Generalized hypergeometric function 1F2: Specific values (formula 07.22.03.a95w)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁},{b₁,b₂},z]

Specific values

For rational parameters with larger denominators and fixed z

For fixed z and a₁=-11/4, b_1`>=-11/2

For fixed z and a₁=-11/4, b_1`=7/2

http://functions.wolfram.com/07.22.03.a95w.01

Input Form

HypergeometricPFQ[{-(11/4)}, {7/2, 19/4}, z] == -(((2 Sqrt[z] (3714726015 - 30875644800 z - 44606225664 z^2 - 122475085824 z^3 + 32444841984 z^4 - 1577058304 z^5 + 16777216 z^6) BesselI[-(1/4), Sqrt[z]]^2 + (-11144178045 + 75645329760 z + 82383485184 z^2 + 104650850304 z^3 - 31487361024 z^4 + 1566572544 z^5 - 16777216 z^6) BesselI[-(1/4), Sqrt[z]] BesselI[3/4, Sqrt[z]] + 2 Sqrt[z] (11144178045 + 54014216256 z + 65141556480 z^2 + 114949029888 z^3 - 32057131008 z^4 + 1572864000 z^5 - 16777216 z^6) BesselI[3/4, Sqrt[z]]^2) Gamma[3/4]^2)/ (142343405568 Sqrt[2] z^(13/4)))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["11", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "2"], ",", FractionBox["19", "4"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["3714726015", "-", RowBox[List["30875644800", " ", "z"]], "-", RowBox[List["44606225664", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["122475085824", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["32444841984", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["1577058304", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["16777216", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "11144178045"]], "+", RowBox[List["75645329760", " ", "z"]], "+", RowBox[List["82383485184", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["104650850304", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["31487361024", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1566572544", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["16777216", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["11144178045", "+", RowBox[List["54014216256", " ", "z"]], "+", RowBox[List["65141556480", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["114949029888", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["32057131008", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1572864000", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["16777216", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselI", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], ")"]], "/", RowBox[List["(", RowBox[List["142343405568", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["13", "/", "4"]]]]], ")"]]]]]]]]]]

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> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 11 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 19 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["11", "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[FractionBox["19", "4"], 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> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 16777216 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1577058304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 32444841984 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 122475085824 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 44606225664 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 30875644800 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 3714726015 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 16777216 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1566572544 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 31487361024 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 104650850304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 82383485184 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 75645329760 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 11144178045 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 16777216 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1572864000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 32057131008 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 114949029888 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 65141556480 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 54014216256 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 11144178045 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msub> <mi> I </mi> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 142343405568 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 13 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </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'> 11 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 7 <sep /> 2 </cn> <cn type='rational'> 19 <sep /> 4 </cn> </list> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 16777216 </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'> 1577058304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 32444841984 </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'> 122475085824 </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'> 44606225664 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30875644800 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3714726015 </cn> </apply> <apply> <power /> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -16777216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1566572544 </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'> 31487361024 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 104650850304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 82383485184 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 75645329760 </cn> <ci> z </ci> </apply> <cn type='integer'> -11144178045 </cn> </apply> <apply> <ci> BesselI </ci> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -16777216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1572864000 </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'> 32057131008 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 114949029888 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 65141556480 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 54014216256 </cn> <ci> z </ci> </apply> <cn type='integer'> 11144178045 </cn> </apply> <apply> <power /> <apply> <ci> BesselI </ci> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 142343405568 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </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", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "2"], ",", FractionBox["19", "4"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["3714726015", "-", RowBox[List["30875644800", " ", "z"]], "-", RowBox[List["44606225664", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["122475085824", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["32444841984", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["1577058304", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["16777216", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "11144178045"]], "+", RowBox[List["75645329760", " ", "z"]], "+", RowBox[List["82383485184", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["104650850304", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["31487361024", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1566572544", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["16777216", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["11144178045", "+", RowBox[List["54014216256", " ", "z"]], "+", RowBox[List["65141556480", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["114949029888", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["32057131008", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1572864000", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["16777216", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselI", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["142343405568", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["13", "/", "4"]]]]]]]]]]]]

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,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},{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]