Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ai5k)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]

Specific values

For integer and half-integer parameters and fixed z

For fixed z and a₁=-3/2, a₂>=-3/2

For fixed z and a₁=-3/2, a₂=3, a₃>=3

For fixed z and a₁=-3/2, a₂=3, a₃=4

For fixed z and a₁=-3/2, a₂=3, a₃=4, b₁=3/2

http://functions.wolfram.com/07.27.03.ai5k.01

Input Form

HypergeometricPFQ[{-(3/2), 3, 4}, {3/2, 3/2}, -z] == (1/(32768 Sqrt[z] (1 + z)^2)) (360 Pi^2 + 26120 Sqrt[z] + 13320 Pi^2 z + 178652 z^(3/2) + 58635 Pi^2 z^2 + 285600 z^(5/2) + 78750 Pi^2 z^3 + 132300 z^(7/2) + 33075 Pi^2 z^4) + (1/(8192 (1 + z)^(11/2))) ((-7832 - 131912 z - 652821 z^2 - 1573816 z^3 - 2112127 z^4 - 1619682 z^5 - 666555 z^6 - 114555 z^7) Log[1 + Sqrt[z] - Sqrt[1 + z]]) + (1/(8192 (1 + z)^(11/2))) ((7832 + 131912 z + 652821 z^2 + 1573816 z^3 + 2112127 z^4 + 1619682 z^5 + 666555 z^6 + 114555 z^7) Log[1 - Sqrt[z] + Sqrt[1 + z]]) + (3 (434 + 8295 z + 26825 z^2 + 29925 z^3 + 11025 z^4) Log[Sqrt[z] + Sqrt[1 + z]])/(8192 Sqrt[z] (1 + z)^(5/2)) + (1/(8192 (1 + z)^(11/2))) ((7832 + 131912 z + 652821 z^2 + 1573816 z^3 + 2112127 z^4 + 1619682 z^5 + 666555 z^6 + 114555 z^7) Log[(-1 + Sqrt[z] + Sqrt[1 + z])/(1 + Sqrt[z] + Sqrt[1 + z])]) + (45 (8 + 280 z + 735 z^2) Log[Sqrt[z] + Sqrt[1 + z]] Log[(-1 + Sqrt[z] + Sqrt[1 + z])/(1 + Sqrt[z] + Sqrt[1 + z])])/ (8192 Sqrt[z]) + (45 (8 + 280 z + 735 z^2) PolyLog[2, -(1/(Sqrt[z] + Sqrt[1 + z]))])/(8192 Sqrt[z]) - (45 (8 + 280 z + 735 z^2) PolyLog[2, 1/(Sqrt[z] + Sqrt[1 + z])])/ (8192 Sqrt[z])

Standard Form

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MathML Form

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<mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 131912 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 7832 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> + </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> + </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 735 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 280 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> 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"z"]]]]]]]]]], "]"]]]], RowBox[List["8192", " ", SqrtBox["z"]]]], "-", FractionBox[RowBox[List["45", " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["280", " ", "z"]], "+", RowBox[List["735", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox["1", RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]], "]"]]]], RowBox[List["8192", " ", SqrtBox["z"]]]]]]]]]]

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},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},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]