|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.31.03.0155.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HypergeometricPFQ[{a, a + 1/3, a + 2/3}, {d, d/2, (d + 1)/2, (3/2) a,
(3 a + 1)/2, 1 + 3 a - d, (3 a - d + 1)/2, (3 a - d)/2 + 1}, z] ==
HypergeometricPFQ[{}, {d, 3 a - d + 1}, 8 Sqrt[-(z/27)]]
HypergeometricPFQ[{}, {d, 3 a - d + 1}, -8 Sqrt[-(z/27)]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[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["d", ",", FractionBox["d", "2"], ",", FractionBox[RowBox[List["d", "+", "1"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "a"]], ",", FractionBox[RowBox[List[RowBox[List["3", "a"]], "+", "1"]], "2"], ",", RowBox[List["1", "+", RowBox[List["3", "a"]], "-", "d"]], ",", FractionBox[RowBox[List[RowBox[List["3", "a"]], "-", "d", "+", "1"]], "2"], ",", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", "a"]], "-", "d"]], "2"], "+", "1"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List[RowBox[List["3", "a"]], "-", "d", "+", "1"]]]], "}"]], ",", RowBox[List["8", SqrtBox[RowBox[List["-", FractionBox["z", "27"]]]]]]]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List[RowBox[List["3", "a"]], "-", "d", "+", "1"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "8"]], SqrtBox[RowBox[List["-", FractionBox["z", "27"]]]]]]]], "]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 8 </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> d </mi> <mo> , </mo> <mfrac> <mi> d </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> d </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["8", 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["d", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["d", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["d", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", " ", "a"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", "a"]], "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["3", " ", "a"]], "-", "d", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "a"]], "-", "d", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "a"]], "-", "d"]], "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mrow> <mi> d </mi> <mo> , </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 27 </mn> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["d", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["3", " ", "a"]], "-", "d", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["8", " ", SqrtBox[RowBox[List["-", FractionBox["z", "27"]]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mrow> <mi> d </mi> <mo> , </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 27 </mn> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["d", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["3", " ", "a"]], "-", "d", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", "8"]], " ", SqrtBox[RowBox[List["-", FractionBox["z", "27"]]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <ci> a </ci> <apply> <plus /> <ci> a </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </list> <list> <ci> d </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> HypergeometricPFQ </ci> <list /> <list> <ci> d </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 27 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list /> <list> <ci> d </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -8 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 27 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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["d_", ",", FractionBox["d_", "2"], ",", FractionBox[RowBox[List["d_", "+", "1"]], "2"], ",", FractionBox[RowBox[List["3", " ", "a_"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a_"]], "+", "1"]], ")"]]]], ",", RowBox[List["1", "+", RowBox[List["3", " ", "a_"]], "-", "d_"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a_"]], "-", "d_", "+", "1"]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a_"]], "-", "d_"]], ")"]]]], "+", "1"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List[RowBox[List["3", " ", "a"]], "-", "d", "+", "1"]]]], "}"]], ",", RowBox[List["8", " ", SqrtBox[RowBox[List["-", FractionBox["z", "27"]]]]]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["d", ",", RowBox[List[RowBox[List["3", " ", "a"]], "-", "d", "+", "1"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "8"]], " ", SqrtBox[RowBox[List["-", FractionBox["z", "27"]]]]]]]], "]"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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},{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] | |
|
|
|