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

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=23/4, b_1`>=-11/2

For fixed z and a₁=23/4, b_1`=-11/2

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

Input Form

HypergeometricPFQ[{23/4}, {-(11/2), -(21/4)}, z] == (1 + (508 z)/231 + (1440 z^2)/1309 + (178624 z^3)/595595 + (19456 z^4)/292383 + (32768 z^5)/1686825 + (524288 z^6)/9398025 + (8107589632 z^7)/635218148565 + (67108864 z^8)/756212081625) Cosh[2 Sqrt[z]] + (1/142924083427125) (2 Sqrt[z] (-142924083427125 - 123743795175000 z - 43053135325200 z^2 - 10098730857600 z^3 - 2335931136000 z^4 - 996663951360 z^5 + 3340763136000 z^6 + 111962750976 z^7 + 134217728 z^8) Sinh[2 Sqrt[z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["23", "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", FractionBox["21", "4"]]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["508", " ", "z"]], "231"], "+", FractionBox[RowBox[List["1440", " ", SuperscriptBox["z", "2"]]], "1309"], "+", FractionBox[RowBox[List["178624", " ", SuperscriptBox["z", "3"]]], "595595"], "+", FractionBox[RowBox[List["19456", " ", SuperscriptBox["z", "4"]]], "292383"], "+", FractionBox[RowBox[List["32768", " ", SuperscriptBox["z", "5"]]], "1686825"], "+", FractionBox[RowBox[List["524288", " ", SuperscriptBox["z", "6"]]], "9398025"], "+", FractionBox[RowBox[List["8107589632", " ", SuperscriptBox["z", "7"]]], "635218148565"], "+", FractionBox[RowBox[List["67108864", " ", SuperscriptBox["z", "8"]]], "756212081625"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "142924083427125"], RowBox[List["(", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "142924083427125"]], "-", RowBox[List["123743795175000", " ", "z"]], "-", RowBox[List["43053135325200", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["10098730857600", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["2335931136000", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["996663951360", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["3340763136000", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["111962750976", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["134217728", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", 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> 23 </mn> <mn> 4 </mn> </mfrac> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 21 </mn> <mn> 4 </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["23", "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[RowBox[List["-", FractionBox["11", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["21", "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> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 67108864 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mn> 756212081625 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 8107589632 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mn> 635218148565 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 524288 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mn> 9398025 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 32768 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mn> 1686825 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 19456 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mn> 292383 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 178624 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 595595 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 1440 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 1309 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 508 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 231 </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> 142924083427125 </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> 134217728 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 111962750976 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3340763136000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 996663951360 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2335931136000 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10098730857600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 43053135325200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 123743795175000 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 142924083427125 </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'> 23 <sep /> 4 </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'> 21 <sep /> 4 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 67108864 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <cn type='integer'> 756212081625 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8107589632 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <cn type='integer'> 635218148565 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 524288 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 9398025 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 32768 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 1686825 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 19456 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 292383 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 178624 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 595595 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1440 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 1309 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 508 </cn> <ci> z </ci> <apply> <power /> <cn type='integer'> 231 </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='rational'> 1 <sep /> 142924083427125 </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'> 134217728 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 111962750976 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3340763136000 </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'> 996663951360 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2335931136000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10098730857600 </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'> 43053135325200 </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'> 123743795175000 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -142924083427125 </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> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["23", "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", FractionBox["21", "4"]]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["508", " ", "z"]], "231"], "+", FractionBox[RowBox[List["1440", " ", SuperscriptBox["z", "2"]]], "1309"], "+", FractionBox[RowBox[List["178624", " ", SuperscriptBox["z", "3"]]], "595595"], "+", FractionBox[RowBox[List["19456", " ", SuperscriptBox["z", "4"]]], "292383"], "+", FractionBox[RowBox[List["32768", " ", SuperscriptBox["z", "5"]]], "1686825"], "+", FractionBox[RowBox[List["524288", " ", SuperscriptBox["z", "6"]]], "9398025"], "+", FractionBox[RowBox[List["8107589632", " ", SuperscriptBox["z", "7"]]], "635218148565"], "+", FractionBox[RowBox[List["67108864", " ", SuperscriptBox["z", "8"]]], "756212081625"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "142924083427125"]], "-", RowBox[List["123743795175000", " ", "z"]], "-", RowBox[List["43053135325200", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["10098730857600", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["2335931136000", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["996663951360", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["3340763136000", " ", SuperscriptBox["z", "6"]]], "+", RowBox[List["111962750976", " ", SuperscriptBox["z", "7"]]], "+", RowBox[List["134217728", " ", SuperscriptBox["z", "8"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "142924083427125"]]]]]]]

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]