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StieltjesGamma






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > StieltjesGamma[n] > Identities > Relation to Li's numbers





http://functions.wolfram.com/10.05.17.0001.01









  


  










Input Form





BoxData[\(\(\(Subscript[\(η, n\)]\)  \(\((n + 1)\) * \(Sum[\(\(\((\(Gamma[\(Sum[\(\(Subscript[\(k, j\)]\), \({j, 0, \(n + 1\)}\)\)]\)]\) * \(KroneckerDelta[\(\(n + 1\), \(Sum[\(\(\((1 + j)\) * \(Subscript[\(k, j\)]\)\), \({j, 0, \(n + 1\)}\)\)]\)\)]\))\) * \(Product[\(\(\(\((\((\(-\(\((\(-1\))\)^\(Subscript[\(k, j\)]\)\)\))\) * \((\(StieltjesGamma[j]\)/\(j !\))\))\)^\(Subscript[\(k, j\)]\)\)/\(\(Subscript[\(k, j\)]\) !\)\), \({j, 0, \(n + 1\)}\)\)]\)\), \({\(Subscript[\(k, 0\)]\), 1, \(n + 1\)}\), \({\(Subscript[\(k, 1\)]\), 1, \(n + 1\)}\), …, \({\(Subscript[\(k, \(n + 1\)\)]\), 1, \(n + 1\)}\)\)]\)\)\)/;\(\(n ∈ Integers\) && \(ν ≥ 0\) && \(\(Subscript[\(η, n\)]\)  \(SeriesTerm[\(\(\(-\(\(\(Derivative[1]\)[Zeta]\)[\(1 + s\)]\)\)/\(Zeta[\(1 + s\)]\)\), \({s, 0, n}\)\)]\)\)\)\)]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[Eta]", "n"], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "0"], "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["n", "+", "1"]]], "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "+", "1"]]], SubscriptBox["k", "j"]]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "j"]], ")"]], SubscriptBox["k", "j"]]]]]]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "0"]], RowBox[List["n", "+", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["k", "j"]]]], RowBox[List[RowBox[List["StieltjesGamma", "[", "j", "]"]], "/", RowBox[List["j", "!"]]]]]], ")"]], SubscriptBox["k", "j"]], RowBox[List[SubscriptBox["k", "j"], "!"]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["\[Eta]", "n"], "\[Equal]", RowBox[List["SeriesTerm", "[", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["Zeta", "'"]], "[", RowBox[List["1", "+", "s"]], "]"]]]], "/", RowBox[List["Zeta", "[", RowBox[List["1", "+", "s"]], "]"]]]], ",", " ", RowBox[List["{", RowBox[List["s", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> &#951; </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mi> j </mi> </msub> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> </msup> <mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#951; </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> s </mi> <mi> n </mi> </msup> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;s&quot;, &quot;+&quot;, &quot;1&quot;]], Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> &#951; </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#947; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Gamma]&quot;, StieltjesGamma] </annotation> </semantics> <mi> j </mi> </msub> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> </msup> <mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#951; </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> s </mi> <mi> n </mi> </msup> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;s&quot;, &quot;+&quot;, &quot;1&quot;]], Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SubscriptBox["$Failed", "n_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "0"], "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["n", "+", "1"]]], "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "+", "1"]]], SubscriptBox["k", "j"]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "j"]], ")"]], " ", SubscriptBox["k", "j"]]]]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "0"]], RowBox[List["n", "+", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["k", "j"]], " ", RowBox[List["StieltjesGamma", "[", "j", "]"]]]], RowBox[List["j", "!"]]]]], ")"]], SubscriptBox["k", "j"]], RowBox[List[SubscriptBox["k", "j"], "!"]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["$Failed", "n"], "\[Equal]", RowBox[List["SeriesTerm", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["1", "+", "s"]], "]"]], RowBox[List["Zeta", "[", RowBox[List["1", "+", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]]]]]]]]










References





K. Maslanka, "An Explicit Formula Relating Stieltjes Numbers and Li's Numbers", math.NT/0406312,










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





2007-05-02





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