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

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=5/2, a₂>=5/2

For fixed z and a₁=5/2, a₂=6, b₁>=-11/2

For fixed z and a₁=5/2, a₂=6, b₁=-11/2

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

Input Form

HypergeometricPFQ[{5/2, 6}, {-(11/2), 11/2}, z] == (1/(7920 z^4)) (-190512000 + 79833600 z - 25401600 z^2 + 6471360 z^3 - 1353600 z^4 + 237600 z^5 - 35280 z^6 + 4200 z^7 - 153 z^8 - 336 z^9 + 1040 z^10 + 288 z^11 + 16 z^12) + (1/(15840 z^(9/2))) (E^z Sqrt[Pi] (190512000 - 206841600 z + 112492800 z^2 - 40824000 z^3 + 11113200 z^4 - 2419200 z^5 + 438480 z^6 - 68040 z^7 + 9225 z^8 - 1110 z^9 + 120 z^10 + 2352 z^11 + 592 z^12 + 32 z^13) Erf[Sqrt[z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "2"], ",", "6"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", FractionBox["11", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["7920", " ", SuperscriptBox["z", "4"]]]], RowBox[List["(", RowBox[List[RowBox[List["-", "190512000"]], "+", RowBox[List["79833600", " ", "z"]], "-", RowBox[List["25401600", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["6471360", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1353600", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["237600", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["35280", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["4200", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["153", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["336", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["1040", " ", SuperscriptBox["z", "10"]]], "+", RowBox[List["288", " ", SuperscriptBox["z", "11"]]], "+", RowBox[List["16", " ", SuperscriptBox["z", "12"]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["15840", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["190512000", "-", RowBox[List["206841600", " ", "z"]], "+", RowBox[List["112492800", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["40824000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["11113200", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["2419200", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["438480", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["68040", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["9225", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["1110", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["120", " ", SuperscriptBox["z", "10"]]], "+", RowBox[List["2352", " ", SuperscriptBox["z", "11"]]], "+", RowBox[List["592", " ", SuperscriptBox["z", "12"]]], "+", RowBox[List["32", " ", SuperscriptBox["z", "13"]]]]], ")"]], " ", 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> <mfrac> <mn> 5 </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> 11 </mn> <mn> 2 </mn> </mfrac> </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[FractionBox["5", "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["11", "2"], 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> <mfrac> <mn> 1 </mn> <mrow> <mn> 7920 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 12 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 288 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 11 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1040 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 336 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 153 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 35280 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 237600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1353600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6471360 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 25401600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 79833600 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 190512000 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 15840 </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> <msup> <mi> ⅇ </mi> <mi> z </mi> </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> 13 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 592 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 12 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2352 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 11 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 120 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1110 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9225 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 68040 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 438480 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2419200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11113200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 40824000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 112492800 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 206841600 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 190512000 </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> <cn type='rational'> 5 <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'> 11 <sep /> 2 </cn> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 7920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 12 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 288 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 11 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1040 </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'> 336 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 153 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4200 </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'> 35280 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 237600 </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'> 1353600 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6471360 </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'> 25401600 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 79833600 </cn> <ci> z </ci> </apply> <cn type='integer'> -190512000 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 15840 </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 /> <exponentiale /> <ci> z </ci> </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'> 13 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 592 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 12 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2352 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 11 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 120 </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'> 1110 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9225 </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'> 68040 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 438480 </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'> 2419200 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11113200 </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'> 40824000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 112492800 </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'> 206841600 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 190512000 </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[FractionBox["5", "2"], ",", "6"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", FractionBox["11", "2"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "190512000"]], "+", RowBox[List["79833600", " ", "z"]], "-", RowBox[List["25401600", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["6471360", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1353600", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["237600", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["35280", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["4200", " ", SuperscriptBox["z", "7"]]], "-", RowBox[List["153", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["336", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["1040", " ", SuperscriptBox["z", "10"]]], "+", RowBox[List["288", " ", SuperscriptBox["z", "11"]]], "+", RowBox[List["16", " ", SuperscriptBox["z", "12"]]]]], RowBox[List["7920", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["190512000", "-", RowBox[List["206841600", " ", "z"]], "+", RowBox[List["112492800", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["40824000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["11113200", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["2419200", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["438480", " ", SuperscriptBox["z", "6"]]], "-", RowBox[List["68040", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["9225", " ", SuperscriptBox["z", "8"]]], "-", RowBox[List["1110", " ", SuperscriptBox["z", "9"]]], "+", RowBox[List["120", " ", SuperscriptBox["z", "10"]]], "+", RowBox[List["2352", " ", SuperscriptBox["z", "11"]]], "+", RowBox[List["592", " ", SuperscriptBox["z", "12"]]], "+", RowBox[List["32", " ", SuperscriptBox["z", "13"]]]]], ")"]], " ", RowBox[List["Erf", "[", SqrtBox["z"], "]"]]]], RowBox[List["15840", " ", 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]