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Pochhammer






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Pochhammer[a,n] > Series representations > Generalized power series > Expansions of (-m+Epsilon)n at Epsilon==0/;m-nN && m ∈ N





http://functions.wolfram.com/06.10.06.0021.01









  


  










Input Form





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










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m_"]], "+", "\[Epsilon]_"]], ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]], " ", RowBox[List["(", RowBox[List["\[Epsilon]", "+", RowBox[List[SuperscriptBox["\[Epsilon]", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "6"], " ", SuperscriptBox["\[Epsilon]", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"]]], "-", RowBox[List["6", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]], "2"]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[RowBox[List["-", "m"]], "+", "n"]]]], "]"]]]]]], ")"]]]], "+", SuperscriptBox[RowBox[List["O", "[", "\[Epsilon]", "]"]], "4"]]], ")"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["m", "-", "n"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["m", "-", "n"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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