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KAKEN_26400021seika.pdf
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Title |
Title |
ゼータ関数・テータ関数の多重母関数 : その定式化と挙動解明
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Kana |
ゼータ カンスウ・テータ カンスウ ノ タジュウ ボカンスウ : ソノ ジョウシキカ ト キョドウ カイメイ
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Zeta kansu teta kansu no taju bokansu : sono joshikika to kyodo kaimei
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Multiple hypergeometric type generating functions for the values of Lerch zeta-functions : their formulation and analytic behaviour
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桂田, 昌紀
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カツラダ, マサノリ
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Katsurada, Masanori
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慶應義塾大学・経済学部・教授
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Research team head
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科研費研究者番号 : 90224485
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野田, 工
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ノダ, タクミ
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Noda, Takumi
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日本大学・工学部・教授
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Research team member
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科研費研究者番号 : 10350034
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天羽, 雅昭
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アモウ, マサアキ
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Romanization |
Amo, Masaaki
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群馬大学・大学院理工学府・教授
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Research team member
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科研費研究者番号 : 60201901
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2017
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科学研究費補助金研究成果報告書
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2016
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Abstract |
Lerchゼータ関数の値列を係数に含む多変数超幾何型母関数に関しては, LauricellaのA型多重超幾何関数に付随した形の母関数の定式化がほぼ満足できる形に達成された。本研究代表者はこの母関数に関して, 複素$n$変数が適切な相互order条件を満たしつつ多重扇状領域内を$0$に収束するとき, 及び$¥infty$に発散するとき, それぞれのcaseについて, 完全漸近展開を導出することに成功しており, この成果からは母関数の高階導関数に対して, 変数$s$が整数点にあるときの完全漸近展開や, $s$が非負の整数点のある場合にはclosed formの表示も得られる。
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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研究種目 : 基盤研究(C)(一般)
研究期間 : 2014~2016
課題番号 : 26400021
研究分野 : 解析的整数論
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