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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. 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  1. 学術雑誌掲載済論文
  2. 和雑誌

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/7338
52913fd6-66c1-4c21-b785-6d330499e0f1
名前 / ファイル ライセンス アクション
127deform.pdf 127deform.pdf (191.4 kB)
Item type 学術雑誌論文 / Journal Article(1)
公開日 2009-03-27
タイトル
言語 en
タイトル Sur la degenerescence de quelques formules de connexion pour les fonctions hypergeometriques de Gauss
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

WEKO 37461

en 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
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
著者版フラグ
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
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