Complete asymptotic expansions associated with Epstein zeta-functions II
Complete asymptotic expansions associated with Epstein zeta-functions II
复制标题
与 Epstein zeta 函数 II 相关的完全渐近展开
DOI：
10.1007/s11139-014-9583-6
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Masanori Katsurada
中科院分区：
文献类型：
--
作者：
Kaneda;M.;Ryousuke Fujita;Masanori Katsurada
设是上半平面中的一个带复参数的正定二次型。附接的Epstein ζ-官能最初由下面的（1.3）定义。我们已经建立了在前面的文件Katsurada（Ramanujan J 14：249-275，2007）完全渐近展开的as，以及其加权平均值（关于）的形式的拉普拉斯-梅林变换（1.4）。本文继续我们以前的研究，以证明类似的渐近级数仍然存在的更一般的Epsteinzeta函数定义的（1.2）下面（定理1），也为Riemann-Liouville变换（1.5）（定理2）。在证明这些渐近展开式之前，先利用Mellin-Barnes积分变换（第三节命题1）得到全平面上的亚纯延拓。这个过程与以前已知的解析延拓方法稍有不同，它以双无穷级数的形式给出了的亚纯延拓[见（2.9）和（3.9）与（3.8）]，这最适合于导出所讨论的渐近展开式。在证明的各个方面，使用梅林-巴恩斯型积分（如公式3.3）都是至关重要的;超几何函数的几个变换和连接公式特别适用于这些积分的操作。
Letbe a positive-definite quadratic form with a complex parameterin the upper half-plane. The Epstein zeta-functionattached tois initially defined by (1.3) below. We have established in the preceding paper Katsurada (Ramanujan J 14:249–275, 2007) complete asymptotic expansions ofas, and those of its weighted mean value (with respect to) in the form of a Laplace–Mellin transform (1.4). The present paper proceeds further with our previous study to show that similar asymptotic series still exist for a more general Epstein zeta-functiondefined by (1.2) below (Theorem 1), and also for the Riemann–Liouville transform (1.5) of(Theorem 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation ofover the whole-plane is prepared by means of Mellin–Barnes integral transforms (Proposition 1 in Sect. 3). This procedure differs slightly from other previously known methods of analytic continuation, and provides the meromorphic continuation ofin the form of a double infinite series [see (2.9) and (3.9) with (3.8)], which is most appropriate for deriving the asymptotic expansions in question. The use of Mellin–Barnes type integrals such as in (3.3) is crucial in all aspects of the proofs; several transformation and connection formulae for hypergeometric functions are especially applied with manipulation of these integrals.