RamificationIndex[HypergeometricU[a, p/q, z], z, 0] == q /; Element[p, Integers] && Element[q - 1, Integers] && q - 1 > 0 && GCD[p, q] == 1 && !(Element[-a, Integers] && -a >= 0)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", FractionBox["p", "q"], ",", "z"]], "]"]], ",", "z", ",", "0"]], "]"]], "\[Equal]", "q"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["q", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["q", "-", "1"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["p", ",", "q"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]

<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> &#8475; </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mi> q </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#8475; </ci> <ci> z </ci> </apply> <apply> <ci> HypergeometricU </ci> <ci> a </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <ci> q </ci> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <integers /> </apply> <apply> <in /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <integers /> </apply> <apply> <eq /> <apply> <gcd /> <ci> p </ci> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List["a_", ",", FractionBox["p_", "q_"], ",", "z_"]], "]"]], ",", "z_", ",", "0"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["q", "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["q", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["q", "-", "1"]], ">", "0"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["p", ",", "q"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]

2001-10-29