Generalized hypergeometric function 2F2: Specific values (formula 07.25.03.anry)

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=11/2, a₂>=11/2

For fixed z and a₁=11/2, a₂=6, b₁>=-11/2

For fixed z and a₁=11/2, a₂=6, b₁=-11/2

http://functions.wolfram.com/07.25.03.anry.01

Input Form

HypergeometricPFQ[{11/2, 6}, {-(11/2), 7/2}, -z] == (1/9823275) (9823275 + 16839900 z + 18918900 z^2 + 19656000 z^3 + 23133600 z^4 + 39070080 z^5 + 176964480 z^6 - 219639420 z^7 + 89182500 z^8 - 17289930 z^9 + 1786728 z^10 - 100112 z^11 + 2848 z^12 - 32 z^13) + (1/9823275) ((Sqrt[Pi] z^(13/2) (-257414850 + 257414850 z - 97046775 z^2 + 18136650 z^3 - 1835400 z^4 + 101520 z^5 - 2864 z^6 + 32 z^7) Erfi[Sqrt[z]])/E^z)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["11", "2"], ",", "6"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", FractionBox["7", "2"]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "9823275"], RowBox[List["(", RowBox[List["9823275", "+", RowBox[List["16839900", " ", "z"]], "+", RowBox[List["18918900", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["19656000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["23133600", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["39070080", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["176964480", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["219639420", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["89182500", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["17289930", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["1786728", " ", SuperscriptBox["z", "10"]]], "-", RowBox[List["100112", " ", SuperscriptBox["z", "11"]]], "+", RowBox[List["2848", " ", SuperscriptBox["z", "12"]]], "-", RowBox[List["32", " ", SuperscriptBox["z", "13"]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "9823275"], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["13", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "257414850"]], "+", RowBox[List["257414850", " ", "z"]], "-", RowBox[List["97046775", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["18136650", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1835400", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["101520", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["2864", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["32", " ", 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> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 6 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> </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[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[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["11", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["7", "2"], 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> <mrow> <mfrac> <mn> 1 </mn> <mn> 9823275 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2864 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 101520 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1835400 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18136650 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 97046775 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 257414850 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 257414850 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 13 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 9823275 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 32 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 13 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2848 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 12 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 100112 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 11 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1786728 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 17289930 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 89182500 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 219639420 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 176964480 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 39070080 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 23133600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 19656000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18918900 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16839900 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 9823275 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 11 <sep /> 2 </cn> <cn type='integer'> 6 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 11 <sep /> 2 </cn> </apply> <cn type='rational'> 7 <sep /> 2 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 9823275 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </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'> 2864 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 101520 </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'> 1835400 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18136650 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 97046775 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 257414850 </cn> <ci> z </ci> </apply> <cn type='integer'> -257414850 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 9823275 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 13 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2848 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 12 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 100112 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 11 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1786728 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 17289930 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 89182500 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 219639420 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 176964480 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 39070080 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 23133600 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 19656000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18918900 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16839900 </cn> <ci> z </ci> </apply> <cn type='integer'> 9823275 </cn> </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[FractionBox["11", "2"], ",", "6"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", FractionBox["7", "2"]]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["9823275", "+", RowBox[List["16839900", " ", "z"]], "+", RowBox[List["18918900", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["19656000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["23133600", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["39070080", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["176964480", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["219639420", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["89182500", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["17289930", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["1786728", " ", SuperscriptBox["z", "10"]]], "-", RowBox[List["100112", " ", SuperscriptBox["z", "11"]]], "+", RowBox[List["2848", " ", SuperscriptBox["z", "12"]]], "-", RowBox[List["32", " ", SuperscriptBox["z", "13"]]]]], "9823275"], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["13", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "257414850"]], "+", RowBox[List["257414850", " ", "z"]], "-", RowBox[List["97046775", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["18136650", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1835400", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["101520", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["2864", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["32", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]]]], "9823275"]]]]]]]

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]