorial region |argzn-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≈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).

KW - Mellin-Barnes integral

KW - Shintani zeta-function

KW - asymptotic expansion

UR - http://www.scopus.com/inward/record.url?scp=81755162353&partnerID=8YFLogxK

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U2 - 10.1063/1.3630041

DO - 10.1063/1.3630041

M3 - Conference contribution

AN - SCOPUS:81755162353

SN - 9780735409477

T3 - AIP Conference Proceedings

SP - 58

EP - 72

BT - Diophantine Analysis and Related Fields 2011, DARF - 2011

T2 - Diophantine Analysis and Related Fields 2011, DARF - 2011

Y2 - 3 March 2011 through 5 March 2011

ER -