Binomial coefficients: Inequalities (formula 06.03.29.0004)

Binomial

$Mathematica Notation$

Gamma, Beta, Erf

Binomial[n,k]

Inequalities

http://functions.wolfram.com/06.03.29.0004.01

Input Form

2 Subscript[d, n][\[Lambda] + \[Mu] + \[Nu] + \[Gamma]] Sum[(Subscript[d, n - k][\[Lambda]]/Subscript[d, k][ \[Mu] + \[Nu] + \[Gamma]]) Abs[Sum[Subscript[u, j] Subscript[v, k - j], {j, 0, k}] Sum[Subscript[x, j] Subscript[y, k - j], {j, 0, k}]], {k, 0, n}] <= Sum[(Subscript[d, n - k][\[Lambda] + \[Nu]]/Subscript[d, k][ \[Mu] + \[Gamma]]) Abs[Subscript[u, k]]^2, {k, 0, n}] Sum[(Subscript[d, n - k][\[Lambda] + \[Mu]]/Subscript[d, k][ \[Nu] + \[Gamma]]) Abs[Subscript[y, k]]^2, {k, 0, n}] Sum[(Subscript[d, n - k][\[Lambda] + \[Nu] + \[Gamma]]/ Subscript[d, k][\[Mu]]) Abs[Subscript[x, k]]^2, {k, 0, n}] Sum[(Subscript[d, n - k][\[Lambda] + \[Mu] + \[Gamma]]/ Subscript[d, k][\[Nu]]) Abs[Subscript[\[Nu], k]]^2, {k, 0, n}] /; Subscript[d, k][\[Alpha]] == Binomial[k + \[Alpha] - 1, k] && Element[\[Mu], Reals] && \[Mu] > 0 && Element[\[Nu], Reals] && \[Nu] > 0 && Element[\[Gamma], Reals] && \[Gamma] >= 0 && Element[\[Lambda], Reals] && \[Lambda] >= 0 && Max[Abs[{Subscript[u, 1], Subscript[u, 2], \[Ellipsis], Subscript[u, n]}]] > 0 && Max[Abs[{Subscript[v, 1], Subscript[v, 2], \[Ellipsis], Subscript[v, n]}]] > 0 && Max[Abs[{Subscript[x, 1], Subscript[x, 2], \[Ellipsis], Subscript[x, n]}]] > 0 && Max[Abs[{Subscript[y, 1], Subscript[y, 2], \[Ellipsis], Subscript[y, n]}]] > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["2", RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "\[Nu]", "+", "\[Gamma]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", "\[Lambda]", "]"]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "\[Gamma]"]], "]"]]], RowBox[List["Abs", "[", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["u", "j"], SubscriptBox["v", RowBox[List["k", "-", "j"]]]]]]], ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["x", "j"], SubscriptBox["y", RowBox[List["k", "-", "j"]]]]]]], ")"]]]], "]"]]]]]]]], "\[LessEqual]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Mu]", "+", "\[Gamma]"]], "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["u", "k"], "]"]], "2"]]]]], ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Mu]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Nu]", "+", "\[Gamma]"]], "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["y", "k"], "]"]], "2"]]]]], ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "\[Gamma]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Mu]", "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["x", "k"], "]"]], "2"]]]]], ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "\[Gamma]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Nu]", "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["\[Nu]", "k"], "]"]], "2"]]]]], ")"]]]]]], "/;", 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["\[Mu]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Mu]", ">", "0"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]], "\[And]", RowBox[List["\[Gamma]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Gamma]", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["\[Lambda]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["u", "1"], ",", SubscriptBox["u", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["u", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["v", "1"], ",", SubscriptBox["v", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["v", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["x", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["y", "1"], ",", SubscriptBox["y", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["y", "n"]]], "}"]], "]"]], "]"]], ">", "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> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msub> <mi> d </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msub> <mi> u </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <msub> <mi> v </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msub> <mi> x </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <msub> <mi> y </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> </mrow> </mrow> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> ≤ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> u </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> y </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> μ </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> x </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> ν </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </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> μ </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> ν </mi> <mo> ∈ </mo> <mrow> <msup> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> <mo> + </mo> </msup> <mo> ⁢ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> γ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> γ </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> u </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> u </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> u </mi> <mi> n </mi> </msub> </mrow> <mo> } </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> v </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> v </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> v </mi> <mi> n </mi> </msub> </mrow> <mo> } </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> x </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> x </mi> <mi> n </mi> </msub> </mrow> <mo> } </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> y </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> y </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> y </mi> <mi> n </mi> </msub> </mrow> <mo> } </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> n </ci> </apply> <apply> <plus /> <ci> γ </ci> <ci> λ </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <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> <ci> λ </ci> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> γ </ci> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <abs /> <apply> <times /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> u </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> v </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> y </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <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> λ </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> γ </ci> <ci> μ </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> u </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <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> λ </ci> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> γ </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> y </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <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> γ </ci> <ci> λ </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> μ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <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> γ </ci> <ci> λ </ci> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> ν </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </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> μ </ci> <apply> <ci> SuperPlus </ci> <reals /> </apply> </apply> <apply> <in /> <ci> ν </ci> <apply> <times /> <apply> <ci> SuperPlus </ci> <reals /> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <in /> <ci> γ </ci> <reals /> </apply> <apply> <geq /> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> λ </ci> <reals /> </apply> <apply> <geq /> <ci> λ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <max /> <apply> <abs /> <list> <apply> <ci> Subscript </ci> <ci> u </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> u </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> u </ci> <ci> n </ci> </apply> </list> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <max /> <apply> <abs /> <list> <apply> <ci> Subscript </ci> <ci> v </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> v </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> v </ci> <ci> n </ci> </apply> </list> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <max /> <apply> <abs /> <list> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> </list> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <max /> <apply> <abs /> <list> <apply> <ci> Subscript </ci> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> y </ci> <ci> n </ci> </apply> </list> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["2", " ", RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "\[Nu]", "+", "\[Gamma]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", "\[Lambda]", "]"]], " ", RowBox[List["Abs", "[", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["u", "j"], " ", SubscriptBox["v", RowBox[List["k", "-", "j"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["x", "j"], " ", SubscriptBox["y", RowBox[List["k", "-", "j"]]]]]]]]], "]"]]]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "\[Gamma]"]], "]"]]]]]]], "\[LessEqual]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["u", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Mu]", "+", "\[Gamma]"]], "]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["y", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Nu]", "+", "\[Gamma]"]], "]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "\[Gamma]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["x", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Mu]", "]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "\[Gamma]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["\[Nu]", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Nu]", "]"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], "\[Equal]", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], ",", "k"]], "]"]]]], "&&", RowBox[List["\[Mu]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Mu]", ">", "0"]], "&&", RowBox[List["\[Nu]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Nu]", ">", "0"]], "&&", RowBox[List["\[Gamma]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Gamma]", "\[GreaterEqual]", "0"]], "&&", RowBox[List["\[Lambda]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["u", "1"], ",", SubscriptBox["u", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["u", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["v", "1"], ",", SubscriptBox["v", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["v", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["x", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["y", "1"], ",", SubscriptBox["y", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["y", "n"]]], "}"]], "]"]], "]"]], ">", "0"]]]]]]]]

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

2002-12-18