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variants of this functions
PolyGamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Specific values > Values at fixed points





http://functions.wolfram.com/06.15.03.0051.01









  


  








Input Form





PolyGamma[-n, m] == (1/(n - 1)!) Derivative[1, 0][Zeta][1 - n, 1 + m] + (m^(-1 + n)/(n - 1)!) (EulerGamma - (EulerGamma m)/n - Log[m] + PolyGamma[n] + Sum[(Binomial[-1 + n, k]/m^k) (Sum[Binomial[k, j] PolyGamma[k - j + 1] Zeta[j - k, 1 + m] (-m)^j, {j, 0, k}] - PolyGamma[k + 1] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, n - 1}]) /; (Element[n, Integers] && n > 0 && Element[m, Integers] && m > 0) || (Element[n, Integers] && n > 1 && m == -1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative]], "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "+", "m"]]]], "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["m", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["EulerGamma", "-", FractionBox[RowBox[List["EulerGamma", "m"]], "n"], "-", RowBox[List["Log", "[", "m", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "k"]], "]"]], SuperscriptBox["m", "k"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "j", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]]]]]], "\[IndentingNewLine]", ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "1"]], "\[And]", RowBox[List["m", "\[Equal]", RowBox[List["-", "1"]]]]]], ")"]]]]]]]]








MathML Form







<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> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mi> m </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;1&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;k&quot;]], Zeta, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]], Zeta, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;k&quot;]], Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> Zeta </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> m </mi> <mi> k </mi> </msup> </mfrac> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#63449; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> m </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> m </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> j </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> m </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <eulergamma /> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", FractionBox[RowBox[List["EulerGamma", " ", "m"]], "n"], "-", RowBox[List["Log", "[", "m", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "j", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]], SuperscriptBox["m", "k"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "1"]], "&&", RowBox[List["m", "\[Equal]", RowBox[List["-", "1"]]]]]], ")"]]]]]]]]]]


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





2007-05-02