Generalized hypergeometric function 2F2: Specific values (formula 07.25.03.4932)

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=7/2

http://functions.wolfram.com/07.25.03.4932.01

Input Form

HypergeometricPFQ[{-(9/2), 4}, {7/2, 6}, z] == -((120 (12 + 19 z))/(46189 z^5)) + (1/(496624128 z^5)) (E^z (15482880 + 9031680 z - 16773120 z^2 - 33971805 z^3 - 61638780 z^4 + 253084356 z^5 - 131191584 z^6 + 24191376 z^7 - 1802944 z^8 + 45760 z^9)) + (1/(993248256 z^(5/2))) (Sqrt[Pi] (43648605 + 87297210 z + 290990700 z^2 - 617253000 z^3 + 284886000 z^4 - 50139936 z^5 + 3651648 z^6 - 91520 z^7) Erfi[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["7", "2"], ",", "6"]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["120", " ", RowBox[List["(", RowBox[List["12", "+", RowBox[List["19", " ", "z"]]]], ")"]]]], RowBox[List["46189", " ", SuperscriptBox["z", "5"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["496624128", " ", SuperscriptBox["z", "5"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List["15482880", "+", RowBox[List["9031680", " ", "z"]], "-", RowBox[List["16773120", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["33971805", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["61638780", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["253084356", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["131191584", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["24191376", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["1802944", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["45760", " ", SuperscriptBox["z", "9"]]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["993248256", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["43648605", "+", RowBox[List["87297210", " ", "z"]], "+", RowBox[List["290990700", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["617253000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["284886000", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["50139936", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["3651648", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["91520", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["Erfi", "[", 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> 7 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 6 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </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["7", "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["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> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 120 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 19 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 12 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 46189 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 496624128 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 45760 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1802944 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 24191376 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 131191584 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 253084356 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 61638780 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 33971805 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 16773120 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9031680 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 15482880 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 993248256 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </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> <mrow> <mo> - </mo> <mn> 91520 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3651648 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 50139936 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 284886000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 617253000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 290990700 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 87297210 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 43648605 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> erfi </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'> 7 <sep /> 2 </cn> <cn type='integer'> 6 </cn> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 19 </cn> <ci> z </ci> </apply> <cn type='integer'> 12 </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> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 496624128 </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 /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 45760 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1802944 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 24191376 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 131191584 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 253084356 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 61638780 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 33971805 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16773120 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9031680 </cn> <ci> z </ci> </apply> <cn type='integer'> 15482880 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 993248256 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <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'> -91520 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3651648 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 50139936 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 284886000 </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'> 617253000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 290990700 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 87297210 </cn> <ci> z </ci> </apply> <cn type='integer'> 43648605 </cn> </apply> <apply> <ci> Erfi </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["7", "2"], ",", "6"]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["120", " ", RowBox[List["(", RowBox[List["12", "+", RowBox[List["19", " ", "z"]]]], ")"]]]], RowBox[List["46189", " ", SuperscriptBox["z", "5"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List["15482880", "+", RowBox[List["9031680", " ", "z"]], "-", RowBox[List["16773120", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["33971805", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["61638780", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["253084356", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["131191584", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["24191376", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["1802944", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["45760", " ", SuperscriptBox["z", "9"]]]]], ")"]]]], RowBox[List["496624128", " ", SuperscriptBox["z", "5"]]]], "+", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["43648605", "+", RowBox[List["87297210", " ", "z"]], "+", RowBox[List["290990700", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["617253000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["284886000", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["50139936", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["3651648", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["91520", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]]]], RowBox[List["993248256", " ", SuperscriptBox["z", RowBox[List["5", "/", "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]