Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











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/06.03.06.0017.01









  


  










Input Form





Binomial[n, k] == Gamma[Subscript[k, 0] - n] Sum[(Pi^(m - 1)/m!) Sum[Binomial[m, j] Sin[Pi (Subscript[k, 0] - n + (m - j)/2)] j! HypergeometricPFQRegularized[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, m + 1], -n}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, m + 1]}, 1] (-(Gamma[Subscript[k, 0] - n]/Pi))^j (k - Subscript[k, 0])^m, {j, 0, m}], {m, 0, Infinity}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, m + 1] == Subscript[k, 0] - n










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["k", "0"], "-", "n"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["\[Pi]", RowBox[List["m", "-", "1"]]], RowBox[List["m", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[SubscriptBox["k", "0"], "-", "n", "+", FractionBox[RowBox[List["m", "-", "j"]], "2"]]], ")"]]]], "]"]], " ", RowBox[List["j", "!"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["m", "+", "1"]]], ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["m", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["k", "0"], "-", "n"]], "]"]], "\[Pi]"]]], ")"]], "j"], SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", SubscriptBox["k", "0"]]], ")"]], "m"]]]]]]]]]]]]], "/;", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["m", "+", "1"]]], "\[Equal]", RowBox[List[SubscriptBox["k", "0"], "-", "n"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> &#960; </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;m&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mi> j </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> k </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;2&quot;]]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;2&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;n&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;;&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <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> <mi> m </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mo> &#8230; </mo> <mo> &#10869; </mo> <msub> <mi> a </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <times /> <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> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> m </ci> <ci> j </ci> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> j </ci> </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> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <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> </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["Gamma", "[", RowBox[List[SubscriptBox["kk", "0"], "-", "n"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["m", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[SubscriptBox["kk", "0"], "-", "n", "+", FractionBox[RowBox[List["m", "-", "j"]], "2"]]], ")"]]]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["m", "+", "1"]]], ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["m", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["kk", "0"], "-", "n"]], "]"]], "\[Pi]"]]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", SubscriptBox["kk", "0"]]], ")"]], "m"]]]]]]], RowBox[List["m", "!"]]]]]]], "/;", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["m", "+", "1"]]], "\[Equal]", RowBox[List[SubscriptBox["kk", "0"], "-", "n"]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.