Gauss hypergeometric function 2F1: Representations through more general functions (formula 07.23.26.0205)

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/07.23.26.0205.01

Input Form

(Sqrt[z] + Sqrt[1 + z])^(1 - c) Hypergeometric2F1[a, 1/2 + a, c, (Sqrt[z] - Sqrt[1 + z])/(2 Sqrt[z])] Hypergeometric2F1Regularized[ 1 + a - c, 3/2 + a - c, 3/2 + 2 a - c, (Sqrt[z] - Sqrt[1 + z])/ (2 Sqrt[z])] == ((2^(-(1/2) + 4 a - c) Gamma[c])/(Pi Gamma[2 a])) MeijerG[{{(3 - c)/2}, {2 - (3 c)/2 + 2 a, 1 - c/2 + 2 a, (1 + c)/2}}, {{(1 - c)/2 + a, (3 - 2 c)/4 + a, 1 - c/2 + a, (5 - 2 c)/4 + a}, {}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]

Standard Form

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MathML Form

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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["1", "-", "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["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "a_", "-", "c_"]], ",", RowBox[List[FractionBox["3", "2"], "+", "a_", "-", "c_"]], ",", RowBox[List[FractionBox["3", "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["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", RowBox[List["4", " ", "a"]], "-", "c"]]], " ", RowBox[List["Gamma", "[", "c", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["3", "-", "c"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", FractionBox[RowBox[List["3", " ", "c"]], "2"], "+", RowBox[List["2", " ", "a"]]]], ",", RowBox[List["1", "-", FractionBox["c", "2"], "+", RowBox[List["2", " ", "a"]]]], ",", FractionBox[RowBox[List["1", "+", "c"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "c"]], "2"], "+", "a"]], ",", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "c"]]]], ")"]]]], "+", "a"]], ",", RowBox[List["1", "-", FractionBox["c", "2"], "+", "a"]], ",", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "c"]]]], ")"]]]], "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "a"]], "]"]]]]], "/;", 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]