Nielsen polylogarithm: Identities (formula 10.09.17.0002)

PolyLog

$Mathematica Notation$

Zeta Functions and Polylogarithms

PolyLog[nu,p,z]

Identities

Functional identities

http://functions.wolfram.com/10.09.17.0002.01

Input Form

PolyLog[n, p, z] == (-1)^n Sum[(-1)^k Sum[(Log[-z]^j Binomial[n + k - j - 1, k - j] PolyLog[n + k - j, p - k, 1/z])/j!, {j, 0, k}], {k, 0, p - 1}] + (-1)^p (Sum[(Log[-z]^j c[n - j, p])/j!, {j, 0, n - 1}] + Log[-z]^(n + p)/(n + p)!) /; c[n, p] == (1 - (-1)^n) (-1)^p PolyLog[n, p, -1] - (-1)^n Sum[Binomial[n + j - 1, j] (-1)^(p - j) PolyLog[n + j, p - j, -1], {j, 1, p - 1}] && Element[n, Integers] && n > 0 && Element[p, Integers] && p > 0 && !IntervalMemberQ[Interval[{0, 1}], z]

Standard Form

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MathML Form

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</mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> c </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "j", "+", "k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> <mrow> <mi> p </mi> <mo> - 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</mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> j </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mi> j </mi> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> j </mi> </mrow> </msubsup> </mstyle> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <semantics> <mi> S </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mi> p </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mrow> <mi> n </mi> <mo> + </mo> <mi> p </mi> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> c </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> j </mi> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "j", "+", "k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> c </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mi> n </mi> <mi> p </mi> </msubsup> </mstyle> <mo> ( </mo> <mrow> <mo> - 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</mo> <mi> j </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mi> j </mi> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> j </mi> </mrow> </msubsup> </mstyle> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

PolyLog[nu,z]
PolyLog[2,z]