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Binomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Binomial[n,k] > Series representations > Generalized power series > Expansions at generic point k==k0 > For the function itself





http://functions.wolfram.com/06.03.06.0015.01









  


  










Input Form





Binomial[n, k] \[Proportional] Binomial[n, Subscript[k, 0]] (1 + (-1 + Cos[Pi (n - Subscript[k, 0])] Gamma[1 + n - Subscript[k, 0]]) (HarmonicNumber[Subscript[k, 0]] - HarmonicNumber[ -1 - n + Subscript[k, 0]]) (k - Subscript[k, 0]) - ((-2 Pi Cot[Pi (n - Subscript[k, 0])] Gamma[-n + Subscript[k, 0]] + Pi^2 - 2 Gamma[-n + Subscript[k, 0]]^2)/ (4 Gamma[-n + Subscript[k, 0]]^2)) ((HarmonicNumber[Subscript[k, 0]] - HarmonicNumber[ -1 - n + Subscript[k, 0]])^2 - PolyGamma[1, 1 + Subscript[k, 0]] + PolyGamma[1, -n + Subscript[k, 0]]) (k - Subscript[k, 0])^2 + \[Ellipsis]) /; (k -> Subscript[k, 0])










Standard Form





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MathML Form







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<annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </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> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </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> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> k </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <cot /> <apply> <times /> <pi /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> 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type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", SubscriptBox["kk", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["n", "-", SubscriptBox["kk", "0"]]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "n", "-", SubscriptBox["kk", "0"]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HarmonicNumber", "[", SubscriptBox["kk", "0"], "]"]], "-", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "-", "n", "+", SubscriptBox["kk", "0"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["k", "-", SubscriptBox["kk", "0"]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["n", "-", SubscriptBox["kk", "0"]]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["kk", "0"]]], "]"]]]], "+", SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["kk", "0"]]], "]"]], "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["HarmonicNumber", "[", SubscriptBox["kk", "0"], "]"]], "-", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "-", "n", "+", SubscriptBox["kk", "0"]]], "]"]]]], ")"]], "2"], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["kk", "0"]]]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["kk", "0"]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", SubscriptBox["kk", "0"]]], ")"]], "2"]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["kk", "0"]]], "]"]], "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["(", RowBox[List["k", "\[Rule]", SubscriptBox["kk", "0"]]], ")"]]]]]]]]










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





2007-05-02





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