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http://functions.wolfram.com/06.17.20.0002.01
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D[HarmonicNumber[z, r], {z, 2}] ==
r (1 + r) (HarmonicNumber[z, 2 + r] - Zeta[2 + r])
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]]]], "\[Equal]", RowBox[List["r", " ", RowBox[List["(", RowBox[List["1", "+", "r"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", RowBox[List["2", "+", "r"]]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List["2", "+", "r"]], "]"]]]], ")"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <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> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> - </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["r", "+", "2"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> </apply> <apply> <times /> <ci> r </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "2"]], "}"]]]]], RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["r", " ", RowBox[List["(", RowBox[List["1", "+", "r"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", RowBox[List["2", "+", "r"]]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List["2", "+", "r"]], "]"]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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