Polygamma function: Specific values (formula 06.15.03.0040)

PolyGamma

$Mathematica Notation$

http://functions.wolfram.com/06.15.03.0040.01

Input Form

PolyGamma[-10, z] == (-(1/(18289152000 Pi^8))) (Pi^8 z^2 (-1785 + z^2 (13750 + z^2 (-35126 + z^2 (106935 + 14258 (-5 + z) z - 453600 Log[Glaisher] - 25200 z Log[2 Pi])))) + 12600 (4 Pi^8 z^9 Log[z] - 36 Pi^6 z^7 Zeta[3] + 378 Pi^4 z^5 Zeta[5] - 1890 Pi^2 z^3 Zeta[7] + 2835 z Zeta[9] + 4 Pi^8 (Derivative[1][Zeta][-9] + 36 z^2 Derivative[1][Zeta][-7] + 126 z^4 Derivative[1][Zeta][-5] + 84 z^6 Derivative[1][Zeta][-3])) - 50400 Pi^8 Derivative[1, 0][Zeta][-9, 1 + z])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "10"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["18289152000", " ", SuperscriptBox["\[Pi]", "8"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", "8"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1785"]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["13750", "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "35126"]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["106935", "+", RowBox[List["14258", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "z"]], ")"]], " ", "z"]], "-", RowBox[List["453600", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "-", RowBox[List["25200", " ", "z", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["12600", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Pi]", "8"], " ", SuperscriptBox["z", "9"], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["36", " ", SuperscriptBox["\[Pi]", "6"], " ", SuperscriptBox["z", "7"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List["378", " ", SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "5"], " ", RowBox[List["Zeta", "[", "5", "]"]]]], "-", RowBox[List["1890", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Zeta", "[", "7", "]"]]]], "+", RowBox[List["2835", " ", "z", " ", RowBox[List["Zeta", "[", "9", "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Pi]", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "9"]], "]"]], "+", RowBox[List["36", " ", SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "7"]], "]"]]]], "+", RowBox[List["126", " ", SuperscriptBox["z", "4"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]]]], "+", RowBox[List["84", " ", SuperscriptBox["z", "6"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["50400", " ", SuperscriptBox["\[Pi]", "8"], " ", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "9"]], ",", RowBox[List["1", "+", "z"]]]], "]"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 10 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 18289152000 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 8 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> π </mi> <mn> 8 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 14258 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 25200 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mn> 453600 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox["A", Function[List[], Glaisher]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 106935 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 35126 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 13750 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1785 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 12600 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 8 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 6 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 378 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["5", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1890 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 7 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["7", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2835 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 9 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["9", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 8 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 84 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 126 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 50400 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 8 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> , </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> -10 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 18289152000 </cn> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 14258 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -5 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 25200 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 453600 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> </apply> </apply> <cn type='integer'> 106935 </cn> </apply> </apply> <cn type='integer'> -35126 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 13750 </cn> </apply> </apply> <cn type='integer'> -1785 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 12600 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 378 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1890 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2835 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 9 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 84 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -3 </cn> </apply> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 126 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> -5 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -7 </cn> </apply> <cn type='integer'> -7 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -9 </cn> </apply> <cn type='integer'> -9 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 50400 </cn> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <cn type='integer'> -9 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "10"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", "8"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1785"]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["13750", "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "35126"]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["106935", "+", RowBox[List["14258", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "z"]], ")"]], " ", "z"]], "-", RowBox[List["453600", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "-", RowBox[List["25200", " ", "z", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["12600", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Pi]", "8"], " ", SuperscriptBox["z", "9"], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["36", " ", SuperscriptBox["\[Pi]", "6"], " ", SuperscriptBox["z", "7"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List["378", " ", SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "5"], " ", RowBox[List["Zeta", "[", "5", "]"]]]], "-", RowBox[List["1890", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Zeta", "[", "7", "]"]]]], "+", RowBox[List["2835", " ", "z", " ", RowBox[List["Zeta", "[", "9", "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Pi]", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "9"]], "]"]], "+", RowBox[List["36", " ", SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "7"]], "]"]]]], "+", RowBox[List["126", " ", SuperscriptBox["z", "4"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]]]], "+", RowBox[List["84", " ", SuperscriptBox["z", "6"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["50400", " ", SuperscriptBox["\[Pi]", "8"], " ", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "9"]], ",", RowBox[List["1", "+", "z"]]]], "]"]]]]]], RowBox[List["18289152000", " ", SuperscriptBox["\[Pi]", "8"]]]]]]]]]]

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

2007-05-02

PolyGamma[z]