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variants of this functions
Factorial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Series representations > Generalized power series > Expansions at n==-m





http://functions.wolfram.com/06.01.06.0006.01









  


  










Input Form





n! \[Proportional] (-1)^(m - 1)/((m - 1)! (n + m)) + ((-1)^(m - 1)/(m - 1)!) (PolyGamma[m] + (1/6) (Pi^2 + 3 PolyGamma[m]^2 - 3 PolyGamma[1, m]) (n + m) + (1/6) (PolyGamma[m]^3 + PolyGamma[m] (Pi^2 - 3 PolyGamma[1, m]) + PolyGamma[2, m]) (n + m)^2 + (1/360) (7 Pi^4 + 15 (PolyGamma[m]^4 + 2 PolyGamma[m]^2 (Pi^2 - 3 PolyGamma[1, m]) + PolyGamma[1, m] (3 PolyGamma[1, m] - 2 Pi^2) + 4 PolyGamma[m] PolyGamma[2, m]) - 15 PolyGamma[3, m]) (n + m)^3) + O[(n + m)^4] /; (n -> -m) && Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; 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</mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <factorial /> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 360 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <apply> <ci> PolyGamma </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", "m", "]"]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", "m", "]"]], "2"]]], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "m"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", "m", "]"]], "3"], "+", RowBox[List[RowBox[List["PolyGamma", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "m"]], "]"]]]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "m"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "2"]]], "+", RowBox[List[FractionBox["1", "360"], " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", SuperscriptBox["\[Pi]", "4"]]], "+", RowBox[List["15", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", "m", "]"]], "4"], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", "m", "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "m"]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "m"]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", "m", "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "m"]], "]"]]]]]], ")"]]]], "-", RowBox[List["15", " ", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", "m"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "3"]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["n", "+", "m"]], "]"]], "4"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", RowBox[List["-", "m"]]]], ")"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]










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





2001-10-29





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