HarmonicNumber[z, r] == ((-1)^r/(r - 1)!) ((D[LogGamma[z + 1], {z, r}] /. {z -> 0}) - D[LogGamma[z + 1], {z, r}]) /; Element[r, Integers] && r > 0

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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> <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> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> r </mi> </msup> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> r </mi> </msup> <mstyle scriptlevel='0'> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </mfrac> <mo> /. </mo> <mtext> &#8203; </mtext> <mrow> <mo> { </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> r </mi> </msup> <mstyle scriptlevel='0'> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> r </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </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> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> ReplaceAll </ci> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> r </ci> </degree> </bvar> <apply> <ci> LogGamma </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <list> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </list> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> r </ci> </degree> </bvar> <apply> <ci> LogGamma </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02