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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Polynomials > LaguerreL[n,lambda,z] > Differentiation > Symbolic differentiation > With respect to lambda





http://functions.wolfram.com/05.08.20.0009.02









  


  








Input Form





D[LaguerreL[n, \[Lambda], z], {\[Lambda], m}] == (((-1)^n z^n)/n!) KroneckerDelta[m] + (1/n!) Sum[((Pochhammer[-n, k] z^k)/k!) Sum[(-1)^(j + n - k) StirlingS1[n - k, j] Pochhammer[j - m + 1, m] (\[Lambda] + k + 1)^(j - m), {j, 1, n - k}], {k, 0, n}] /; 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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> m </mi> </msup> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#955; </mi> <mi> m </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> m </mi> </msub> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;n&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;m&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;m&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> &#955; </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> &#955; </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <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> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Lambda]_", ",", "m_"]], "}"]]]]], RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", "n"]]], ")"]], " ", RowBox[List["KroneckerDelta", "[", "m", "]"]]]], RowBox[List["n", "!"]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n", "-", "k"]]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["j", "-", "m", "+", "1"]], ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "k", "+", "1"]], ")"]], RowBox[List["j", "-", "m"]]]]]]]]], RowBox[List["k", "!"]]]]], RowBox[List["n", "!"]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29