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http://functions.wolfram.com/06.17.06.0007.01
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HarmonicNumber[z, r] == HarmonicNumber[z] +
Sum[(Log[k + z]/(k + z) - Log[k]/k) (r - 1), {k, 1, Infinity}] +
(1/2) Sum[(Log[k + z]^2/(k + z) - Log[k]^2/k) (r - 1)^2,
{k, 1, Infinity}] + \[Ellipsis]
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Cell[BoxData[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["k", "+", "z"]], "]"]], RowBox[List["k", "+", "z"]]], "-", FractionBox[RowBox[List["Log", "[", "k", "]"]], "k"]]], ")"]], RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]]]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "z"]], "]"]], "2"], RowBox[List["k", "+", "z"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "k", "]"]], "2"], "k"]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]], "2"]]]]]]], "+", "\[Ellipsis]"]]]]]]
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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> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <ci> k </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> … </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["k", "+", "z"]], "]"]], RowBox[List["k", "+", "z"]]], "-", FractionBox[RowBox[List["Log", "[", "k", "]"]], "k"]]], ")"]], " ", RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["k", "+", "z"]], "]"]], "2"], RowBox[List["k", "+", "z"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "k", "]"]], "2"], "k"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]], "2"]]]]]]], "+", "\[Ellipsis]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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