Generalized hypergeometric function 2F2: Specific values (formula 07.25.03.4950)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]

Specific values

For integer and half-integer parameters and fixed z

For fixed z and a₁=-9/2, a₂>=-9/2

For fixed z and a₁=-9/2, a₂=4, b₁>=-11/2

For fixed z and a₁=-9/2, a₂=4, b₁=11/2

http://functions.wolfram.com/07.25.03.4950.01

Input Form

HypergeometricPFQ[{-(9/2), 4}, {11/2, 6}, -z] == (7560 (4 + 19 z))/(46189 z^5) + (1/(9081126912 z^5)) ((-5945425920 - 11270681415 z + 3705397920 z^2 - 2191303800 z^3 + 2029255200 z^4 + 3873021504 z^5 + 1288197888 z^6 + 167350656 z^7 + 9298432 z^8 + 183040 z^9)/E^z) + (1/(18162253824 z^(9/2))) (Sqrt[Pi] (-22915517625 - 19641872250 z + 6285399120 z^2 - 4190266080 z^3 + 6983776800 z^4 + 8888443200 z^5 + 2734905600 z^6 + 343816704 z^7 + 18779904 z^8 + 366080 z^9) Erf[Sqrt[z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["9", "2"]]], ",", "4"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["11", "2"], ",", "6"]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["7560", " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["19", " ", "z"]]]], ")"]]]], RowBox[List["46189", " ", SuperscriptBox["z", "5"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["9081126912", " ", SuperscriptBox["z", "5"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5945425920"]], "-", RowBox[List["11270681415", " ", "z"]], "+", RowBox[List["3705397920", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2191303800", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2029255200", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3873021504", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["1288197888", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["167350656", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["9298432", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["183040", " ", SuperscriptBox["z", "9"]]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["18162253824", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "22915517625"]], "-", RowBox[List["19641872250", " ", "z"]], "+", RowBox[List["6285399120", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["4190266080", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["6983776800", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["8888443200", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["2734905600", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["343816704", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["18779904", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["366080", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["Erf", "[", SqrtBox["z"], "]"]]]], ")"]]]]]]]]]]

MathML Form

<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> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mn> 4 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 6 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["9", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["4", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["11", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["6", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", "z"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> <mo>  </mo> <mrow> <mfrac> <mrow> <mn> 7560 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 19 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 46189 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 9081126912 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 183040 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9298432 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 167350656 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1288197888 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3873021504 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2029255200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2191303800 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3705397920 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 11270681415 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 5945425920 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 18162253824 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 9 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 366080 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18779904 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 343816704 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2734905600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8888443200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6983776800 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4190266080 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6285399120 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 19641872250 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 22915517625 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 9 <sep /> 2 </cn> </apply> <cn type='integer'> 4 </cn> </list> <list> <cn type='rational'> 11 <sep /> 2 </cn> <cn type='integer'> 6 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 7560 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 19 </cn> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 46189 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 9081126912 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 183040 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9298432 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 167350656 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1288197888 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3873021504 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2029255200 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2191303800 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3705397920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11270681415 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -5945425920 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 18162253824 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 366080 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18779904 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 343816704 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2734905600 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8888443200 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6983776800 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4190266080 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6285399120 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19641872250 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -22915517625 </cn> </apply> <apply> <ci> Erf </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </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["9", "2"]]], ",", "4"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["11", "2"], ",", "6"]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["7560", " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["19", " ", "z"]]]], ")"]]]], RowBox[List["46189", " ", SuperscriptBox["z", "5"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5945425920"]], "-", RowBox[List["11270681415", " ", "z"]], "+", RowBox[List["3705397920", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2191303800", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["2029255200", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3873021504", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["1288197888", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["167350656", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["9298432", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["183040", " ", SuperscriptBox["z", "9"]]]]], ")"]]]], RowBox[List["9081126912", " ", SuperscriptBox["z", "5"]]]], "+", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "22915517625"]], "-", RowBox[List["19641872250", " ", "z"]], "+", RowBox[List["6285399120", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["4190266080", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["6983776800", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["8888443200", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["2734905600", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["343816704", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["18779904", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["366080", " ", SuperscriptBox["z", "9"]]]]], ")"]], " ", RowBox[List["Erf", "[", SqrtBox["z"], "]"]]]], RowBox[List["18162253824", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]]]]]]]]

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

2007-05-02

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,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]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]