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http://functions.wolfram.com/06.02.16.0020.01
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(m!! n!!)/(m + n + 2)!! ==
(1/2) (Pi/2)^((1/4) (Cos[m Pi] + Cos[n Pi] - Cos[Pi (m + n)] - 1))
Beta[m/2 + 1, n/2 + 1]
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Cell[BoxData[RowBox[List[FractionBox[RowBox[List[RowBox[List["m", "!!"]], " ", RowBox[List["n", "!!"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "+", "2"]], ")"]], "!!"]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", "2"], ")"]], RowBox[List[FractionBox["1", "4"], RowBox[List["(", " ", RowBox[List[RowBox[List["Cos", "[", RowBox[List["m", " ", "\[Pi]"]], "]"]], "+", " ", RowBox[List["Cos", "[", RowBox[List["n", " ", "\[Pi]"]], "]"]], "-", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List["m", "+", "n"]], ")"]]]], "]"]], "-", "1"]], ")"]]]]], RowBox[List["Beta", "[", RowBox[List[RowBox[List[FractionBox["m", "2"], "+", "1"]], ",", RowBox[List[FractionBox["n", "2"], "+", "1"]]]], "]"]]]]]]]]
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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> <mfrac> <mrow> <mrow> <mi> m </mi> <mo> !! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> !! </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mi> Β </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> Factorial2 </ci> <ci> m </ci> </apply> <apply> <ci> Factorial2 </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> m </ci> <pi /> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> n </ci> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <pi /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["m_", "!!"]], " ", RowBox[List["n_", "!!"]]]], RowBox[List[RowBox[List["(", RowBox[List["m_", "+", "n_", "+", "2"]], ")"]], "!!"]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", "2"], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["m", " ", "\[Pi]"]], "]"]], "+", RowBox[List["Cos", "[", RowBox[List["n", " ", "\[Pi]"]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["m", "+", "n"]], ")"]]]], "]"]], "-", "1"]], ")"]]]]], " ", RowBox[List["Beta", "[", RowBox[List[RowBox[List[FractionBox["m", "2"], "+", "1"]], ",", RowBox[List[FractionBox["n", "2"], "+", "1"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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