Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0188)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Representations through more general functions

Through Meijer G

Classical cases involving algebraic functions and ₂F^~₁with algebraic arguments

http://functions.wolfram.com/07.23.26.0188.01

Input Form

(2 + z + 2 Sqrt[1 + z])^(a + b - c) Hypergeometric2F1[a, b, c, -((2 (1 + Sqrt[1 + z]))/z)] Hypergeometric2F1Regularized[-a + c, -b + c, c, (2 (-1 + Sqrt[1 + z]))/z] == ((2^(1 - c) Sqrt[Pi] Gamma[c])/(Gamma[a] Gamma[b] Gamma[-a + c] Gamma[-b + c])) MeijerG[{{1 + a + b - c}, {a + b - c/2, (1 - c)/2 + a + b, a + b}}, {{a, b, 2 a + b - c, a + 2 b - c}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]

Standard Form

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MathML Form

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</mo> <mrow> <mi> z </mi> <mo> ∉ </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["2", "+", "z_", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z_"]]]]]]], ")"]], RowBox[List["a_", "+", "b_", "-", "c_"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]]]], "z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "a_"]], "+", "c_"]], ",", RowBox[List[RowBox[List["-", "b_"]], "+", "c_"]], ",", "c_", ",", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox[RowBox[List["1", "+", "z_"]]]]], ")"]]]], "z_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "c"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "c", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "b", "-", FractionBox["c", "2"]]], ",", RowBox[List[FractionBox[RowBox[List["1", "-", "c"]], "2"], "+", "a", "+", "b"]], ",", RowBox[List["a", "+", "b"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", "c"]], ",", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "-", "c"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "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

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]