HarmonicNumber[z, r] \[Proportional] -(1/(z + m)^r) + (-1)^(r - 1) HarmonicNumber[m, r] - ((-1)^r/(r - 1)!) Sum[(1/k!) (PolyGamma[r + k - 1, 1] + Gamma[r + k] HarmonicNumber[m, r + k]) (z + m)^k, {k, 1, Infinity}] /; (z -> -m) && Element[m, Integers] && m > 0 && Element[r, Integers] && r > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "m"]], ")"]], "r"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "-", "1"]]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", "r"]], "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], RowBox[List[RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["r", "+", "k", "-", "1"]], ",", "1"]], "]"]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["r", "+", "k"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", RowBox[List["r", "+", "k"]]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "m"]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "m"]]]], ")"]], "\[And]", RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["r", ",", "Integers"]], "]"]], "\[And]", RowBox[List["r", ">", "0"]]]]]]]]

<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> &#8733; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> r </mi> </msup> </mfrac> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> m </mi> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> </mrow> <mo> - </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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> m </mi> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> r </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> m </ci> </apply> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HarmonicNumber </ci> <ci> m </ci> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <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> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <ci> r </ci> </apply> </apply> <apply> <ci> HarmonicNumber </ci> <ci> m </ci> <apply> <plus /> <ci> k </ci> <ci> r </ci> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> m </ci> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </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["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "m"]], ")"]], "r"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "-", "1"]]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", "r"]], "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["r", "+", "k", "-", "1"]], ",", "1"]], "]"]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["r", "+", "k"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", RowBox[List["r", "+", "k"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "m"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["r", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "m"]]]], ")"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", ">", "0"]]]]]]]]]]

2001-10-29