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http://functions.wolfram.com/09.25.20.0007.01
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D[JacobiCD[z, m], {z, n}] == JacobiCD[z, m] KroneckerDelta[n] +
(m - 1) Sum[Binomial[n - 1, j] D[JacobiSD[z, m], {z, j}]
D[JacobiND[z, m], {z, n - 1 - j}], {j, 0, n - 1}] /;
Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["JacobiCD", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["JacobiCD", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "j"]], "}"]]], RowBox[List["JacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", RowBox[List["n", "-", "1", "-", "j"]]]], "}"]]], RowBox[List["JacobiND", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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<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> ∂ </mo> <mi> n </mi> </msup> <mrow> <mi> cd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mrow> <mrow> <mi> cd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> j </mi> </msup> <mrow> <mi> sd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> nd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> JacobiCD </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCD </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> j </ci> </degree> </bvar> <apply> <ci> JacobiSD </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </degree> </bvar> <apply> <ci> JacobiND </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["JacobiCD", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["JacobiCD", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", "j"]], "}"]]]]], RowBox[List["JacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", RowBox[List["n", "-", "1", "-", "j"]]]], "}"]]]]], RowBox[List["JacobiND", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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