HarmonicNumber[z] + HarmonicNumber[z + 1/2] == 2 HarmonicNumber[2 z + 1] - 2 Log[2]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "+", RowBox[List["HarmonicNumber", "[", RowBox[List["z", "+", FractionBox["1", "2"]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["2", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], "]"]]]], "-", RowBox[List["2", RowBox[List["Log", "[", "2", "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> + </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["HarmonicNumber", "[", "z_", "]"]], "+", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", "+", FractionBox["1", "2"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]]]]]]]]

2007-05-02