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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Specific values > Specialized values > For fixed z





http://functions.wolfram.com/06.15.03.0042.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "n"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["z", "-", "n"]], ")"]]]], "+", RowBox[List["n", " ", RowBox[List["PolyGamma", "[", "n", "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "1"]]]]], RowBox[List["n", "!"]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "k"]], ")"]], RowBox[List["n", "-", "1"]]]]], RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["z", "k"]]], "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", "n"], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "m"]], "]"]], " "]], RowBox[List["m", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["m", "+", "1"]]], RowBox[List[FractionBox["1", "j"], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "k"]]], ")"]], RowBox[List["j", "-", "m", "-", "1"]]]]]]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










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> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> + </mo> <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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mtext> </mtext> </mrow> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;n&quot;]], &quot;)&quot;]], &quot;m&quot;], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> j </mi> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mi> k </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <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> </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> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> PolyGamma </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </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> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "n"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["z", "-", "n"]], ")"]]]], "+", RowBox[List["n", " ", RowBox[List["PolyGamma", "[", "n", "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "1"]]]]], RowBox[List["n", "!"]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "k"]], ")"]], RowBox[List["n", "-", "1"]]]]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["z", "k"]]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["m", "+", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "k"]]], ")"]], RowBox[List["j", "-", "m", "-", "1"]]], "j"]]]]], RowBox[List["m", "!"]]]]]]], "k"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02





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