Product[(1 - q^k)^(DivisorSum[MoebiusMu[Subscript[d, j]] r^(k/Subscript[d, j]), {Subscript[d, j], k}]/k), {k, 1, Infinity}] == 1 - r q /; Element[r, Integers] && r >= 0 && Element[Subscript[d, j], Divisors[k]]

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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> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <munder> <mo> &#8721; </mo> <mrow> <mi> d </mi> <mo> | </mo> <mi> k </mi> </mrow> </munder> <mrow> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mfrac> <mi> k </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> </msup> </mrow> </mrow> <mi> k </mi> </mfrac> </msup> </mrow> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> r </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> r </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <mrow> <mi> divisors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </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> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> q </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> | </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#956; </ms> <ci> MoebiusMu </ci> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> r </ms> <apply> <ci> FractionBox </ci> <ms> k </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </apply> </apply> </list> </apply> </list> </apply> <ms> k </ms> </apply> </apply> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> q </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> &#8712; </ms> <apply> <ci> RowBox </ci> <list> <ms> divisors </ms> <ms> ( </ms> <ms> k </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q_", "k"]]], ")"]], FractionBox[RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["MoebiusMu", "[", SubscriptBox["d_", "j"], "]"]], " ", SuperscriptBox["r_", FractionBox["k", SubscriptBox["d_", "j"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d_", "j"], ",", "k"]], "}"]]]], "]"]], "k"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "-", RowBox[List["r", " ", "q"]]]], "/;", RowBox[List[RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["d", "j"], "\[Element]", RowBox[List["Divisors", "[", "k", "]"]]]]]]]]]]]]

S.‐.J. Kang, M.‐H. Kim, "Free Lie Algebras, Generalized Witt Formula, and the Denominator Identity", Journal of Algebra, v. 183, pp. 560-594 (1996)

2002-12-18