Generalized hypergeometric function 1F2: Specific values (formula 07.22.03.0311)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁},{b₁,b₂},z]

Specific values

Specialized values

For fixed b₁, b₂, z

http://functions.wolfram.com/07.22.03.0311.01

Input Form

HypergeometricPFQ[{5 + Subscript[b, 1]}, {Subscript[b, 1], Subscript[b, 2]}, z] == (z^((1/2) (1 - Subscript[b, 2])) Gamma[Subscript[b, 2]] (BesselI[-1 + Subscript[b, 2], 2 Sqrt[z]] (5 (7 + 2 z) Subscript[b, 1]^3 + 10 Subscript[b, 1]^4 + Subscript[b, 1]^5 - 10 Subscript[b, 1]^2 (-5 - 8 z + z Subscript[b, 2]) + Subscript[b, 1] (24 + 200 z + 5 z^2 - 55 z Subscript[b, 2] + 5 z Subscript[b, 2]^2) - z (-2 (77 + 8 z) + (71 + 2 z) Subscript[b, 2] - 14 Subscript[b, 2]^2 + Subscript[b, 2]^3)) + Sqrt[z] BesselI[Subscript[b, 2], 2 Sqrt[z]] (120 + 36 z + z^2 + 5 (5 + Subscript[b, 1])^4 + (96 + 19 z) Subscript[b, 2] + (46 + 3 z) Subscript[b, 2]^2 + 11 Subscript[b, 2]^3 + Subscript[b, 2]^4 - 10 (5 + Subscript[b, 1])^3 (5 + Subscript[b, 2]) + 5 (5 + Subscript[b, 1])^2 (35 + 2 z + 14 Subscript[b, 2] + 2 Subscript[b, 2]^2) - 5 (5 + Subscript[b, 1]) (50 + 8 z + 2 (15 + z) Subscript[b, 2] + 9 Subscript[b, 2]^2 + Subscript[b, 2]^3))))/ (Subscript[b, 1] (1 + Subscript[b, 1]) (2 + Subscript[b, 1]) (3 + Subscript[b, 1]) (4 + Subscript[b, 1]))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["5", "+", SubscriptBox["b", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SubscriptBox["b", "2"]]], ")"]]]]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["b", "2"]]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", SubsuperscriptBox["b", "1", "3"]]], "+", RowBox[List["10", " ", SubsuperscriptBox["b", "1", "4"]]], "+", SubsuperscriptBox["b", "1", "5"], "-", RowBox[List["10", " ", SubsuperscriptBox["b", "1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["8", " ", "z"]], "+", RowBox[List["z", " ", SubscriptBox["b", "2"]]]]], ")"]]]], "+", RowBox[List[SubscriptBox["b", "1"], " ", RowBox[List["(", RowBox[List["24", "+", RowBox[List["200", " ", "z"]], "+", RowBox[List["5", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["55", " ", "z", " ", SubscriptBox["b", "2"]]], "+", RowBox[List["5", " ", "z", " ", SubsuperscriptBox["b", "2", "2"]]]]], ")"]]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["77", "+", RowBox[List["8", " ", "z"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["71", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", SubscriptBox["b", "2"]]], "-", RowBox[List["14", " ", SubsuperscriptBox["b", "2", "2"]]], "+", SubsuperscriptBox["b", "2", "3"]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[SubscriptBox["b", "2"], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["120", "+", RowBox[List["36", " ", "z"]], "+", SuperscriptBox["z", "2"], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", SubscriptBox["b", "1"]]], ")"]], "4"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["96", "+", RowBox[List["19", " ", "z"]]]], ")"]], " ", SubscriptBox["b", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["46", "+", RowBox[List["3", " ", "z"]]]], ")"]], " ", SubsuperscriptBox["b", "2", "2"]]], "+", RowBox[List["11", " ", SubsuperscriptBox["b", "2", "3"]]], "+", SubsuperscriptBox["b", "2", "4"], "-", RowBox[List["10", " ", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", SubscriptBox["b", "1"]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List["5", "+", SubscriptBox["b", "2"]]], ")"]]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", SubscriptBox["b", "1"]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["35", "+", RowBox[List["2", " ", "z"]], "+", RowBox[List["14", " ", SubscriptBox["b", "2"]]], "+", RowBox[List["2", " ", SubsuperscriptBox["b", "2", "2"]]]]], ")"]]]], "-", RowBox[List["5", " ", RowBox[List["(", RowBox[List["5", "+", SubscriptBox["b", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["50", "+", RowBox[List["8", " ", "z"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["15", "+", "z"]], ")"]], " ", SubscriptBox["b", "2"]]], "+", RowBox[List["9", " ", SubsuperscriptBox["b", "2", "2"]]], "+", SubsuperscriptBox["b", "2", "3"]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", SubscriptBox["b", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", SubscriptBox["b", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", SubscriptBox["b", "1"]]], ")"]]]], ")"]]]]]]]]

