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   http://functions.wolfram.com/07.28.04.0015.01
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    RamificationIndex[HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], 
     Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], 
     Subscript[b, 3]}, z], z, ComplexInfinity] == 
  LCM[Subscript[s, 1], Subscript[s, 2], Subscript[s, 3], Subscript[s, 4]] /; 
 Subscript[a, l] == Subscript[r, l]/Subscript[s, l] && 
  Element[{Subscript[r, l], Subscript[s, l]}, Integers] && 
  Subscript[s, l] > 1 && GCD[Subscript[r, l], Subscript[s, l]] == 1 && 
  1 <= l <= 4 && NonTerminatingHypergeometricSeriesQ[
   {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}] 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], ",", "z", ",", "ComplexInfinity"]], "]"]], "\[Equal]", RowBox[List["LCM", "[", RowBox[List[SubscriptBox["s", "1"], ",", SubscriptBox["s", "2"], ",", SubscriptBox["s", "3"], ",", SubscriptBox["s", "4"]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "\[Equal]", FractionBox[SubscriptBox["r", "l"], SubscriptBox["s", "l"]]]], "\[And]", RowBox[List["Element", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["r", "l"], ",", SubscriptBox["s", "l"]]], "}"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[SubscriptBox["s", "l"], ">", "1"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["r", "l"], ",", SubscriptBox["s", "l"]]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["1", "\[LessEqual]", "l", "\[LessEqual]", "4"]], "\[And]", RowBox[List["NonTerminatingHypergeometricSeriesQ", "[", RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], "]"]]]]]]]] 
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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>  <mrow>  <mrow>  <msub>  <mi> ℛ </mi>  <mi> z </mi>  </msub>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 4 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 3 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <semantics>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 4 </mn>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation>  </semantics>  <mo> ; </mo>  <semantics>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 3 </mn>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation>  </semantics>  <mo> ; </mo>  <semantics>  <mi> z </mi>  <annotation encoding='Mathematica'> TagBox["z", HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> , </mo>  <mover>  <mi> ∞ </mi>  <mo> ~ </mo>  </mover>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mi> lcm </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <msub>  <mi> s </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mn> 4 </mn>  </msub>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mi> l </mi>  </msub>  <mo> ⩵ </mo>  <mfrac>  <msub>  <mi> r </mi>  <mi> l </mi>  </msub>  <msub>  <mi> s </mi>  <mi> l </mi>  </msub>  </mfrac>  </mrow>  <mo> ∧ </mo>  <mrow>  <mrow>  <mo> { </mo>  <mrow>  <msub>  <mi> r </mi>  <mi> l </mi>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mi> l </mi>  </msub>  </mrow>  <mo> } </mo>  </mrow>  <mo> ∈ </mo>  <semantics>  <mi> ℤ </mi>  <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation>  </semantics>  </mrow>  <mo> ∧ </mo>  <mrow>  <msub>  <mi> s </mi>  <mi> l </mi>  </msub>  <mo> > </mo>  <mn> 1 </mn>  </mrow>  <mo> ∧ </mo>  <mrow>  <mrow>  <mi> gcd </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <msub>  <mi> r </mi>  <mi> l </mi>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mi> l </mi>  </msub>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⩵ </mo>  <mn> 1 </mn>  </mrow>  <mo> ∧ </mo>  <mrow>  <mn> 1 </mn>  <mo> ≤ </mo>  <mi> l </mi>  <mo> ≤ </mo>  <mn> 4 </mn>  </mrow>  <mo> ∧ </mo>  <mrow>  <mi> 𝒩𝒯 </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mo> { </mo>  <mrow>  <msub>  <mi> s </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> s </mi>  <mn> 4 </mn>  </msub>  </mrow>  <mo> } </mo>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> FormBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> ℛ </ms>  <ms> z </ms>  </apply>  <ms> [ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <apply>  <ci> ErrorBox </ci>  <ms>  </ms>  </apply>  <apply>  <ci> FormBox </ci>  <ms> 4 </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  <apply>  <ci> SubscriptBox </ci>  <ms> F </ms>  <apply>  <ci> FormBox </ci>  <ms> 3 </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  </list>  </apply>  <ms> ⁡ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 1 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 2 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 3 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 4 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <false />  </apply>  </apply>  <ms> ; </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 1 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 2 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 3 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <false />  </apply>  </apply>  <ms> ; </ms>  <apply>  <ci> TagBox </ci>  <ms> z </ms>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> OverscriptBox </ci>  <ms> ∞ </ms>  <ms> ~ </ms>  </apply>  <ms> ] </ms>  </list>  </apply>  <ms> ⩵ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> lcm </ms>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 1 </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 2 </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 3 </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 4 </ms>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  <ms> /; </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> l </ms>  </apply>  <ms> ⩵ </ms>  <apply>  <ci> FractionBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> r </ms>  <ms> l </ms>  </apply>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> l </ms>  </apply>  </apply>  </list>  </apply>  <ms> ∧ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> { </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> r </ms>  <ms> l </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> l </ms>  </apply>  </list>  </apply>  <ms> } </ms>  </list>  </apply>  <ms> ∈ </ms>  <apply>  <ci> TagBox </ci>  <ms> ℤ </ms>  <apply>  <ci> Function </ci>  <integers />  </apply>  </apply>  </list>  </apply>  <ms> ∧ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> l </ms>  </apply>  <ms> > </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> ∧ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> gcd </ms>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> r </ms>  <ms> l </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> l </ms>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  <ms> ⩵ </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> ∧ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> 1 </ms>  <ms> ≤ </ms>  <ms> l </ms>  <ms> ≤ </ms>  <ms> 4 </ms>  </list>  </apply>  <ms> ∧ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> 𝒩𝒯 </ms>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> { </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 1 </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 2 </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 3 </ms>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> s </ms>  <ms> 4 </ms>  </apply>  </list>  </apply>  <ms> } </ms>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", SubscriptBox["a_", "3"], ",", SubscriptBox["a_", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"]]], "}"]], ",", "z_"]], "]"]], ",", "z_", ",", "ComplexInfinity"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["s", "1"], ",", SubscriptBox["s", "2"], ",", SubscriptBox["s", "3"], ",", SubscriptBox["s", "4"]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "\[Equal]", FractionBox[SubscriptBox["r", "l"], SubscriptBox["s", "l"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["r", "l"], ",", SubscriptBox["s", "l"]]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["s", "l"], ">", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["r", "l"], ",", SubscriptBox["s", "l"]]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["1", "\[LessEqual]", "l", "\[LessEqual]", "4"]], "&&", RowBox[List["NonTerminatingHypergeometricSeriesQ", "[", RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], "]"]]]]]]]]]]  |  
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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},{b1,b2},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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