Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aavi)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-17/4, b>=a

For fixed z and a=-17/4, b=19/4

http://functions.wolfram.com/07.23.03.aavi.01

Input Form

Hypergeometric2F1[-(17/4), 19/4, -(9/2), z] == (1/(9313920 Pi^(3/2))) (((1/(-1 + z)^5) (2 (-2328480 + 905520 z + 1055670 z^2 + 1897049 z^3 + 5393388 z^4 + 41144565 z^5 - 235431040 z^6 + 370882560 z^7 - 243597312 z^8 + 58720256 z^9) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^5) (2 (-2328480 + 905520 z + 1055670 z^2 + 1897049 z^3 + 5393388 z^4 + 41144565 z^5 - 235431040 z^6 + 370882560 z^7 - 243597312 z^8 + 58720256 z^9) EllipticE[(1/2) (1 + Sqrt[z])]) - (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^4)) ((-2328480 + 1164240 Sqrt[z] - 258720 z + 614460 z^(3/2) + 441210 z^2 + 75845 z^(5/2) + 1821204 z^3 - 841302 z^(7/2) + 6234690 z^4 - 3447675 z^(9/2) + 44592240 z^5 - 85164160 z^(11/2) - 150266880 z^6 + 228096000 z^(13/2) + 142786560 z^7 - 199557120 z^(15/2) - 44040192 z^8 + 58720256 z^(17/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^4 (1 + Sqrt[z])^5)) ((-2328480 - 1164240 Sqrt[z] - 258720 z - 614460 z^(3/2) + 441210 z^2 - 75845 z^(5/2) + 1821204 z^3 + 841302 z^(7/2) + 6234690 z^4 + 3447675 z^(9/2) + 44592240 z^5 + 85164160 z^(11/2) - 150266880 z^6 - 228096000 z^(13/2) + 142786560 z^7 + 199557120 z^(15/2) - 44040192 z^8 - 58720256 z^(17/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 243597312 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 370882560 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 235431040 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 41144565 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5393388 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1897049 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1055670 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 905520 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Rule Form

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SuperscriptBox["z", RowBox[List["17", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["z"]]], ")"]], "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]], "5"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], "2"]]], RowBox[List["9313920", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]

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

2007-05-02

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]