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variants of this functions
BellB




Mathematica Notation

Traditional Notation









Polynomials > BellB[n,z] > Differentiation > Low-order differentiation





http://functions.wolfram.com/05.15.20.0002.01









  


  








Input Form





D[BellB[n, z], {z, 2}] == BellB[n, z] - ((1 + 2 z)/z^2) BellB[1 + n, z] + (1/z^2) BellB[2 + n, z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["BellB", "[", RowBox[List["n", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["BellB", "[", RowBox[List["n", ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "z"]]]], SuperscriptBox["z", "2"]], " ", RowBox[List["BellB", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox["z", "2"]], RowBox[List["BellB", "[", RowBox[List[RowBox[List["2", "+", "n"]], ",", "z"]], "]"]]]]]]]]]]








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> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BellB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <msub> <mo> &#63449; </mo> <mi> n </mi> </msub> <mrow> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BellB] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BellB] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#8706; </ms> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> BellB </ci> </apply> <ms> n </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#63449; </ms> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> z </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> BellB </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> BellB </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "2"]], "}"]]]]], RowBox[List["BellB", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["BellB", "[", RowBox[List["n", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["BellB", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List["BellB", "[", RowBox[List[RowBox[List["2", "+", "n"]], ",", "z"]], "]"]], SuperscriptBox["z", "2"]]]]]]]]


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





2007-05-02