  
| 
 | 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   http://functions.wolfram.com/07.23.03.0669.01
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
 | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
    Hypergeometric2F1[a, -(5/6) + (5 a)/3, 2 a, 4/GoldenRatio^3] == 
 3^a 5^(1/6 - (5 a)/6) GoldenRatio^(-3 + 5 a) Gamma[1/6 + a/3] 
  Gamma[1/2 + a/3] (GoldenRatio/(Gamma[7/30 + a/3] Gamma[13/30 + a/3]) - 
   1/(Gamma[1/30 + a/3] Gamma[19/30 + a/3])) 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List[RowBox[List["-", FractionBox["5", "6"]]], "+", FractionBox[RowBox[List["5", " ", "a"]], "3"]]], ",", RowBox[List["2", " ", "a"]], ",", FractionBox["4", SuperscriptBox["GoldenRatio", "3"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["3", "a"], " ", SuperscriptBox["5", RowBox[List[FractionBox["1", "6"], "-", FractionBox[RowBox[List["5", " ", "a"]], "6"]]]], " ", SuperscriptBox["GoldenRatio", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", "a"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "6"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["GoldenRatio", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["7", "30"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["13", "30"], "+", FractionBox["a", "3"]]], "]"]]]]], "-", FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "30"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["19", "30"], "+", FractionBox["a", "3"]]], "]"]]]]]]], ")"]]]]]]]] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
 
  | 
   
   <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>  <mfrac>  <mrow>  <mn> 5 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 3 </mn>  </mfrac>  <mo> - </mo>  <mfrac>  <mn> 5 </mn>  <mn> 6 </mn>  </mfrac>  </mrow>  </mrow>  <mo> ; </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> ; </mo>  <mfrac>  <mn> 4 </mn>  <msup>  <mi> ϕ </mi>  <mn> 3 </mn>  </msup>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["5", " ", "a"]], "3"], "-", FractionBox["5", "6"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", " ", "a"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["4", SuperscriptBox[TagBox["\[Phi]", Function[List[], GoldenRatio]], "3"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation>  </semantics>  <mo>  </mo>  <mrow>  <msup>  <mn> 3 </mn>  <mi> a </mi>  </msup>  <mo> ⁢ </mo>  <msup>  <mn> 5 </mn>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 6 </mn>  </mfrac>  <mo> - </mo>  <mfrac>  <mrow>  <mn> 5 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 6 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation>  </semantics>  <mrow>  <mrow>  <mn> 5 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mn> 3 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mi> a </mi>  <mn> 3 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 6 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mi> a </mi>  <mn> 3 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mfrac>  <semantics>  <mi> ϕ </mi>  <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation>  </semantics>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mi> a </mi>  <mn> 3 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mn> 7 </mn>  <mn> 30 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mi> a </mi>  <mn> 3 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mn> 13 </mn>  <mn> 30 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mi> a </mi>  <mn> 3 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 30 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mfrac>  <mi> a </mi>  <mn> 3 </mn>  </mfrac>  <mo> + </mo>  <mfrac>  <mn> 19 </mn>  <mn> 30 </mn>  </mfrac>  </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'> 5 </cn>  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <cn type='rational'> 5 <sep /> 6 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <power />  <apply>  <power />  <ci> GoldenRatio </ci>  <cn type='integer'> 3 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <cn type='integer'> 3 </cn>  <ci> a </ci>  </apply>  <apply>  <power />  <cn type='integer'> 5 </cn>  <apply>  <plus />  <cn type='rational'> 1 <sep /> 6 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 5 </cn>  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 6 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <power />  <ci> GoldenRatio </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 5 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> -3 </cn>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 6 </cn>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <ci> GoldenRatio </ci>  <apply>  <power />  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 7 <sep /> 30 </cn>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 13 <sep /> 30 </cn>  </apply>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 30 </cn>  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <ci> a </ci>  <apply>  <power />  <cn type='integer'> 3 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <cn type='rational'> 19 <sep /> 30 </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["-", FractionBox["5", "6"]]], "+", FractionBox[RowBox[List["5", " ", "a_"]], "3"]]], ",", RowBox[List["2", " ", "a_"]], ",", FractionBox["4", SuperscriptBox["GoldenRatio", "3"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["3", "a"], " ", SuperscriptBox["5", RowBox[List[FractionBox["1", "6"], "-", FractionBox[RowBox[List["5", " ", "a"]], "6"]]]], " ", SuperscriptBox["GoldenRatio", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", "a"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "6"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["GoldenRatio", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["7", "30"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["13", "30"], "+", FractionBox["a", "3"]]], "]"]]]]], "-", FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "30"], "+", FractionBox["a", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["19", "30"], "+", FractionBox["a", "3"]]], "]"]]]]]]], ")"]]]]]]]]  |  
  |  
  |   
  |  
  |  
  | 
 
  |   
  |  
  |  
  |   
  |  |  
  |   
  |  
  |  
  |   
  |  |  
  |   
  |  
  |  
  | 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   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] |  |   |  
  |  
  
  
  
 |  
 
 |