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http://functions.wolfram.com/06.03.29.0005.01
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Subscript[d, n][\[Alpha] + \[Beta] + \[Gamma]] Subscript[d, k][\[Alpha]]
Subscript[d, j][\[Alpha]] Subscript[d, n - k - j][\[Gamma]] <=
Subscript[d, j + k][\[Alpha] + \[Beta]] Subscript[d, n - j][
\[Alpha] + \[Gamma]] Subscript[d, n - k][\[Beta] + \[Gamma]] /;
Subscript[d, k][\[Alpha]] == Binomial[k + \[Alpha] - 1, k] &&
Element[\[Alpha], Reals] && \[Alpha] > 0 && Element[\[Beta], Reals] &&
\[Beta] > 0 && Element[\[Gamma], Reals] && \[Gamma] > 0 &&
Element[j, Integers] && j >= 0 && Element[k, Integers] && k >= 0 &&
Element[n, Integers] && n >= 1 && j + k <= n
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Alpha]", "+", "\[Beta]", "+", "\[Gamma]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], RowBox[List[SubscriptBox["d", "j"], "[", "\[Alpha]", "]"]], RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k", "-", "j"]]], "[", "\[Gamma]", "]"]]]], "\[LessEqual]", RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["j", "+", "k"]]], "[", RowBox[List["\[Alpha]", "+", "\[Beta]"]], "]"]], RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "j"]]], "[", RowBox[List["\[Alpha]", "+", "\[Gamma]"]], "]"]], RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Beta]", "+", "\[Gamma]"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], "\[Equal]", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], ",", "k"]], "]"]]]], "\[And]", RowBox[List["\[Alpha]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Alpha]", ">", "0"]], "\[And]", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Beta]", ">", "0"]], "\[And]", RowBox[List["\[Gamma]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Gamma]", ">", "0"]], "\[And]", RowBox[List["j", "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List[RowBox[List["j", "+", "k"]], "\[LessEqual]", "n"]]]]]]]]
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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> <mrow> <mrow> <msub> <mi> d </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> β </mi> <mo> + </mo> <mi> γ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> d </mi> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> </mrow> <mo> ≤ </mo> <mrow> <mrow> <msub> <mi> d </mi> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> β </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> γ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> β </mi> <mo> + </mo> <mi> γ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> β </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> γ </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> j </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> n </ci> </apply> <apply> <plus /> <ci> α </ci> <ci> β </ci> <ci> γ </ci> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> α </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> j </ci> </apply> <ci> α </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> </apply> </apply> <ci> γ </ci> </apply> </apply> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <plus /> <ci> α </ci> <ci> β </ci> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <plus /> <ci> α </ci> <ci> γ </ci> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> β </ci> <ci> γ </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> α </ci> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> k </ci> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> <apply> <in /> <ci> α </ci> <apply> <ci> SuperPlus </ci> <reals /> </apply> </apply> <apply> <in /> <ci> β </ci> <apply> <ci> SuperPlus </ci> <reals /> </apply> </apply> <apply> <in /> <ci> γ </ci> <apply> <ci> SuperPlus </ci> <reals /> </apply> </apply> <apply> <in /> <ci> j </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> k </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <leq /> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Alpha]", "+", "\[Beta]", "+", "\[Gamma]"]], "]"]], " ", RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], " ", RowBox[List[SubscriptBox["d", "j"], "[", "\[Alpha]", "]"]], " ", RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k", "-", "j"]]], "[", "\[Gamma]", "]"]]]], "\[LessEqual]", RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["j", "+", "k"]]], "[", RowBox[List["\[Alpha]", "+", "\[Beta]"]], "]"]], " ", RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "j"]]], "[", RowBox[List["\[Alpha]", "+", "\[Gamma]"]], "]"]], " ", RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Beta]", "+", "\[Gamma]"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], "\[Equal]", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], ",", "k"]], "]"]]]], "&&", RowBox[List["\[Alpha]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Alpha]", ">", "0"]], "&&", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Beta]", ">", "0"]], "&&", RowBox[List["\[Gamma]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Gamma]", ">", "0"]], "&&", RowBox[List["j", "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[GreaterEqual]", "0"]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List[RowBox[List["j", "+", "k"]], "\[LessEqual]", "n"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
