Generalized hypergeometric function 5F4: Specific values (formula 07.29.03.0026)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]

Specific values

Values at other z

Values at z==4

http://functions.wolfram.com/07.29.03.0026.01

Input Form

HypergeometricPFQ[{-n, b, c, (2 b)/3 + 1, b - c + 1/2}, {(2 b)/3, b + n/2 + 1, 2 c, 2 b - 2 c + 1}, 4] == ((1 + (-1)^n) n! Pochhammer[b + 1, n/2])/(2^(n + 1) (n/2)! Pochhammer[c + 1/2, n/2] Pochhammer[b - c + 1, n/2]) /; Element[n, Integers] && n >= 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", "b", ",", "c", ",", RowBox[List[FractionBox[RowBox[List["2", "b"]], "3"], "+", "1"]], ",", RowBox[List["b", "-", "c", "+", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "b"]], "3"], ",", RowBox[List["b", "+", FractionBox["n", "2"], "+", "1"]], ",", RowBox[List["2", "c"]], ",", RowBox[List[RowBox[List["2", "b"]], "-", RowBox[List["2", "c"]], "+", "1"]]]], "}"]], ",", "4"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], RowBox[List["n", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "+", "1"]], ",", FractionBox["n", "2"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["(", FractionBox["n", "2"], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["c", "+", FractionBox["1", "2"]]], ",", FractionBox["n", "2"]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "-", "c", "+", "1"]], ",", FractionBox["n", "2"]]], "]"]]]], ")"]]]]]], "/;", 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> 5 </mn> </msub> <msub> <mi> F </mi> <mn> 4 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["5", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["c", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "b"]], "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["2", " ", "b"]], "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["n", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", " ", "c"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["2", " ", "c"]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["4", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "1"]], ")"]], FractionBox["n", "2"]], Pochhammer] </annotation> </semantics> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "+", FractionBox["1", "2"]]], ")"]], FractionBox["n", "2"]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "+", "1"]], ")"]], FractionBox["n", "2"]], Pochhammer] </annotation> </semantics> </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> <ci> n </ci> </apply> <ci> b </ci> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <list> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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["-", "n_"]], ",", "b_", ",", "c_", ",", RowBox[List[FractionBox[RowBox[List["2", " ", "b_"]], "3"], "+", "1"]], ",", RowBox[List["b_", "-", "c_", "+", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", " ", "b_"]], "3"], ",", RowBox[List["b_", "+", FractionBox["n_", "2"], "+", "1"]], ",", RowBox[List["2", " ", "c_"]], ",", RowBox[List[RowBox[List["2", " ", "b_"]], "-", RowBox[List["2", " ", "c_"]], "+", "1"]]]], "}"]], ",", "4"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "+", "1"]], ",", FractionBox["n", "2"]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["n", "+", "1"]]], " ", RowBox[List[FractionBox["n", "2"], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["c", "+", FractionBox["1", "2"]]], ",", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "-", "c", "+", "1"]], ",", FractionBox["n", "2"]]], "]"]]]]], "/;", 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},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]