2016 Fiscal Year Final Research Report
Multiple hypergeometric type generating functions for the values of Lerch zeta-functions--their formulation and analytic behaviour--
| Project/Area Number |
26400021
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Keio University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | generating function / zeta-function |
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| Outline of Final Research Achievements |
As for the multiple hypergeometric type generating functions for the values of Lerch zeta-functions, the head investigator has succeeded in formulating the expected generating functions (of several complex variables) for the values of Lerch zeta-functions, in the form of Lauricella (type A) multiple hypergeometric series. The major achievements of the present research include complete asymptotic expansions for these multiple generating functions when the variables $(z_1,\ldots,z_n)$ tend to $0$ and to $\infty$, while suitable mutual order conditions on $z_j$'s are imposed, through an appropriate poly-sector. These asymptotic expansions further yield: 1) asymptotics for higher derivatives of the generating functions when the variable $s$ is at any integer point; 2) closed form evaluation of the generating functions when $s$ is at any non-positive integer point; 3) asymptotics for two variable analogues of the classical trigonometric sums treated in [Hardy-Littlewood (1936)].
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| Free Research Field |
解析的整数論
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