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http://functions.wolfram.com/07.23.04.0015.01
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RamificationIndex[Hypergeometric2F1[a, b, c, z], z, ComplexInfinity] ==
LCM[s, u] /; a == r/s && b == t/u && Element[{r, s, t, u}, Integers] &&
s > 1 && u > 1 && GCD[r, s] == 1 && GCD[t, u] == 1 &&
NonTerminatingHypergeometricSeriesQ[{a, b}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], ",", "z", ",", "ComplexInfinity"]], "]"]], "\[Equal]", RowBox[List["LCM", "[", RowBox[List["s", ",", "u"]], "]"]]]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", FractionBox["r", "s"]]], "\[And]", RowBox[List["b", "\[Equal]", FractionBox["t", "u"]]], "\[And]", RowBox[List["Element", "[", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s", ",", "t", ",", "u"]], "}"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List["s", ">", "1"]], "\[And]", RowBox[List["u", ">", "1"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["t", ",", "u"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["NonTerminatingHypergeometricSeriesQ", "[", RowBox[List["{", RowBox[List["a", ",", "b"]], "}"]], "]"]]]]]]]]
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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> <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> <mi> b </mi> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]] </annotation> </semantics> <mo> , </mo> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> lcm </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> ⩵ </mo> <mfrac> <mi> r </mi> <mi> s </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> ⩵ </mo> <mfrac> <mi> t </mi> <mi> u </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> u </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> 𝒩𝒯 </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> { </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> } </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> ℛ </ci> <ci> z </ci> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <ci> a </ci> <ci> b </ci> </list> <list> <ci> c </ci> </list> <ci> z </ci> </apply> <apply> <ci> OverTilde </ci> <infinity /> </apply> </apply> <apply> <lcm /> <ci> s </ci> <ci> u </ci> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> a </ci> <apply> <times /> <ci> r </ci> <apply> <power /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> b </ci> <apply> <times /> <ci> t </ci> <apply> <power /> <ci> u </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <list> <ci> r </ci> <ci> s </ci> <ci> t </ci> <ci> u </ci> </list> <integers /> </apply> <apply> <gt /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> u </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <gcd /> <ci> r </ci> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <gcd /> <ci> t </ci> <ci> u </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> 𝒩𝒯 </ci> <list> <ci> a </ci> <ci> b </ci> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", "z_"]], "]"]], ",", "z_", ",", "ComplexInfinity"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["LCM", "[", RowBox[List["s", ",", "u"]], "]"]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", FractionBox["r", "s"]]], "&&", RowBox[List["b", "\[Equal]", FractionBox["t", "u"]]], "&&", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s", ",", "t", ",", "u"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["s", ">", "1"]], "&&", RowBox[List["u", ">", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["t", ",", "u"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["NonTerminatingHypergeometricSeriesQ", "[", RowBox[List["{", RowBox[List["a", ",", "b"]], "}"]], "]"]]]]]]]]]] |
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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,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] | |
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