HarmonicNumber[z] == D[LogGamma[z + 1], z] + EulerGamma

Cell[BoxData[RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["LogGamma", "[", RowBox[List["z", "+", "1"]], "]"]]]], "+", "EulerGamma"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <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> <mi> z </mi> </mrow> </mfrac> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> LogGamma </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <eulergamma /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z"]]], RowBox[List["LogGamma", "[", RowBox[List["z", "+", "1"]], "]"]]]], "+", "EulerGamma"]]]]]]

2001-10-29