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http://functions.wolfram.com/07.27.26.0012.01
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HypergeometricPFQ[{a, a + 1/3, a + 2/3}, {b, 3 a - b + 3/2},
(27 z^2)/(4 + z)^3]/(4 + z)^(3 a) ==
((Gamma[b] Gamma[3 a - b + 3/2])/(2 Pi Gamma[3 a]))
MeijerG[{{1 - 3 a, 3/2 - b, b - 3 a}, {}},
{{0}, {2 - 2 b, -1 + 2 b - 6 a}}, z] /;
!IntervalMemberQ[Interval[{-1, 0}], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "3"]], " ", "a"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", RowBox[List["a", "+", FractionBox["1", "3"]]], ",", RowBox[List["a", "+", FractionBox["2", "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", RowBox[List[RowBox[List["3", " ", "a"]], "-", "b", "+", FractionBox["3", "2"]]]]], "}"]], ",", FractionBox[RowBox[List["27", " ", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z"]], ")"]], "3"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["3", " ", "a"]], "-", "b", "+", FractionBox["3", "2"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["3", " ", "a"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["3", " ", "a"]]]], ",", RowBox[List[FractionBox["3", "2"], "-", "b"]], ",", RowBox[List["b", "-", RowBox[List["3", " ", "a"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]], "-", RowBox[List["6", " ", "a"]]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["2", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["3", " ", "a"]], "-", "b", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["27", " ", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "4"]], ")"]], "3"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - 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</mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["1", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", RowBox[List["3", " ", "a"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "2"], "-", "b"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", RowBox[List["3", " ", "a"]]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["6", "a"]], "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - 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</mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z_"]], ")"]], RowBox[List[RowBox[List["-", "3"]], " ", "a_"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", RowBox[List["a_", "+", FractionBox["1", "3"]]], ",", RowBox[List["a_", "+", FractionBox["2", "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", RowBox[List[RowBox[List["3", " ", "a_"]], "-", "b_", "+", FractionBox["3", "2"]]]]], "}"]], ",", FractionBox[RowBox[List["27", " ", SuperscriptBox["z_", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z_"]], ")"]], "3"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["3", " ", "a"]], "-", "b", "+", FractionBox["3", "2"]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["3", " ", "a"]]]], ",", RowBox[List[FractionBox["3", "2"], "-", "b"]], ",", RowBox[List["b", "-", RowBox[List["3", " ", "a"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]], "-", RowBox[List["6", " ", "a"]]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["3", " ", "a"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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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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