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variants of this functions
Hypergeometric2F1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Identities > Functional identities > Confluent general cases





http://functions.wolfram.com/07.23.17.0150.01









  


  










Input Form





Hypergeometric2F1[a, b, c, z] == (Gamma[c]/((-z)^\[Mu] (Gamma[a] Gamma[b]))) (Sum[(Gamma[\[Mu] + k] (n - k - 1)!)/(z^k (k! (c - \[Mu] - k - 1)!)), {k, 0, n - 1}] + (-1)^n Sum[(Gamma[\[Mu] + k]/((-z)^k (k! (k - n)! (c - \[Mu] - k - 1)!))) (PolyGamma[k + 1] + PolyGamma[k - n + 1] - PolyGamma[\[Mu] + k] - PolyGamma[c - \[Mu] - k] + Log[-z]), {k, n, c - \[Mu] - 1}] + (-1)^n (-z)^(-c + \[Mu]) Gamma[c] HypergeometricPFQRegularized[{1, 1, c}, {1 + c - \[Mu], 1 + c - \[Mu] - n}, 1/z]) /; Abs[z] > 1 && Element[b - a, Integers] && n == Abs[b - a] && Element[c - a, Integers] && \[Mu] == a + Min[0, b - a]










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <ci> &#956; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <integers /> </apply> <apply> <eq /> <ci> n </ci> <apply> <abs /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <integers /> </apply> <apply> <eq /> <ci> &#956; </ci> <apply> <plus /> <ci> a </ci> <apply> <min /> <cn type='integer'> 0 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02





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