html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Binomial

 http://functions.wolfram.com/06.03.29.0003.01

 Input Form

 Subscript[d, n][\[Lambda] + \[Alpha] + \[Beta]] Sum[(Subscript[d, n - k][\[Lambda]]/Subscript[d, k][\[Alpha] + \[Beta]]) Abs[Sum[Subscript[a, j] Subscript[b, k - j], {j, 0, k}]]^2, {k, 0, n}] <= Sum[(Subscript[d, n - k][\[Lambda] + \[Beta]]/ Subscript[d, k][\[Alpha]]) Abs[Subscript[a, k]]^2, {k, 0, n}] Sum[(Subscript[d, n - k][\[Lambda] + \[Alpha]]/Subscript[d, k][\[Beta]]) Abs[Subscript[b, k]]^2, {k, 0, n}] /; Subscript[d, k][\[Alpha]] == Binomial[k + \[Alpha] - 1, k] && Element[\[Alpha], Reals] && \[Alpha] > 0 && Element[\[Beta], Reals] && \[Beta] > 0 && Element[\[Lambda], Reals] && \[Lambda] >= 0 && Max[Abs[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n]}]] > 0 && Max[Abs[{Subscript[b, 1], Subscript[b, 2], \[Ellipsis], Subscript[b, n]}]] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]", "+", "\[Beta]"]], "]"]], 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["\[Alpha]", "+", "\[Beta]"]], "]"]]], SuperscriptBox[RowBox[List["Abs", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "j"], SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]], "]"]], "2"]]]]]]], "\[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]", "+", "\[Beta]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["a", "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]", "+", "\[Alpha]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Beta]", "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["b", "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["\[Alpha]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Alpha]", ">", "0"]], "\[And]", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Beta]", ">", "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["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "n"]]], "}"]], "]"]], "]"]], ">", "0"]]]]]]]]

 MathML Form

 d n ( α + β + λ ) k = 0 n d n - k ( λ ) d k ( α + β ) "\[LeftBracketingBar]" j = 0 k a j b k - j "\[RightBracketingBar]" 2 ( k = 0 n d n - k ( β + λ ) d k ( α ) "\[LeftBracketingBar]" a k "\[RightBracketingBar]" 2 ) k = 0 n d n - k ( α + λ ) d k ( β ) "\[LeftBracketingBar]" b k "\[RightBracketingBar]" 2 /; d k ( α ) ( k + α - 1 k ) 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]] α TagBox["\[DoubleStruckCapitalR]", Function[Reals]] + β TagBox["\[DoubleStruckCapitalR]", Function[Reals]] + λ λ 0 max ( "\[LeftBracketingBar]" { a 1 , a 2 , , a n } "\[RightBracketingBar]" ) > 0 max ( "\[LeftBracketingBar]" { b 1 , b 2 , , b n } "\[RightBracketingBar]" ) > 0 Condition Subscript d n α β λ k 0 n Subscript d n -1 k λ Subscript d k α β -1 j 0 k Subscript a j Subscript b k -1 j 2 k 0 n Subscript d n -1 k β λ Subscript d k α -1 Subscript a k 2 k 0 n Subscript d n -1 k α λ Subscript d k β -1 Subscript b k 2 Subscript d k α Binomial k α -1 k α SuperPlus β SuperPlus λ λ 0 Subscript a 1 Subscript a 2 Subscript a n 0 Subscript b 1 Subscript b 2 Subscript b n 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]", "+", "\[Beta]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", "\[Lambda]", "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "j"], " ", SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Alpha]", "+", "\[Beta]"]], "]"]]]]]]], "\[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]", "+", "\[Beta]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["a", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["b", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Beta]", "]"]]]]]]]]], "/;", 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["\[Lambda]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "n"]]], "}"]], "]"]], "]"]], ">", "0"]]]]]]]]

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

 2002-12-18