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variants of this functions
Factorial2




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial2[n] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/06.02.16.0018.01









  


  








Input Form





n!!/m!! == 2^((n - m)/2) (2/Pi)^((1/4) Cos[m Pi] - (1/4) Cos[n Pi]) Pochhammer[m/2 + 1, (n - m)/2]








Standard Form





Cell[BoxData[RowBox[List[FractionBox[RowBox[List["n", "!!"]], RowBox[List["m", "!!"]]], "\[Equal]", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["n", "-", "m"]], "2"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", "\[Pi]"], ")"]], RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Cos", "[", RowBox[List["m", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Cos", "[", RowBox[List["n", " ", "\[Pi]"]], "]"]]]]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["m", "2"], "+", "1"]], ",", FractionBox[RowBox[List["n", "-", "m"]], "2"]]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> !! </mo> </mrow> <mrow> <mi> m </mi> <mo> !! </mo> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <msup> <mn> 2 </mn> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[FractionBox[&quot;m&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], FractionBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;m&quot;]], &quot;2&quot;]], Pochhammer] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> Factorial2 </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> Factorial2 </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> n </ci> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </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> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["n_", "!!"]], RowBox[List["m_", "!!"]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["n", "-", "m"]], "2"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", "\[Pi]"], ")"]], RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Cos", "[", RowBox[List["m", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Cos", "[", RowBox[List["n", " ", "\[Pi]"]], "]"]]]]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["m", "2"], "+", "1"]], ",", FractionBox[RowBox[List["n", "-", "m"]], "2"]]], "]"]]]]]]]]


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





2001-10-29