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 PolyGamma

 http://functions.wolfram.com/06.15.06.0050.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], "-", "EulerGamma"]], ")"]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["m", "+", "1"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", " "]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "\[Nu]", ",", RowBox[List["2", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["k", "z"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], "-", "EulerGamma"]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["m", "+", "1"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "\[Nu]", ",", RowBox[List["2", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["k", "z"]]]]], "]"]], "k"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02