### 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 | Diophantine Analysis and Related Fields 2011, DARF - 2011 |

Pages | 58-72 |

Number of pages | 15 |

DOIs | |

Publication status | Published - 2011 Nov 25 |

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

### Publication series

Name | AIP Conference Proceedings |
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Volume | 1385 |

ISSN (Print) | 0094-243X |

ISSN (Electronic) | 1551-7616 |

### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)

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## Cite this

*Diophantine Analysis and Related Fields 2011, DARF - 2011*(pp. 58-72). (AIP Conference Proceedings; Vol. 1385). https://doi.org/10.1063/1.3630041