Stieltjes constant: Identities (formula 10.05.17.0001)

StieltjesGamma

$Mathematica Notation$

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[\(Î·, n$]\) ï± $\((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, $n + 1$}\), â€¦, ${\(Subscript[\(k, \(n + 1$\)]\), 1, $n + 1$}\)\)]\)\)\)/;$\(n âˆˆ Integers$ && $Î½ â‰¥ 0$ && $\(Subscript[\(Î·, n$]\) ï± $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> η </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </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> ∑ </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> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </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> Γ </mi> <mo> ⁡ </mo> <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> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </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> ∑ </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> ⁢ </mo> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <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> <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> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", 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> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> η </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> s </mi> <mi> n </mi> </msup> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["s", "+", "1"]], Zeta, Rule[Editable, True]], ")"]], 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> η </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </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> ∑ </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> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </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> Γ </mi> <mo> ⁡ </mo> <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> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </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> ∑ </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> ⁢ </mo> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <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> <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> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", 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> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> η </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> s </mi> <mi> n </mi> </msup> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["s", "+", "1"]], Zeta, Rule[Editable, True]], ")"]], 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