HarmonicNumber[z, r] == Zeta[r] - ZetaClassical[r, z + 1] /; r != 1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Zeta", "[", "r", "]"]], "-", RowBox[List["ZetaClassical", "[", RowBox[List["r", ",", RowBox[List["z", "+", "1"]]]], "]"]]]]]], "/;", RowBox[List["r", "\[NotEqual]", "1"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> &#10869; </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;r&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox[&quot;\[Zeta]&quot;, &quot;^&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;r&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> r </mi> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <plus /> <apply> <ci> Zeta </ci> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <ci> r </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <neq /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "r", "]"]], "-", RowBox[List["ZetaClassical", "[", RowBox[List["r", ",", RowBox[List["z", "+", "1"]]]], "]"]]]], "/;", RowBox[List["r", "\[NotEqual]", "1"]]]]]]]]

2001-10-29