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| Hypergeometric Functions  HypergeometricPFQ[{a1,a2,a3},{b1,b2},z]  Specific values  For integer and half-integer parameters and fixed z  For fixed z and a1=-1/2, a2>=-1/2  For fixed z and a1=-1/2, a2=1/2, a3>=1/2  For fixed z and a1=-1/2, a2=1/2, a3=2  For fixed z and a1=-1/2, a2=1/2, a3=2, b1=-7/2   |  |  
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 | | http://functions.wolfram.com/07.27.03.ajgm.01 | 
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 | | HypergeometricPFQ[{-(1/2), 1/2, 2}, {-(7/2), 3/2}, z] == 
 (9 - 58 z + 121 z^2 - 96 z^3)/(105 (-1 + z)^4) + 
  (32 ArcTanh[Sqrt[z]])/(35 Sqrt[z]) | 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], ",", FractionBox["3", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["9", "-", RowBox[List["58", " ", "z"]], "+", RowBox[List["121", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["96", " ", SuperscriptBox["z", "3"]]]]], RowBox[List["105", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"]]]], "+", FractionBox[RowBox[List["32", " ", RowBox[List["ArcTanh", "[", SqrtBox["z"], "]"]]]], RowBox[List["35", " ", SqrtBox["z"]]]]]]]]]] | 
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   <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> 3 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 7 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 3 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["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["7", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["3", "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>  <mfrac>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mn> 96 </mn>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mn> 3 </mn>  </msup>  </mrow>  <mo> + </mo>  <mrow>  <mn> 121 </mn>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mrow>  <mo> - </mo>  <mrow>  <mn> 58 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> + </mo>  <mn> 9 </mn>  </mrow>  <mrow>  <mn> 105 </mn>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mn> 4 </mn>  </msup>  </mrow>  </mfrac>  <mo> + </mo>  <mfrac>  <mrow>  <mn> 32 </mn>  <mo> ⁢ </mo>  <mrow>  <msup>  <mi> tanh </mi>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ( </mo>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mn> 35 </mn>  <mo> ⁢ </mo>  <msqrt>  <mi> z </mi>  </msqrt>  </mrow>  </mfrac>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <ci> HypergeometricPFQ </ci>  <list>  <apply>  <times />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  <cn type='integer'> 2 </cn>  </list>  <list>  <apply>  <times />  <cn type='integer'> -1 </cn>  <cn type='rational'> 7 <sep /> 2 </cn>  </apply>  <cn type='rational'> 3 <sep /> 2 </cn>  </list>  <ci> z </ci>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> -96 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 3 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 121 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 58 </cn>  <ci> z </ci>  </apply>  </apply>  <cn type='integer'> 9 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> 105 </cn>  <apply>  <power />  <apply>  <plus />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  <cn type='integer'> 4 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 32 </cn>  <apply>  <arctanh />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> 35 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], ",", FractionBox["3", "2"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["9", "-", RowBox[List["58", " ", "z"]], "+", RowBox[List["121", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["96", " ", SuperscriptBox["z", "3"]]]]], RowBox[List["105", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"]]]], "+", FractionBox[RowBox[List["32", " ", RowBox[List["ArcTanh", "[", SqrtBox["z"], "]"]]]], RowBox[List["35", " ", SqrtBox["z"]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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 | | HypergeometricPFQ[{},{},z] |  | HypergeometricPFQ[{},{b},z] |  | HypergeometricPFQ[{a},{},z] |  | HypergeometricPFQ[{a},{b},z] |  | HypergeometricPFQ[{a1},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2},{b1},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2,b3},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,...,ap},{b1,...,bq},z] |  |  | 
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