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> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List[SubscriptBox["b", "1"], "+", "5"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 14 </mn> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 35 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 50 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> <mo> + </mo> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 46 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 19 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 96 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mn> 120 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> I </mi> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 5 </mn> </msubsup> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 55 </mn> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 200 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 24 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> <mo> - </mo> <mrow> <mn> 14 </mn> <mo> ⁢ </mo> <msubsup> <mi> b </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 71 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 77 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> BesselI </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 35 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> 15 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> <cn type='integer'> 50 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <cn type='integer'> 46 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 36 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 19 </cn> <ci> z </ci> </apply> <cn type='integer'> 96 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 120 </cn> </apply> </apply> <apply> <times /> <apply> <ci> BesselI </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -5 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 55 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 200 </cn> <ci> z </ci> </apply> <cn type='integer'> 24 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 71 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> <cn type='integer'> 77 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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["5", "+", SubscriptBox["b_", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SubscriptBox["bb", "2"]]], ")"]]]]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["bb", "2"]]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", SubsuperscriptBox["bb", "1", "3"]]], "+", RowBox[List["10", " ", SubsuperscriptBox["bb", "1", "4"]]], "+", SubsuperscriptBox["bb", "1", "5"], "-", RowBox[List["10", " ", SubsuperscriptBox["bb", "1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["8", " ", "z"]], "+", RowBox[List["z", " ", SubscriptBox["bb", "2"]]]]], ")"]]]], "+", RowBox[List[SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["24", "+", RowBox[List["200", " ", "z"]], "+", RowBox[List["5", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["55", " ", "z", " ", SubscriptBox["bb", "2"]]], "+", RowBox[List["5", " ", "z", " ", SubsuperscriptBox["bb", "2", "2"]]]]], ")"]]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["77", "+", RowBox[List["8", " ", "z"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["71", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", SubscriptBox["bb", "2"]]], "-", RowBox[List["14", " ", SubsuperscriptBox["bb", "2", "2"]]], "+", SubsuperscriptBox["bb", "2", "3"]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SqrtBox["z"], " ", RowBox[List["BesselI", "[", RowBox[List[SubscriptBox["bb", "2"], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["120", "+", RowBox[List["36", " ", "z"]], "+", SuperscriptBox["z", "2"], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", SubscriptBox["bb", "1"]]], ")"]], "4"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["96", "+", RowBox[List["19", " ", "z"]]]], ")"]], " ", SubscriptBox["bb", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["46", "+", RowBox[List["3", " ", "z"]]]], ")"]], " ", SubsuperscriptBox["bb", "2", "2"]]], "+", RowBox[List["11", " ", SubsuperscriptBox["bb", "2", "3"]]], "+", SubsuperscriptBox["bb", "2", "4"], "-", RowBox[List["10", " ", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", SubscriptBox["bb", "1"]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List["5", "+", SubscriptBox["bb", "2"]]], ")"]]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", SubscriptBox["bb", "1"]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["35", "+", RowBox[List["2", " ", "z"]], "+", RowBox[List["14", " ", SubscriptBox["bb", "2"]]], "+", RowBox[List["2", " ", SubsuperscriptBox["bb", "2", "2"]]]]], ")"]]]], "-", RowBox[List["5", " ", RowBox[List["(", RowBox[List["5", "+", SubscriptBox["bb", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["50", "+", RowBox[List["8", " ", "z"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["15", "+", "z"]], ")"]], " ", SubscriptBox["bb", "2"]]], "+", RowBox[List["9", " ", SubsuperscriptBox["bb", "2", "2"]]], "+", SubsuperscriptBox["bb", "2", "3"]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", SubscriptBox["bb", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", SubscriptBox["bb", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", SubscriptBox["bb", "1"]]], ")"]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2002-12-18

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]