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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=3, a3>=3
For fixed z and a1=-1/2, a2=3, a3=3
For fixed z and a1=-1/2, a2=3, a3=3, b1=1/2
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http://functions.wolfram.com/07.27.03.alrd.01
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HypergeometricPFQ[{-(1/2), 3, 3}, {1/2, 7/2}, -z] ==
(5 (-96 - 104 z + 700 z^2 + 225 Pi^2 z^(5/2) + 900 z^3 + 225 Pi^2 z^(7/2)))/
(4096 z^2 (1 + z)) - (1/(6144 (1 + z)^(13/2)))
(5 (2304 + 88122 z + 198490 z^2 + 400070 z^3 + 456790 z^4 + 304929 z^5 +
111072 z^6 + 17115 z^7) Log[1 + Sqrt[z] - Sqrt[1 + z]]) +
(1/(6144 (1 + z)^(13/2))) (5 (2304 + 88122 z + 198490 z^2 + 400070 z^3 +
456790 z^4 + 304929 z^5 + 111072 z^6 + 17115 z^7)
Log[1 - Sqrt[z] + Sqrt[1 + z]]) +
(15 (8 + 14 z + 15 z^2 + 100 z^3 + 75 z^4) Log[Sqrt[z] + Sqrt[1 + z]])/
(1024 z^(5/2) (1 + z)^(3/2)) + (1/(6144 (1 + z)^(13/2)))
(5 (2304 + 88122 z + 198490 z^2 + 400070 z^3 + 456790 z^4 + 304929 z^5 +
111072 z^6 + 17115 z^7) Log[(-1 + Sqrt[z] + Sqrt[1 + z])/
(1 + Sqrt[z] + Sqrt[1 + z])]) +
(1125 Sqrt[z] Log[Sqrt[z] + Sqrt[1 + z]]
Log[(-1 + Sqrt[z] + Sqrt[1 + z])/(1 + Sqrt[z] + Sqrt[1 + z])])/1024 +
(1125 Sqrt[z] PolyLog[2, -(1/(Sqrt[z] + Sqrt[1 + z]))])/1024 -
(1125 Sqrt[z] PolyLog[2, 1/(Sqrt[z] + Sqrt[1 + z])])/1024
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<msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> + </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 1024 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 1125 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> + </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mn> 1024 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> 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<ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 104 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -96 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4096 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6144 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 17115 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 111072 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 304929 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 456790 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 400070 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 198490 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 88122 </cn> <ci> z </ci> </apply> <cn type='integer'> 2304 </cn> </apply> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> 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/> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 1024 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times 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Date Added to functions.wolfram.com (modification date)
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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,...,ap},{b1,...,bq},z] | |
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