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Binomial






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.03.06.0004.01









  


  










Input Form





Binomial[n, k] \[Proportional] ((Gamma[n + 1] Sin[Pi (k - n)])/Pi) k^(-n - 1) Sum[((((-1)^j Pochhammer[n + 1, j])/j!) B[j, -n, -n])/k^j, {j, 0, Infinity}] /; (Abs[k] -> Infinity) && B[n, \[Alpha], z] == n! SeriesTerm[(t^\[Alpha] E^(t z))/(E^t - 1)^\[Alpha], {t, 0, n}] && Abs[Arg[k - n]] < Pi










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["k", "-", "n"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox["k", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["B", "[", RowBox[List["j", ",", RowBox[List["-", "n"]], ",", RowBox[List["-", "n"]]]], "]"]], " ", SuperscriptBox["k", RowBox[List["-", "j"]]]]], RowBox[List["j", "!"]]]]]]], "\[Pi]"], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "k", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["B", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["t", "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["t", " ", "z"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "t"], "-", "1"]], ")"]], "\[Alpha]"]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["k", "-", "n"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]]










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





2001-10-29





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