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

1 引用 (Scopus)

### 抄録

This is a summarized version of the forthcoming paper . Let m and n be any positive integers. We write e(x)=e√-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 |argzn0|<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).

元の言語 English AIP Conference Proceedings 58-72 15 1385 https://doi.org/10.1063/1.3630041 Published - 2011 Diophantine Analysis and Related Fields 2011, DARF - 2011 - Musashino, Tokyo, Japan継続期間: 2011 3 3 → 2011 3 5

### Other

Other Diophantine Analysis and Related Fields 2011, DARF - 2011 Japan Musashino, Tokyo 11/3/3 → 11/3/5

### Fingerprint

expansion
theorems
asymptotic series
complex variables
hypergeometric functions
integers
coding
sectors

### ASJC Scopus subject areas

• Physics and Astronomy(all)

### これを引用

AIP Conference Proceedings. 巻 1385 2011. p. 58-72.

Katsurada, M 2011, Asymptotic expansions for double Shintani zeta-functions of several variables. ： AIP Conference Proceedings. 巻. 1385, pp. 58-72, Diophantine Analysis and Related Fields 2011, DARF - 2011, Musashino, Tokyo, Japan, 11/3/3. https://doi.org/10.1063/1.3630041
@inproceedings{b288c2c6ab3a4d878b598cdb25f9609f,
title = "Asymptotic expansions for double Shintani zeta-functions of several variables",
abstract = "This is a summarized version of the forthcoming paper . 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).",
keywords = "asymptotic expansion, Mellin-Barnes integral, Shintani zeta-function",
year = "2011",
doi = "10.1063/1.3630041",
language = "English",
isbn = "9780735409477",
volume = "1385",
pages = "58--72",
booktitle = "AIP Conference Proceedings",

}

TY - GEN

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

PY - 2011

Y1 - 2011

N2 - This is a summarized version of the forthcoming paper . 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 . 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 - asymptotic expansion

KW - Mellin-Barnes integral

KW - Shintani zeta-function

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

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

U2 - 10.1063/1.3630041

DO - 10.1063/1.3630041

M3 - Conference contribution

AN - SCOPUS:81755162353

SN - 9780735409477

VL - 1385

SP - 58

EP - 72

BT - AIP Conference Proceedings

ER -