Asymptotic expansions for double Shintani zeta-functions of several variables

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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 publicationAIP Conference Proceedings
Pages58-72
Number of pages15
Volume1385
DOIs
Publication statusPublished - 2011
EventDiophantine Analysis and Related Fields 2011, DARF - 2011 - Musashino, Tokyo, Japan
Duration: 2011 Mar 32011 Mar 5

Other

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

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expansion
theorems
asymptotic series
complex variables
hypergeometric functions
integers
coding
sectors

Keywords

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

ASJC Scopus subject areas

  • Physics and Astronomy(all)

Cite this

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

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Katsurada, M 2011, Asymptotic expansions for double Shintani zeta-functions of several variables. in 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
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