Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.am6r)

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=-1/2, a₂>=-1/2

For fixed z and a₁=-1/2, a₂=4, a₃>=4

For fixed z and a₁=-1/2, a₂=4, a₃=4

For fixed z and a₁=-1/2, a₂=4, a₃=4, b₁=1/2

http://functions.wolfram.com/07.27.03.am6r.01

Input Form

HypergeometricPFQ[{-(1/2), 4, 4}, {1/2, 5/2}, -z] == (1/(24576 z (1 + z)^4)) (-72 + 22500 z + 11025 Pi^2 z^(3/2) + 133536 z^2 + 44100 Pi^2 z^(5/2) + 230104 z^3 + 66150 Pi^2 z^(7/2) + 166600 z^4 + 44100 Pi^2 z^(9/2) + 44100 z^5 + 11025 Pi^2 z^(11/2)) + (1/(102400 (1 + z)^(15/2))) ((-159744 - 2243130 z - 11122260 z^2 - 27197160 z^3 - 39404640 z^4 - 35416564 z^5 - 19491900 z^6 - 6042375 z^7 - 810775 z^8) Log[1 + Sqrt[z] - Sqrt[1 + z]]) + (1/(102400 (1 + z)^(15/2))) ((159744 + 2243130 z + 11122260 z^2 + 27197160 z^3 + 39404640 z^4 + 35416564 z^5 + 19491900 z^6 + 6042375 z^7 + 810775 z^8) Log[1 - Sqrt[z] + Sqrt[1 + z]]) + ((6 + 177 z + 6611 z^2 + 20370 z^3 + 26460 z^4 + 15925 z^5 + 3675 z^6) Log[Sqrt[z] + Sqrt[1 + z]])/(2048 z^(3/2) (1 + z)^(9/2)) + (1/(102400 (1 + z)^(15/2))) ((159744 + 2243130 z + 11122260 z^2 + 27197160 z^3 + 39404640 z^4 + 35416564 z^5 + 19491900 z^6 + 6042375 z^7 + 810775 z^8) Log[(-1 + Sqrt[z] + Sqrt[1 + z])/ (1 + Sqrt[z] + Sqrt[1 + z])]) + (3675 Sqrt[z] Log[Sqrt[z] + Sqrt[1 + z]] Log[(-1 + Sqrt[z] + Sqrt[1 + z])/(1 + Sqrt[z] + Sqrt[1 + z])])/2048 + (3675 Sqrt[z] PolyLog[2, -(1/(Sqrt[z] + Sqrt[1 + z]))])/2048 - (3675 Sqrt[z] PolyLog[2, 1/(Sqrt[z] + Sqrt[1 + z])])/2048

Standard Form

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MathML Form

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<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> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 102400 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 15 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 810775 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6042375 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 19491900 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 35416564 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 39404640 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 27197160 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11122260 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2243130 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 159744 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3675 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 15925 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 26460 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> 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SqrtBox[RowBox[List["1", "+", "z"]]]]]], "]"]]]], "2048"], "+", FractionBox[RowBox[List["3675", " ", SqrtBox["z"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox["1", RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]]]], "]"]]]], "2048"], "-", FractionBox[RowBox[List["3675", " ", SqrtBox["z"], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox["1", RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]], "]"]]]], "2048"]]]]]]]

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]