Asymptotic expansions for double Shintani zeta-functions of several variables

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Abstract

This is a summarized version of the forthcoming paper [19]. 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).

Original languageEnglish
Title of host publicationDiophantine Analysis and Related Fields 2011, DARF - 2011
Pages58-72
Number of pages15
DOIs
Publication statusPublished - 2011 Nov 25
EventDiophantine Analysis and Related Fields 2011, DARF - 2011 - Musashino, Tokyo, Japan
Duration: 2011 Mar 32011 Mar 5

Publication series

NameAIP Conference Proceedings
Volume1385
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Other

OtherDiophantine Analysis and Related Fields 2011, DARF - 2011
CountryJapan
CityMusashino, Tokyo
Period11/3/311/3/5

Keywords

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

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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    Katsurada, M. (2011). Asymptotic expansions for double Shintani zeta-functions of several variables. In Diophantine Analysis and Related Fields 2011, DARF - 2011 (pp. 58-72). (AIP Conference Proceedings; Vol. 1385). https://doi.org/10.1063/1.3630041