Generalized hypergeometric function 1F2: Specific values (formula 07.22.03.5528)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁},{b₁,b₂},z]

Specific values

For integer and half-integer parameters and fixed z

For fixed z and a₁=11/2, b_1`>=-11/2

For fixed z and a₁=11/2, b₁=-11/2

http://functions.wolfram.com/07.22.03.5528.01

Input Form

HypergeometricPFQ[{11/2}, {-(11/2), -(9/2)}, z] == (1 + (20 z)/9 + (656 z^2)/567 + (6784 z^3)/19845 + (256 z^4)/2835 + (2048 z^5)/42525 - (8192 z^6)/93555 - (2818048 z^7)/618866325 - (65536 z^8)/9282994875) Cosh[2 Sqrt[z]] - (1/3094331625) (2 Sqrt[z] (3094331625 + 2750517000 z + 1012888800 z^2 + 266112000 z^3 + 79833600 z^4 + 81285120 z^5 + 48087040 z^6 + 458752 z^7) Sinh[2 Sqrt[z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["11", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", FractionBox["9", "2"]]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["20", " ", "z"]], "9"], "+", FractionBox[RowBox[List["656", " ", SuperscriptBox["z", "2"]]], "567"], "+", FractionBox[RowBox[List["6784", " ", SuperscriptBox["z", "3"]]], "19845"], "+", FractionBox[RowBox[List["256", " ", SuperscriptBox["z", "4"]]], "2835"], "+", FractionBox[RowBox[List["2048", " ", SuperscriptBox["z", "5"]]], "42525"], "-", FractionBox[RowBox[List["8192", " ", SuperscriptBox["z", "6"]]], "93555"], "-", FractionBox[RowBox[List["2818048", " ", SuperscriptBox["z", "7"]]], "618866325"], "-", FractionBox[RowBox[List["65536", " ", SuperscriptBox["z", "8"]]], "9282994875"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "3094331625"], RowBox[List["(", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["3094331625", "+", RowBox[List["2750517000", " ", "z"]], "+", RowBox[List["1012888800", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["266112000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["79833600", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["81285120", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["48087040", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["458752", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", 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> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> </mrow> </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[FractionBox["11", "2"], 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[RowBox[List["-", FractionBox["9", "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> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 65536 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mn> 9282994875 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2818048 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mn> 618866325 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 8192 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mn> 93555 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 2048 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mn> 42525 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 256 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mn> 2835 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 6784 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 19845 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 656 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 567 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 20 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 9 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3094331625 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 458752 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48087040 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 81285120 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 79833600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 266112000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1012888800 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2750517000 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 3094331625 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <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'> 11 <sep /> 2 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 11 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 65536 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <cn type='integer'> 9282994875 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2818048 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <cn type='integer'> 618866325 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8192 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 93555 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2048 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 42525 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 2835 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6784 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 19845 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 656 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 567 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <ci> z </ci> <apply> <power /> <cn type='integer'> 9 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 3094331625 </cn> <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'> 458752 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 48087040 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 81285120 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 79833600 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 266112000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1012888800 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2750517000 </cn> <ci> z </ci> </apply> <cn type='integer'> 3094331625 </cn> </apply> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </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["{", FractionBox["11", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", FractionBox["9", "2"]]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["20", " ", "z"]], "9"], "+", FractionBox[RowBox[List["656", " ", SuperscriptBox["z", "2"]]], "567"], "+", FractionBox[RowBox[List["6784", " ", SuperscriptBox["z", "3"]]], "19845"], "+", FractionBox[RowBox[List["256", " ", SuperscriptBox["z", "4"]]], "2835"], "+", FractionBox[RowBox[List["2048", " ", SuperscriptBox["z", "5"]]], "42525"], "-", FractionBox[RowBox[List["8192", " ", SuperscriptBox["z", "6"]]], "93555"], "-", FractionBox[RowBox[List["2818048", " ", SuperscriptBox["z", "7"]]], "618866325"], "-", FractionBox[RowBox[List["65536", " ", SuperscriptBox["z", "8"]]], "9282994875"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "-", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["3094331625", "+", RowBox[List["2750517000", " ", "z"]], "+", RowBox[List["1012888800", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["266112000", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["79833600", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["81285120", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["48087040", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["458752", " ", SuperscriptBox["z", "7"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "3094331625"]]]]]]]

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]