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http://functions.wolfram.com/05.16.17.0003.01
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NorlundB[n, n - z] == ((-1)^n/Binomial[z, n])
Sum[Binomial[n - z, n + k] Binomial[n + z, n - k] Binomial[n + k - 1, n]
NorlundB[n, k + n], {k, 0, n}] /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["NorlundB", "[", RowBox[List["n", ",", RowBox[List["n", "-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["Binomial", "[", RowBox[List["z", ",", "n"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "z"]], ",", RowBox[List["n", "+", "k"]]]], "]"]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "z"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "1"]], ",", "n"]], "]"]], RowBox[List["NorlundB", "[", RowBox[List["n", ",", RowBox[List["k", "+", "n"]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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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> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", NorlundB] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo>  </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> z </mi> </mtd> </mtr> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> z </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "z"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["k", "+", "n"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> + </mo> <mi> z </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "+", "z"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["n", "-", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", NorlundB] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> NorlundB </ci> <ci> n </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> Binomial </ci> <ci> z </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <ci> z </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> <apply> <ci> NorlundB </ci> <ci> n </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["NorlundB", "[", RowBox[List["n_", ",", RowBox[List["n_", "-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "z"]], ",", RowBox[List["n", "+", "k"]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "z"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "1"]], ",", "n"]], "]"]], " ", RowBox[List["NorlundB", "[", RowBox[List["n", ",", RowBox[List["k", "+", "n"]]]], "]"]]]]]]]], RowBox[List["Binomial", "[", RowBox[List["z", ",", "n"]], "]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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