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
 











Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > General characteristics > Poles and essential singularities > With respect to nk





http://functions.wolfram.com/06.04.04.0006.01









  


  










Input Form





Residue[Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]], {Subscript[n, k], -OverTilde[Subscript[N, k]] - j}] == (-1)^(j - 1)/(Gamma[-OverTilde[Subscript[N, k]] - j + 1] Product[Gamma[Subscript[n, r] + 1] Product[Gamma[Subscript[n, r] + 1] (j - 1)!, {r, k + 1, m}], {r, 1, k - 1}]) /; OverTilde[Subscript[N, k]] == Sum[Subscript[n, r], {r, 1, k - 1}] + Sum[Subscript[n, r], {r, k + 1, m}] && Element[j, Integers] && j > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", "k"], ",", " ", RowBox[List[RowBox[List["-", OverscriptBox[SubscriptBox["N", "k"], "~"]]], "-", "j"]]]], "}"]]]], "]"]], " ", "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", OverscriptBox[SubscriptBox["N", "k"], "~"]]], "-", "j", "+", "1"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["n", "r"], "+", "1"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", RowBox[List["k", "+", "1"]]]], "m"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["n", "r"], "+", "1"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["j", "-", "1"]], ")"]], "!"]]]]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List[OverscriptBox[SubscriptBox["N", "k"], "~"], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List["k", "-", "1"]]], SubscriptBox["n", "r"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", RowBox[List["k", "+", "1"]]]], "m"], SubscriptBox["n", "r"]]]]]]], "\[And]", RowBox[List["j", "\[Element]", "Integers"]], "\[And]", RowBox[List["j", ">", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> res </mi> <msub> <mi> n </mi> <mi> k </mi> </msub> </msub> <mo> ( </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;;&quot;, SubscriptBox[&quot;n&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mover> <msub> <mi> N </mi> <mi> k </mi> </msub> <mo> ~ </mo> </mover> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mover> <msub> <mi> N </mi> <mi> k </mi> </msub> <mo> ~ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> r </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> r </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mover> <msub> <mi> N </mi> <mi> k </mi> </msub> <mo> ~ </mo> </mover> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> n </mi> <mi> r </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> r </mi> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> j </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> </apply> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> OverTilde </ci> <apply> <ci> Subscript </ci> <ci> N </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> OverTilde </ci> <apply> <ci> Subscript </ci> <ci> N </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> OverTilde </ci> <apply> <ci> Subscript </ci> <ci> N </ci> <ci> k </ci> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> r </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> j </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Residue", "[", RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n_", "k_"], ",", RowBox[List[RowBox[List["-", OverscriptBox[SubscriptBox["N", "k_"], "~"]]], "-", "j_"]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", OverscriptBox[SubscriptBox["N", "k"], "~"]]], "-", "j", "+", "1"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["nn", "r"], "+", "1"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", RowBox[List["k", "+", "1"]]]], "m"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["nn", "r"], "+", "1"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "-", "1"]], ")"]], "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[OverscriptBox[SubscriptBox["N", "k"], "~"], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List["k", "-", "1"]]], SubscriptBox["nn", "r"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", RowBox[List["k", "+", "1"]]]], "m"], SubscriptBox["nn", "r"]]]]]]], "&&", RowBox[List["j", "\[Element]", "Integers"]], "&&", RowBox[List["j", ">", "0"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.