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 PolyGamma

 http://functions.wolfram.com/06.15.02.0007.01

 Input Form

 PolyGamma[\[Nu], z] == Piecewise[{{-(1/z) - EulerGamma + z Sum[1/(k (k + z)), {k, 1, Infinity}], \[Nu] == 0}, {(-1)^(\[Nu] + 1) \[Nu]! Sum[1/(k + z)^(\[Nu] + 1), {k, 0, Infinity}], Element[\[Nu], Integers] && \[Nu] > 0}, {((\[Nu] Log[z] - EulerGamma (z + \[Nu]) - \[Nu] PolyGamma[-\[Nu]])/ Gamma[1 - \[Nu]]) z^(-1 - \[Nu]) + z^(1 - \[Nu]) Sum[(1/k^2) Hypergeometric2F1Regularized[1, 2, 2 - \[Nu], -(z/k)], {k, 1, Infinity}], True}}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "z"]]], "-", "EulerGamma", " ", "+", RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["k", RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]]]]]]]]]]], ",", RowBox[List["\[Nu]", "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["\[Nu]", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]]]]]]], ",", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["z", "+", "\[Nu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], " ", "+", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["k", "2"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "2", ",", RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["-", FractionBox["z", "k"]]]]], "]"]]]]]]]]]], ",", "True"]], "}"]]]], "}"]], "]"]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( ν ) ( z ) - 1 z - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + z k = 1 1 k ( k + z ) ν 0 ( - 1 ) ν + 1 ν ! k = 0 1 ( k + z ) ν + 1 ν + - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( z + ν ) + ν log ( z ) - ν ψ TagBox["\[Psi]", PolyGamma] ( - ν ) Γ ( 1 - ν ) z - ν - 1 + z 1 - ν k = 1 1 k 2 2 F ~ 1 ( 1 , 2 ; 2 - ν ; - z k ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["2", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["z", "k"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] True PolyGamma ν z Piecewise -1 1 z -1 -1 z k 1 1 k k z -1 ν 0 -1 ν 1 ν k 0 1 k z ν 1 -1 ν SuperPlus -1 z ν ν z -1 ν PolyGamma -1 ν Gamma 1 -1 ν -1 z -1 ν -1 z 1 -1 ν k 1 1 k 2 -1 Hypergeometric2F1Regularized 1 2 2 -1 ν -1 z k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[RowBox[List["-", FractionBox["1", "z"]]], "-", "EulerGamma", "+", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]]]]]]]]]]], RowBox[List["\[Nu]", "\[Equal]", "0"]]], List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["\[Nu]", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]]]]]]], RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]]], List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["z", "+", "\[Nu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "2", ",", RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["-", FractionBox["z", "k"]]]]], "]"]], SuperscriptBox["k", "2"]]]]]]]], "True"]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02