RamificationIndex[JacobiP[\[Nu], a, b, z], z, ComplexInfinity] == LCM[s, u] /; \[Nu] == r/s && a + b + \[Nu] == t/u && Element[{r, s, t, u}, Integers] && s > 1 && u > 1 && GCD[r, s] == 1 && GCD[t, u] == 1 && NonTerminatingHypergeometricSeriesQ[ {-\[Nu], a + b + \[Nu] + 1}]

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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> &#8475; </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mover> <mi> &#8734; </mi> <mo> ~ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#10869; </mo> <mfrac> <mi> r </mi> <mi> s </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> &#10869; </mo> <mfrac> <mi> t </mi> <mi> u </mi> </mfrac> </mrow> <mo> &#8743; </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> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> u </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#119977;&#119983; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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> &#8475; </ci> <ci> z </ci> </apply> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> a </ci> <ci> b </ci> <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> &#957; </ci> <apply> <times /> <ci> r </ci> <apply> <power /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> </apply> <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> &#119977;&#119983; </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29