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http://functions.wolfram.com/06.15.06.0036.01
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PolyGamma[n, -m + \[Epsilon]] == ((-1)^(n - 1) n!)/\[Epsilon]^(n + 1) +
Sum[(1/k!) (PolyGamma[n + k, 1] + Gamma[n + k + 1]
HarmonicNumber[m, n + k + 1]) \[Epsilon]^k, {k, 0, Infinity}] /;
Abs[\[Epsilon]] < 1 && Element[m, Integers] && m >= 0 &&
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["-", "m"]], "+", "\[Epsilon]"]]]], "]"]], "\[Equal]", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["n", "+", "k"]], ",", "1"]], "]"]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "k", "+", "1"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", RowBox[List["n", "+", "k", "+", "1"]]]], "]"]]]]]], ")"]], SuperscriptBox["\[Epsilon]", "k"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "\[Epsilon]", "]"]], "<", "1"]], "\[And]", 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"]]]]]]]]
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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> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", 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> ϵ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </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> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <msup> <mi> ϵ </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </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> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ϵ </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ϵ </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyGamma </ci> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> ϵ </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> ϵ </ci> <apply> <plus /> <ci> n </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'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> HarmonicNumber </ci> <ci> m </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> ϵ </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> ϵ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["n", "+", "k"]], ",", "1"]], "]"]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "k", "+", "1"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["m", ",", RowBox[List["n", "+", "k", "+", "1"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "k"]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "\[Epsilon]", "]"]], "<", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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