Sum[(-1)^k (1/(\[Nu] - k) - 1/(k + \[Nu] + 1)) LegendreP[k, x], {k, 0, Infinity}] == (Pi/Sin[\[Nu] Pi]) LegendreP[\[Nu], x] /; Element[x, Reals] && Inequality[-1, Less, x, LessEqual, 1] && !Element[\[Nu], Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["\[Nu]", "-", "k"]]], "-", FractionBox["1", RowBox[List["k", "+", "\[Nu]", "+", "1"]]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["k", ",", "x"]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "<", "x", "\[LessEqual]", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> k </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &#8804; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <ci> x </ci> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> &#957; </ci> <pi /> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> -1 </cn> <lt /> <ci> x </ci> <leq /> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <ci> &#957; </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k_"], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["\[Nu]_", "-", "k_"]]], "-", FractionBox["1", RowBox[List["k_", "+", "\[Nu]_", "+", "1"]]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["k_", ",", "x_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "x", "\[LessEqual]", "1"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

2001-10-29