### Abstract

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=(x_{1},...,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}-θ_{0}|<π/2 for any angle θ_{0} with |θ_{0}|<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).

Original language | English |
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Title of host publication | AIP Conference Proceedings |

Pages | 58-72 |

Number of pages | 15 |

Volume | 1385 |

DOIs | |

Publication status | Published - 2011 |

Event | Diophantine Analysis and Related Fields 2011, DARF - 2011 - Musashino, Tokyo, Japan Duration: 2011 Mar 3 → 2011 Mar 5 |

### Other

Other | Diophantine Analysis and Related Fields 2011, DARF - 2011 |
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Country | Japan |

City | Musashino, Tokyo |

Period | 11/3/3 → 11/3/5 |

### Fingerprint

### Keywords

- asymptotic expansion
- Mellin-Barnes integral
- Shintani zeta-function

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*AIP Conference Proceedings*(Vol. 1385, pp. 58-72) https://doi.org/10.1063/1.3630041

**Asymptotic expansions for double Shintani zeta-functions of several variables.** / Katsurada, Masanori.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*AIP Conference Proceedings.*vol. 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

}

TY - GEN

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

AU - Katsurada, Masanori

PY - 2011

Y1 - 2011

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 - 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

SN - 9780735409477

VL - 1385

SP - 58

EP - 72

BT - AIP Conference Proceedings

ER -