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http://functions.wolfram.com/10.08.13.0001.01
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Nest[Function[f, z D[f, z] + f], w[z], n + 1] -
D[Nest[Function[f, z D[f, z] + f], w[z], n], z] == 0 /;
w[z] == PolyLog[n, z]/z && Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["f", ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", "f"]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["n", "+", "1"]]]], "]"]], "-", RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["Nest", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["f", ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", "f"]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", "n"]], "]"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", FractionBox[RowBox[List["PolyLog", "[", RowBox[List["n", ",", "z"]], "]"]], "z"]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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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> <mtext> </mtext> <mrow> <mstyle scriptlevel='0'> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> /; </mo> </mstyle> <mrow> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <plus /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> PolyLog </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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