Generalized hypergeometric function 1F2: Specific values (formula 07.22.03.a891)

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=-15/4, b_1`>=-11/2

For fixed z and a₁=-15/4, b_1`=7/2

http://functions.wolfram.com/07.22.03.a891.01

Input Form

HypergeometricPFQ[{-(15/4)}, {7/2, -(13/4)}, -z] == (1/(3257982000 Sqrt[2] z^(7/4))) ((4 z (1748906775 + 1582131600 z - 800467200 z^2 + 130928640 z^3 - 12124160 z^4 + 1048576 z^5) BesselJ[-(1/4), Sqrt[z]]^2 - 4 Sqrt[z] (5246720325 + 1303533000 z - 457430400 z^2 + 70133760 z^3 - 6389760 z^4 + 524288 z^5) BesselJ[-(1/4), Sqrt[z]] BesselJ[3/4, Sqrt[z]] + (15740160975 + 912473100 z + 7501032000 z^2 - 3359462400 z^3 + 537477120 z^4 - 49545216 z^5 + 4194304 z^6) BesselJ[3/4, Sqrt[z]]^2) Gamma[3/4]^2)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["15", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "2"], ",", RowBox[List["-", FractionBox["13", "4"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["3257982000", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["7", "/", "4"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["1748906775", "+", RowBox[List["1582131600", " ", "z"]], "-", RowBox[List["800467200", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["130928640", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["12124160", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1048576", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], "2"]]], "-", RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["5246720325", "+", RowBox[List["1303533000", " ", "z"]], "-", RowBox[List["457430400", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["70133760", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["6389760", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["524288", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["15740160975", "+", RowBox[List["912473100", " ", "z"]], "+", RowBox[List["7501032000", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["3359462400", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["537477120", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["49545216", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4194304", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], ")"]]]]]]]]

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> 15 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 13 </mn> <mn> 4 </mn> </mfrac> </mrow> </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]", "1"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["15", "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[RowBox[List["-", FractionBox["13", "4"]]], 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> <mn> 1 </mn> <mrow> <mn> 3257982000 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1048576 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 12124160 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 130928640 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 800467200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1582131600 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1748906775 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msub> <mi> J </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> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 524288 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6389760 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 70133760 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 457430400 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1303533000 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 5246720325 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> J </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> J </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> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4194304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 49545216 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 537477120 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3359462400 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7501032000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 912473100 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 15740160975 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <msub> <mi> J </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 15 <sep /> 4 </cn> </apply> </list> <list> <cn type='rational'> 7 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 13 <sep /> 4 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3257982000 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 1048576 </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'> 12124160 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 130928640 </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'> 800467200 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1582131600 </cn> <ci> z </ci> </apply> <cn type='integer'> 1748906775 </cn> </apply> <apply> <power /> <apply> <ci> BesselJ </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 /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 524288 </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'> 6389760 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 70133760 </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'> 457430400 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1303533000 </cn> <ci> z </ci> </apply> <cn type='integer'> 5246720325 </cn> </apply> <apply> <ci> BesselJ </ci> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselJ </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> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4194304 </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'> 49545216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 537477120 </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'> 3359462400 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7501032000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 912473100 </cn> <ci> z </ci> </apply> <cn type='integer'> 15740160975 </cn> </apply> <apply> <power /> <apply> <ci> BesselJ </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> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["15", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "2"], ",", RowBox[List["-", FractionBox["13", "4"]]]]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["1748906775", "+", RowBox[List["1582131600", " ", "z"]], "-", RowBox[List["800467200", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["130928640", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["12124160", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1048576", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], "2"]]], "-", RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["5246720325", "+", RowBox[List["1303533000", " ", "z"]], "-", RowBox[List["457430400", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["70133760", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["6389760", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["524288", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", SqrtBox["z"]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["15740160975", "+", RowBox[List["912473100", " ", "z"]], "+", RowBox[List["7501032000", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["3359462400", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["537477120", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["49545216", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["4194304", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List[FractionBox["3", "4"], ",", SqrtBox["z"]]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["3257982000", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["7", "/", "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]