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http://functions.wolfram.com/06.17.16.0010.01
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HarmonicNumber[z, r] + HarmonicNumber[z + 1/2, r] ==
2^r HarmonicNumber[2 z + 1, r] + (2 - 2^r) Zeta[r] /; Re[z] > -(1/2)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "+", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["z", "+", FractionBox["1", "2"]]], ",", "r"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", "r"], RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], ",", "r"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", SuperscriptBox["2", "r"]]], ")"]], RowBox[List["Zeta", "[", "r", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]]]]]]
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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> <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> + </mo> <msubsup> <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> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mi> r </mi> </msup> <mo> ⁢ </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["r", Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> r </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> r </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "+", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["z_", "+", FractionBox["1", "2"]]], ",", "r_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", "r"], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], ",", "r"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", SuperscriptBox["2", "r"]]], ")"]], " ", RowBox[List["Zeta", "[", "r", "]"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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