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Pochhammer






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Pochhammer[a,n] > Series representations > Asymptotic series expansions > Expansions at n==infinity





http://functions.wolfram.com/06.10.06.0007.01









  


  










Input Form





Pochhammer[a, n] \[Proportional] ((Sqrt[2 Pi] n^(a + n - 1/2))/ (E^n Gamma[a])) (1 + (1 - 6 a + 6 a^2)/(12 n) + (1 - 36 a + 120 a^2 - 120 a^3 + 36 a^4)/(288 n^2) + (1/(51840 n^3)) (-139 - 450 a + 8190 a^2 - 20160 a^3 + 18900 a^4 - 7560 a^5 + 1080 a^6) + (1/(2488320 n^4)) (-571 + 23352 a + 34464 a^2 - 465840 a^3 + 945000 a^4 - 828576 a^5 + 362880 a^6 - 77760 a^7 + 6480 a^8) + O[1/n^5]) /; (Abs[n] -> Infinity) && Abs[Arg[a + n]] < Pi










Standard Form





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MathML Form







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type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> a </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 120 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 288 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1080 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7560 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18900 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20160 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8190 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 450 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -139 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 51840 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2488320 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6480 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 77760 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 362880 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 828576 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 945000 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 465840 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 34464 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 23352 </cn> <ci> a </ci> </apply> <cn type='integer'> -571 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> n </ci> </apply> <infinity /> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Pochhammer", "[", RowBox[List["a_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "n"]]], " ", SuperscriptBox["n", RowBox[List["a", "+", "n", "-", FractionBox["1", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", RowBox[List["6", " ", "a"]], "+", RowBox[List["6", " ", SuperscriptBox["a", "2"]]]]], RowBox[List["12", " ", "n"]]], "+", FractionBox[RowBox[List["1", "-", RowBox[List["36", " ", "a"]], "+", RowBox[List["120", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["120", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["36", " ", SuperscriptBox["a", "4"]]]]], RowBox[List["288", " ", SuperscriptBox["n", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["-", "139"]], "-", RowBox[List["450", " ", "a"]], "+", RowBox[List["8190", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["20160", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["18900", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["7560", " ", SuperscriptBox["a", "5"]]], "+", RowBox[List["1080", " ", SuperscriptBox["a", "6"]]]]], RowBox[List["51840", " ", SuperscriptBox["n", "3"]]]], "+", FractionBox[RowBox[List[RowBox[List["-", "571"]], "+", RowBox[List["23352", " ", "a"]], "+", RowBox[List["34464", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["465840", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List["945000", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["828576", " ", SuperscriptBox["a", "5"]]], "+", RowBox[List["362880", " ", SuperscriptBox["a", "6"]]], "-", RowBox[List["77760", " ", SuperscriptBox["a", "7"]]], "+", RowBox[List["6480", " ", SuperscriptBox["a", "8"]]]]], RowBox[List["2488320", " ", SuperscriptBox["n", "4"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["n", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "5"]], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", "a", "]"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["a", "+", "n"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]]










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





2001-10-29





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