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Sur la degenerescence de quelques formules de connexion pour les fonctions hypergeometriques de Gauss
https://kitami-it.repo.nii.ac.jp/records/7338
https://kitami-it.repo.nii.ac.jp/records/733852913fd6-66c1-4c21-b785-6d330499e0f1
| 名前 / ファイル | ライセンス | アクション |
|---|---|---|
127deform.pdf (191.4 kB)
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| Item type | 学術雑誌論文 / Journal Article(1) | |||||||
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| 公開日 | 2009-03-27 | |||||||
| タイトル | ||||||||
| タイトル | Sur la degenerescence de quelques formules de connexion pour les fonctions hypergeometriques de Gauss | |||||||
| 言語 | en | |||||||
| 言語 | ||||||||
| 言語 | eng | |||||||
| キーワード | ||||||||
| 主題Scheme | Other | |||||||
| 主題 | Gauss hypergeometric function | |||||||
| キーワード | ||||||||
| 主題Scheme | Other | |||||||
| 主題 | confluent hypergeometric function | |||||||
| キーワード | ||||||||
| 主題Scheme | Other | |||||||
| 主題 | Pochhammer integral representation | |||||||
| キーワード | ||||||||
| 主題Scheme | Other | |||||||
| 主題 | connection formula | |||||||
| 資源タイプ | ||||||||
| 資源 | http://purl.org/coar/resource_type/c_6501 | |||||||
| タイプ | journal article | |||||||
| アクセス権 | ||||||||
| アクセス権 | open access | |||||||
| アクセス権URI | http://purl.org/coar/access_right/c_abf2 | |||||||
| 著者 |
Watanabe, Humihiko
× Watanabe, Humihiko
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| 抄録 | ||||||||
| 内容記述タイプ | Abstract | |||||||
| 内容記述 | It is well known that there exists an operation of limit that makes the degeneration of Gauss hypergeometric differential equation to the confluent hypergeometric differential equation. With this operation, we calculate the degeneration of connection formulas of Gauss hypergeometric function explicitely. This kind of degeneration connects the theory of analytic continuation of Gauss hypergeometric function with that of confluent hypergeometric function. In the case of the connection formula between 0 and $$∖infty$$ ($$∖S$$2), we would like to calculate the limit of the formula:1.4$$F(∖alpha, ∖beta, ∖gamma, z/∖beta) = {∖frac{∖Gamma(∖gamma)∖Gamma(∖beta-∖alpha)}{∖Gamma(∖beta)∖Gamma(∖gamma-∖alpha)}}(e^{-∖pi i}z/∖beta)^{-∖alpha}F(∖alpha, 1 ∖, +∖, ∖alpha ∖, -∖, ∖gamma, 1∖,+∖,∖alpha ∖, - ∖,∖beta, ∖beta/z) +{∖frac{∖Gamma(∖gamma)∖Gamma(∖alpha-∖beta)}{∖Gamma(∖alpha)∖Gamma(∖gamma-∖beta)}}(e^{-∖pi i}z/∖beta)^{-∖beta}F(∖beta, 1+∖beta - ∖gamma, 1+ ∖beta -∖alpha , ∖beta/z)$$as $$∖beta∖rightarrow∖infty$$. It is the problem that the right-hand side of (1.4) diverges if we take the limit $$∖beta∖rightarrow∖infty$$ for it directly. The problem will be resolved if we follow the following procedure: First, we replace $$(e^{-∖pi i}Z/∖beta)^{-∖beta}F(∖beta , 1+∖beta - ∖gamma ,1+∖beta - ∖alpha ,∖beta /z)$$ in the right-hand side of (1.4) by, after Kummer [4], the function $$(e^{-∖pi i}z/∖beta)^{∖alpha -∖gamma}(1+e^{-∖pi i}z/∖beta)^{∖gamma-∖alpha-∖beta}F(1-∖alpha,∖gamma - ∖alpha , ∖beta-∖alpha+1,∖beta/z)$$. Next, we replace the two functions $$F(∖alpha,1+∖alpha -∖gamma,1+∖alpha-∖beta,∖beta/z)$$ and $$F(1-∖alpha,∖gamma-∖alpha,∖beta-∖alpha+1,∖beta/z)$$ by suitable Pochhammer integral representations. After these replacements, if we take the limit $$∖beta∖rightarrow e^{i ∖theta}∖infty$$, then we have (Théorème 2.5)2.9$$F(∖alpha,∖gamma,z)={∖frac{∖Gamma(∖gamma)z^{-∖alpha}}{∖Gamma(∖gamma-∖alpha)∖Gamma(∖alpha)(e^{∖pi i ∖alpha}-e^{-∖pi i ∖alpha})}}∖int_{e^{i (∖theta-∖pi)}∖infty}^{(0 + )} e^{-∖upsilon}∖upsilon^{∖alpha-1}∖left(1+{∖frac{∖upsilon}{z}} ∖right)^{∖gamma-∖alpha-1}d∖upsilon∖, +∖, {∖frac{∖Gamma(∖gamma)z^{∖alpha-∖gamma}e^z}{∖Gamma(∖alpha)∖Gamma(∖gamma-∖alpha)(e^{2∖pi i (∖gamma-∖alpha)}-1)}}∖int_{e^{i (∖theta-∖pi)}∖infty}^{(0 + )} e^{-∖upsilon}∖upsilon^{∖gamma-∖alpha-1}∖left(1-∖frac{∖upsilon}{z} ∖right)^{∖alpha-1}d∖upsilon$$(if $$∖frac{1}{2}∖pi < ∖theta < ∖pi ∖, ∖text{or}∖,∖pi < ∖theta < ∖frac{3}{2}∖pi$$)under certain conditions for the parameters $$∖alpha, ∖beta, ∖gamma$$ and the independent variable z. A similar procedure works also for the case of the connection formula between 0 and 1 ($$∖S$$3, Théorème 3.5). | |||||||
| 書誌情報 |
Aequationes Mathematicae 巻 73, 号 3, p. 201-221, 発行日 2007-07 |
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| DOI | ||||||||
| 識別子タイプ | DOI | |||||||
| 関連識別子 | http://doi.org/10.1007/s00010-006-2848-4 | |||||||
| フォーマット | ||||||||
| 内容記述タイプ | Other | |||||||
| 内容記述 | application/pdf | |||||||
| 出版者 | ||||||||
| 出版者 | Springer Verlag | |||||||
| 言語 | en | |||||||
| 著者版フラグ | ||||||||
| 言語 | en | |||||||
| 値 | author | |||||||
| 出版タイプ | ||||||||
| 出版タイプ | AM | |||||||
| 出版タイプResource | http://purl.org/coar/version/c_ab4af688f83e57aa | |||||||
