|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Hypergeometric Functions
HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z]
Specific values
Values at z=1
For fixed a1, a2, a3, a4, b1, b2, b3
|
|
|
|
|
|
|
http://functions.wolfram.com/07.28.03.0002.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3],
Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]},
1] == ((Gamma[Subscript[\[Psi], 3]] Product[Gamma[Subscript[b, k]],
{k, 1, 3}])/Product[Gamma[Subscript[a, k]], {k, 3, 4}])
Sum[(Pochhammer[Subscript[\[Psi], 3], k]
HypergeometricPFQExpansionCoefficient[{Subscript[a, 1],
Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]},
{Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, k])/
(Gamma[Subscript[\[Psi], 3] + Subscript[a, 1] + k]
Gamma[Subscript[\[Psi], 3] + Subscript[a, 2] + k]),
{k, 0, Infinity}] /; Subscript[\[Psi], 3] ==
Sum[Subscript[b, j], {j, 1, 3}] - Sum[Subscript[a, j], {j, 1, 4}] &&
Re[Subscript[\[Psi], 3]] > 0 && Re[Subscript[a, 3]] > 0 &&
Re[Subscript[a, 4]] > 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["\[Psi]", "3"], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "3"]], "4"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], ",", "k"]], "]"]], RowBox[List["HypergeometricPFQExpansionCoefficient", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["a", "1"], "+", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["a", "2"], "+", "k"]], "]"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", RowBox[List[RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "3"], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["b", "j"]]], "-", RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "4"], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["a", "j"]]]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Psi]", "3"], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "3"], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "4"], "]"]], ">", "0"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mn> 4 </mn> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msub> <mrow> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℰ </mi> <mi> k </mi> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "3"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "3", ")"]]], "(", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]]]], ")"]]]], RowBox[List[RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo movablelimits='false'> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo movablelimits='false'> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mn> 4 </mn> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msub> <mrow> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℰ </mi> <mi> k </mi> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "3"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "3", ")"]]], "(", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]]]], ")"]]]], RowBox[List[RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo movablelimits='false'> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo movablelimits='false'> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", SubscriptBox["a_", "3"], ",", SubscriptBox["a_", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["\[Psi]", "3"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], ",", "k"]], "]"]], " ", RowBox[List["HypergeometricPFQExpansionCoefficient", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", SubscriptBox["aa", "3"], ",", SubscriptBox["aa", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"], ",", SubscriptBox["bb", "3"]]], "}"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["aa", "1"], "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", SubscriptBox["aa", "2"], "+", "k"]], "]"]]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "3"]], "4"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "3"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "4"], SubscriptBox["a", "j"]]]]]]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Psi]", "3"], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "3"], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "4"], "]"]], ">", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
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,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] | |
|
|
|