Limit[SubStar[m][n]/n - ProductLog[n], n -> Infinity] == 0 /; StirlingS2[n, SubStar[m][n]] >= StirlingS2[n, m] && 0 <= m <= n

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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> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msub> <mi> m </mi> <mo> * </mo> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mi> n </mi> </mfrac> <mo> - </mo> <mrow> <mi> W </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mo> * </mo> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> &#8805; </mo> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <plus /> <apply> <times /> <apply> <apply> <ci> SubStar </ci> <ci> m </ci> </apply> <ci> n </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> W </ci> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <geq /> <apply> <ci> StirlingS2 </ci> <ci> n </ci> <apply> <apply> <ci> SubStar </ci> <ci> m </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <ci> StirlingS2 </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29