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Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Identities > Recurrence identities > Generalized Cauchy summation





http://functions.wolfram.com/06.04.17.0003.01









  


  










Input Form





BoxData[\(\(\(Sum[\(\(\(Multinomial[\(\(p - \(Sum[\(\(Subscript[\(j, n\)]\), \({h, 1, n}\)\)]\)\), \(Subscript[\(j, 1\)]\), \(Subscript[\(j, 2\)]\), …, \(Subscript[\(j, n\)]\)\)]\) * \(Multinomial[\(\(q + \(Sum[\(\(Subscript[\(j, n\)]\), \({h, 1, n}\)\)]\)\), \(\(Subscript[\(k, 1\)]\) - \(Subscript[\(j, 1\)]\)\), \(\(Subscript[\(k, 2\)]\) - \(Subscript[\(j, 2\)]\)\), …, \(\(Subscript[\(k, n\)]\) - \(Subscript[\(j, n\)]\)\)\)]\)\), \({\(Subscript[\(j, 1\)]\), 0, \(Subscript[\(k, 1\)]\)}\), \({\(Subscript[\(j, 2\)]\), 0, \(Subscript[\(k, 2\)]\)}\), …\ ,\(\ \ \)\({\(Subscript[\(j, n\)]\), 0, \(Subscript[\(k, n\)]\)}\)\)]\)  \(Multinomial[\(\(p + q\), \(Subscript[\(k, 1\)]\), \(Subscript[\(k, 2\)]\), …, \(Subscript[\(k, n\)]\)\)]\)\)/;\(\(\(Subscript[\(k, 1\)]\) ∈ Integers\) && \(\(Subscript[\(k, 2\)]\) ∈ Integers\) && … && \(\(Subscript[\(k, n\)]\) ∈ Integers\) && \(\(Subscript[\(k, 1\)]\) ≥ 0\) && \(\(Subscript[\(k, 2\)]\) ≥ 0\) && … && \(\(Subscript[\(k, n\)]\) ≥ 0\) && \(p ∈ Integers\) && \(p ≥ 0\) && \(q ∈ Integers\) && \(q ≥ 0\) && \(n ∈ Integers\) && \(n > 0\)\)\)]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], SubscriptBox["k", "1"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], SubscriptBox["k", "2"]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", SubscriptBox["j", RowBox[List["n", "=", "0"]]], SubscriptBox["k", "n"]], RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["p", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "n"], SubscriptBox["j", "n"]]]]], ",", SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["j", "n"]]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["q", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "n"], SubscriptBox["j", "n"]]]]], ",", RowBox[List[SubscriptBox["k", "1"], "-", SubscriptBox["j", "1"]]], ",", RowBox[List[SubscriptBox["k", "2"], "-", SubscriptBox["j", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["k", "n"], "-", SubscriptBox["j", "n"]]]]], "]"]]]]]]]]]]]], "\[Equal]", " ", RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["p", "+", "q"]], ",", SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["k", "n"]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["k", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "2"], "\[Element]", "Integers"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["k", "n"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "1"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["k", "2"], "\[GreaterEqual]", "0"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["k", "n"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]










MathML Form







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</ms> <ms> + </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










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





2002-12-18





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