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http://functions.wolfram.com/06.04.06.0001.01
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Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis],
Subscript[n, m]] \[Proportional]
(Subscript[n, 1]^(a - 1)/Product[Gamma[Subscript[n, k] + 1], {k, 2, m}])
Sum[((((-1)^k Pochhammer[1 - a, k])/k!) B[k, a, a])/Subscript[n, 1]^k,
{k, 0, Infinity}] /; (Abs[Subscript[n, 1]] -> Infinity) &&
a == 1 + Sum[Subscript[n, k], {k, 2, m}] &&
B[n, \[Alpha], z] == n! SeriesTerm[(t^\[Alpha] E^(t z))/(E^t - 1)^\[Alpha],
{t, 0, n}] && Abs[Arg[Subscript[n, 1] + a]] < Pi
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SubsuperscriptBox["n", "1", RowBox[List["a", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "m"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["n", "k"], "+", "1"]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", "k"]], "]"]], " "]], RowBox[List["k", "!"]]], RowBox[List["B", "[", RowBox[List["k", ",", "a", ",", "a"]], "]"]], " ", SubsuperscriptBox["n", "1", RowBox[List["-", "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["n", "1"], "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["a", "\[Equal]", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "m"], SubscriptBox["n", "k"]]]]]]], "\[And]", 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"]], "}"]]]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List[SubscriptBox["n", "1"], "+", "a"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]
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<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> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> … </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> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["n", "m"]]], ";", SubscriptBox["n", "1"]]], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], ")"]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mfrac> <msubsup> <mi> n </mi> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> B </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> a </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> n </mi> <mn> 1 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> a </mi> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> B </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> α </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mi> t </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <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> … </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> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["n", "m"]]], ";", SubscriptBox["n", "1"]]], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], ")"]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mfrac> <msubsup> <mi> n </mi> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> B </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> a </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> n </mi> <mn> 1 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> a </mi> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> B </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> α </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mi> t </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> t </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["nn", "1", RowBox[List["a", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["B", "[", RowBox[List["k", ",", "a", ",", "a"]], "]"]], " ", SubsuperscriptBox["nn", "1", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "m"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["nn", "k"], "+", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["nn", "1"], "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["a", "\[Equal]", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "m"], SubscriptBox["nn", "k"]]]]]]], "&&", 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[SubscriptBox["nn", "1"], "+", "a"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
