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variants of this functions
Hypergeometric2F1Regularized






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1Regularized[a,b,c,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Case of simple poles





http://functions.wolfram.com/07.24.06.0064.01









  


  










Input Form





Hypergeometric2F1Regularized[a, b, c, z] == Subscript[F, Infinity][z, a, b, c] /; (Subscript[F, n][z, a, b, c] == ((Gamma[b - a]/(Gamma[b] Gamma[c - a])) Sum[(Pochhammer[a, k] Pochhammer[a - c + 1, k])/ (k! Pochhammer[a - b + 1, k])/z^k, {k, 0, n}])/(-z)^a + ((Gamma[a - b]/(Gamma[a] Gamma[c - b])) Sum[(Pochhammer[b, k] Pochhammer[b - c + 1, k])/ (k! Pochhammer[b - a + 1, k])/z^k, {k, 0, n}])/(-z)^b == Hypergeometric2F1Regularized[a, b, c, z] - (Gamma[-a + b] Pochhammer[a, 1 + n] (Pochhammer[1 + a - c, 1 + n]/ ((1 + n)! Gamma[b] Gamma[-a + c] Pochhammer[1 + a - b, 1 + n])) z^(-n - 1) HypergeometricPFQ[{1, 1 + a + n, 2 + a - c + n}, {2 + n, 2 + a - b + n}, 1/z])/(-z)^a - (((Gamma[a - b] Pochhammer[b, 1 + n] Pochhammer[1 + b - c, 1 + n])/ ((1 + n)! Gamma[a] Gamma[-b + c] Pochhammer[1 - a + b, 1 + n])) z^(-n - 1) HypergeometricPFQ[{1, 1 + b + n, 2 + b - c + n}, {2 + n, 2 - a + b + n}, 1/z])/(-z)^b && Element[n, Integers] && n >= 0) && !Element[a - b, Integers]










Standard Form





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MathML Form







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<semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;a&quot;]], &quot;+&quot;, &quot;b&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> 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</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