D[InverseJacobiDS[z, m], {m, n}] == (z^(-2 n - 1)/(2 n + 1)) Sum[Binomial[n, k] Pochhammer[1/2 - k, k] Pochhammer[k - n + 1/2, n - k] AppellF1[1/2 + n, 1/2 - k + n, 1/2 + k, 3/2 + n, (1 - m)/z^2, -(m/z^2)], {k, 0, n}] /; Abs[z] > 1 && Abs[m] > 1 && Element[n, Integers] && n >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "n"]], "}"]]], RowBox[List["InverseJacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "1"]]], " "]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", FractionBox[RowBox[List["1", "-", "m"]], SuperscriptBox["z", "2"]], ",", RowBox[List["-", FractionBox["m", SuperscriptBox["z", "2"]]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], ">", "1"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

<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> n </mi> </msup> <mrow> <msup> <mi> ds </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> m </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </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> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;n&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], &quot;)&quot;]], RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;k&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> InverseJacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </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> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> AppellF1 </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <abs /> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "n_"]], "}"]]]]], RowBox[List["InverseJacobiDS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", FractionBox[RowBox[List["1", "-", "m"]], SuperscriptBox["z", "2"]], ",", RowBox[List["-", FractionBox["m", SuperscriptBox["z", "2"]]]]]], "]"]]]]]]]], RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], ">", "1"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

2001-10-29