Generalized hypergeometric function 4F3: Specific values (formula 07.28.03.0186)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]

Specific values

Values at fixed points

For z==-27 and integer parameters

http://functions.wolfram.com/07.28.03.0186.01

Input Form

HypergeometricPFQ[{-(n/2), (1 - n)/2, 1/3 - n, (22 - 9 n)/21}, {5/6, 4/3, (1 - 9 n)/21}, -27] == (-8)^n/(1 - 9 n) /; Element[n, Integers] && n >= 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List[FractionBox["1", "3"], "-", "n"]], ",", FractionBox[RowBox[List["22", "-", RowBox[List["9", "n"]]]], "21"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", FractionBox["4", "3"], ",", FractionBox[RowBox[List["1", "-", RowBox[List["9", "n"]]]], "21"]]], "}"]], ",", RowBox[List["-", "27"]]]], "]"]], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "n"], RowBox[List["1", "-", RowBox[List["9", "n"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

MathML Form

<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> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 22 </mn> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mn> 21 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mn> 21 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 27 </mn> </mrow> </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[RowBox[List["-", FractionBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "n"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "3"], "-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["22", "-", RowBox[List["9", "n"]]]], "21"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["4", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", RowBox[List["9", "n"]]]], "21"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "27"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 22 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 5 <sep /> 6 </cn> <cn type='rational'> 4 <sep /> 3 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <cn type='integer'> -27 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -8 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n_", "2"]]], ",", FractionBox[RowBox[List["1", "-", "n_"]], "2"], ",", RowBox[List[FractionBox["1", "3"], "-", "n_"]], ",", RowBox[List[FractionBox["1", "21"], " ", RowBox[List["(", RowBox[List["22", "-", RowBox[List["9", " ", "n_"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", FractionBox["4", "3"], ",", RowBox[List[FractionBox["1", "21"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["9", " ", "n_"]]]], ")"]]]]]], "}"]], ",", RowBox[List["-", "27"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "n"], RowBox[List["1", "-", RowBox[List["9", " ", "n"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29

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,...,a_p},{b₁,...,b_q},z]