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variants of this functions
Hypergeometric2F1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Representations through more general functions > Through Meijer G > Classical cases involving products of 2F1 with algebraic arguments





http://functions.wolfram.com/07.23.26.0137.01









  


  








Input Form





(Sqrt[z] + Sqrt[1 + z])^(3/2 + 2 a - 2 c) Hypergeometric2F1[a, 1/2 + a, c, (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] Hypergeometric2F1[1 - a, 1/2 - a, 1/2 - 2 a + c, (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] == ((4^(-a + c - 1) Gamma[c] Gamma[1/2 - 2 a + c])/(Pi Gamma[-1 - 2 a + 2 c])) MeijerG[{{7/4 + a - c}, {3/4 + a, 1/4 + c - a, 5/4 - a}}, {{1/4, 1/2, 3/4, 1}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





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MathML Form







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</mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["z_"], "+", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]], RowBox[List[FractionBox["3", "2"], "+", RowBox[List["2", " ", "a_"]], "-", RowBox[List["2", " ", "c_"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", RowBox[List[FractionBox["1", "2"], "+", "a_"]], ",", "c_", ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", "a_"]], ",", RowBox[List[FractionBox["1", "2"], "-", "a_"]], ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "a_"]], "+", "c_"]], ",", FractionBox[RowBox[List[SqrtBox["z_"], "-", SqrtBox[RowBox[List["1", "+", "z_"]]]]], RowBox[List["2", " ", SqrtBox["z_"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "a"]], "+", "c", "-", "1"]]], " ", RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "a"]], "+", "c"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "4"], "+", "a", "-", "c"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", "a"]], ",", RowBox[List[FractionBox["1", "4"], "+", "c", "-", "a"]], ",", RowBox[List[FractionBox["5", "4"], "-", "a"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "2"], ",", FractionBox["3", "4"], ",", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "c"]]]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29