PolyGamma[n, -m + \[Epsilon]] \[Proportional] ((-1)^(n - 1) n!)/\[Epsilon]^(n + 1) - (-1)^n n! Zeta[n + 1] + n! HarmonicNumber[m, n + 1] (1 + O[\[Epsilon]]) /; Element[m, Integers] && m >= 0 && Element[n, Integers] && n > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["-", "m"]], "+", "\[Epsilon]"]]]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["n", "!"]]]], SuperscriptBox["\[Epsilon]", RowBox[List["n", "+", "1"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["Zeta", "[", RowBox[List["n", "+", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", RowBox[List["n", "+", "1"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "\[Epsilon]", "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> &#1013; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <msup> <mi> &#1013; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mrow> <mi> n </mi> <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> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#1013; </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </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> PolyGamma </ci> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> &#1013; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> &#1013; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> HarmonicNumber </ci> <ci> m </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> &#1013; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> n </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["PolyGamma", "[", RowBox[List["n_", ",", RowBox[List[RowBox[List["-", "m_"]], "+", "\[Epsilon]_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["n", "!"]]]], SuperscriptBox["\[Epsilon]", RowBox[List["n", "+", "1"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["Zeta", "[", RowBox[List["n", "+", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "\[Epsilon]", "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

2007-05-02