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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Differentiation > Fractional integro-differentiation > With respect to z





http://functions.wolfram.com/05.07.20.0007.01









  


  










Input Form





D[LegendreP[n, \[Mu], 2, z], {z, \[Alpha]}] == (\[Mu]/2) z^(1 - \[Alpha]) Sum[(k + 1)! Pochhammer[1 + \[Mu]/2, k] z^k HypergeometricPFQRegularized[{{k + 2}, {-(\[Mu]/2)}, {-n, 1 + n, 1 - \[Mu]/2}}, {{k - \[Alpha] + 2}, {}, {k + 2, 1 - \[Mu]}}, -z, -(z/2)], {k, 0, Infinity}] + (Gamma[1 - \[Mu]/2] Sum[Pochhammer[-(\[Mu]/2), k] (-z)^k HypergeometricPFQRegularized[{{-n, 1 + n, 1 - \[Mu]/2}, {1}, {k + 1}}, {{1, 1 - \[Mu]}, {1 - \[Mu]/2}, {k - \[Alpha] + 1}}, 1/2, -(z/2)], {k, 0, Infinity}])/z^\[Alpha]










Standard Form





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MathML Form







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</mi> </munderover> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </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> &#945; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> P </ms> <ci> LegendreP </ci> </apply> <ms> n </ms> <ms> &#956; 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</ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ; </ms> <ms> 1 </ms> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> &#956; </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> &#945; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </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_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Mu]", " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "-", "\[Alpha]", "+", "2"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["k", "+", "2"]], ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["k", "+", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "}"]]]], "}"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["z", "2"]]]]], "]"]]]]]]]]]]]]]]










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





2001-10-29





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