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variants of this functions
LaguerreL






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LaguerreL[nu,z] > Differentiation > Low-order differentiation > With respect to nu





http://functions.wolfram.com/07.02.20.0003.01









  


  










Input Form





D[LaguerreL[\[Nu], z], {\[Nu], 2}] == (1/\[Nu]!) Sum[((z^k Pochhammer[-\[Nu], k])/k!^2) (Gamma[1 + \[Nu]] PolyGamma[k - \[Nu]] PolyGamma[1 + \[Nu]] + \[Nu]! (PolyGamma[k - \[Nu]]^2 - PolyGamma[k - \[Nu]] (3/\[Nu] + 2 Pi Cot[\[Nu] Pi] + 3 PolyGamma[\[Nu]]) + PolyGamma[1, k - \[Nu]])), {k, 0, Infinity}] + (-Pi^2 + 2 Pi Cot[\[Nu] Pi] PolyGamma[1 + \[Nu]] + PolyGamma[1 + \[Nu]]^2 + PolyGamma[1, 1 + \[Nu]]) LaguerreL[\[Nu], z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "2"]], "}"]]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["\[Nu]", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " "]], RowBox[List[SuperscriptBox[RowBox[List["k", "!"]], "2"], " "]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], "+", RowBox[List[RowBox[List["\[Nu]", "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "2"], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["3", "\[Nu]"], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["k", "-", "\[Nu]"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "\[Nu]"]]]], "]"]]]], ")"]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " "]]]]]]]]










MathML Form







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</mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> LaguerreL </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> LaguerreL </ci> <ci> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "2"]], "}"]]]]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], "+", RowBox[List[RowBox[List["\[Nu]", "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "2"], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["3", "\[Nu]"], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["k", "-", "\[Nu]"]]]], "]"]]]], ")"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]], RowBox[List["\[Nu]", "!"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "\[Nu]"]]]], "]"]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]]]]










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





2001-10-29





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