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Binomial




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Binomial[n,k] > Summation > Multiple sums





http://functions.wolfram.com/06.03.23.0053.01









  


  








Input Form





Sum[(-1)^(k + l) Binomial[n + \[Alpha], k] Binomial[n + \[Beta], n - k] Binomial[m + \[Alpha], l] Binomial[m + \[Beta], m - l] Beta[n + m + \[Alpha] - k - l + 1, \[Beta] + k + l + 1], {k, 0, n}, {l, 0, m}] == KroneckerDelta[n, m] (1/(2 n + \[Alpha] + \[Beta] + 1)) (Beta[n + \[Alpha] + 1, n + \[Beta] + 1]/ Beta[n + 1, n + \[Alpha] + \[Beta] + 1]) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0








Standard Form





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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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#945; </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;\[Alpha]&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#946; </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[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;\[Beta]&quot;]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;k&quot;]], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> m </mi> <mo> + </mo> <mi> &#945; </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> l </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;\[Alpha]&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;l&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> m </mi> <mo> + </mo> <mi> &#946; </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> m </mi> <mo> - </mo> <mi> l </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;\[Beta]&quot;]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;l&quot;]], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> l </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> <mo> + </mo> <mi> &#946; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> </msub> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> &#945; </mi> <mo> + </mo> <mi> &#946; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#946; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mtext> </mtext> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#945; </mi> <mo> + </mo> <mi> &#946; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <ci> l </ci> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <ci> &#945; </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <ci> &#946; </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> m </ci> <ci> &#945; </ci> </apply> <ci> l </ci> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> m </ci> <ci> &#946; </ci> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> <ci> m </ci> <ci> n </ci> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> k </ci> <ci> l </ci> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> &#945; </ci> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Beta </ci> <apply> <plus /> <ci> n </ci> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Beta </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <ci> &#945; </ci> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18