Sum[ZernikeR[m + 2 k, m, z] w^k, {k, 0, Infinity}] == (1 + w - Sqrt[1 + 2 w (1 - 2 z^2) + w^2])^m/ ((2 z w)^m Sqrt[1 + 2 w (1 - 2 z^2) + w^2]) /; Element[m, Integers] && m >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["m", "+", RowBox[List["2", " ", "k"]]]], ",", "m", ",", "z"]], "]"]], " ", SuperscriptBox["w", "k"]]]]], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "+", RowBox[List["2", "w", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", SuperscriptBox["w", "2"]]]]]], ")"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "z", " ", "w"]], ")"]], "m"], SqrtBox[RowBox[List["1", "+", RowBox[List["2", "w", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", SuperscriptBox["w", "2"]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

<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> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> k </mi> </msup> </mrow> </mrow> <mo> &#63449; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> w </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> w </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </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> <ci> ZernikeR </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <ci> m </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> w </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["m_", "+", RowBox[List["2", " ", "k_"]]]], ",", "m_", ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "k_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", "w", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", SuperscriptBox["w", "2"]]]]]], ")"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z", " ", "w"]], ")"]], "m"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["2", " ", "w", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", SuperscriptBox["w", "2"]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

2007-05-02