  
| 
 | 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   http://functions.wolfram.com/07.23.03.0074.01
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
 | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
    Hypergeometric2F1[a, 2 a - 3/2, 4 a - 3, 12 Sqrt[2] - 16] == 
 ((2 + Sqrt[2])^(2 a)/2^(4 a - 2)) Sqrt[Pi] 
  ((3 Gamma[2 a - 1])/(Gamma[a/2] Gamma[(3 a - 1)/2]) - 
   Gamma[2 a - 1]/(Gamma[(a + 1)/2] Gamma[(3 a)/2 - 1])) 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List[RowBox[List["2", "a"]], "-", FractionBox["3", "2"]]], ",", RowBox[List[RowBox[List["4", "a"]], "-", "3"]], ",", RowBox[List[RowBox[List["12", SqrtBox["2"]]], "-", "16"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", SqrtBox["2"]]], ")"]], RowBox[List["2", "a"]]], SuperscriptBox["2", RowBox[List[RowBox[List["4", "a"]], "-", "2"]]]], SqrtBox["\[Pi]"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["3", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "a"]], "-", "1"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox["a", "2"], "]"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List[RowBox[List["3", "a"]], "-", "1"]], "2"], "]"]]]]], "-", FractionBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "a"]], "-", "1"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["a", "+", "1"]], "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["3", "a"]], "2"], "-", "1"]], "]"]]]]]]], ")"]]]]]]]] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
 
  | 
   
   <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> 2 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 1 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mn> 3 </mn>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <mn> 12 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  </mrow>  <mo> - </mo>  <mn> 16 </mn>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", FractionBox["3", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["4", " ", "a"]], "-", "3"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["12", " ", SqrtBox["2"]]], "-", "16"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> + </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <msqrt>  <mi> π </mi>  </msqrt>  </mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msup>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mi> a </mi>  <mn> 2 </mn>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  <mo> - </mo>  <mfrac>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <ci> Hypergeometric2F1 </ci>  <ci> a </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <cn type='rational'> 3 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 4 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> -3 </cn>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 12 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -16 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <times />  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  </apply>  <apply>  <power />  <pi />  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <power />  <cn type='integer'> 2 </cn>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 4 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> -2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 3 </cn>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <times />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <times />  <apply>  <plus />  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 
  |   
  |  
  |  
  |   
  |  |  
  |   
  |  
  |  
  |   
  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", RowBox[List[RowBox[List["2", " ", "a_"]], "-", FractionBox["3", "2"]]], ",", RowBox[List[RowBox[List["4", " ", "a_"]], "-", "3"]], ",", RowBox[List[RowBox[List["12", " ", SqrtBox["2"]]], "-", "16"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", SqrtBox["2"]]], ")"]], RowBox[List["2", " ", "a"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["3", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "a"]], "-", "1"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox["a", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "-", "1"]], ")"]]]], "]"]]]]], "-", FractionBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "a"]], "-", "1"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["a", "+", "1"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["3", " ", "a"]], "2"], "-", "1"]], "]"]]]]]]], ")"]]]], SuperscriptBox["2", RowBox[List[RowBox[List["4", " ", "a"]], "-", "2"]]]]]]]]  |  
  |  
  |   
  |  
  |  
  | 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Date Added to functions.wolfram.com (modification date)
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 | 
  | 
  
| HypergeometricPFQ[{},{},z] |  | HypergeometricPFQ[{},{b},z] |  | HypergeometricPFQ[{a},{},z] |  | HypergeometricPFQ[{a},{b},z] |  | HypergeometricPFQ[{a1},{b1,b2},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,...,ap},{b1,...,bq},z] |  |   |  
  |  
  
  
  
 |  
 
 |