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http://functions.wolfram.com/05.10.23.0010.01
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Sum[((n - m + p)!/Sqrt[(2 n + 1) (n + m)! (n - m)!])
SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] w^(n - m),
{n, m, Infinity}] == (p!/(2^(m + 1) Sqrt[Pi] m!))
(((-Sin[\[CurlyTheta]]) E^(I \[CurlyPhi]))^m/(1 - w Cos[\[CurlyTheta]])^
(p + 1)) Hypergeometric2F1[(p + 1)/2, p/2 + 1, m + 1,
-((w Sin[\[CurlyTheta]])/(1 - w Cos[\[CurlyTheta]]))^2] /;
m >= 0 && Element[p, Integers] && p >= 0 && Element[\[CurlyTheta], Reals] &&
Element[\[CurlyPhi], Reals] && Abs[w] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "m"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m", "+", "p"]], ")"]], "!"]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "!"]]]]]], RowBox[List["SphericalHarmonicY", "[", RowBox[List["n", ",", "m", ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], SuperscriptBox["w", RowBox[List["n", "-", "m"]]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["p", "!"]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], SqrtBox["\[Pi]"], " ", RowBox[List["m", "!"]]]]], " ", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]], ")"]], RowBox[List["p", "+", "1"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "+", "1"]], "2"], ",", RowBox[List[FractionBox["p", "2"], "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["w", " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], RowBox[List["1", "-", RowBox[List["w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]], ")"]], "2"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]
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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> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mi> m </mi> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> p </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["p", "2"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["m", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["w", " ", RowBox[List["sin", "(", "\[CurlyTheta]", ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", RowBox[List["cos", "(", "\[CurlyTheta]", ")"]]]]]]], ")"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> ϑ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> φ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> w </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <ci> m </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SphericalHarmonicY </ci> <ci> n </ci> <ci> m </ci> <ci> ϑ </ci> <ci> φ </ci> </apply> <apply> <power /> <ci> w </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <factorial /> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <ci> ϑ </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> φ </ci> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> p </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <sin /> <ci> ϑ </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <apply> <cos /> <ci> ϑ </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <geq /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> p </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> ϑ </ci> <reals /> </apply> <apply> <in /> <ci> φ </ci> <reals /> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "m_"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_", "+", "p_"]], ")"]], "!"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", SuperscriptBox["w_", RowBox[List["n_", "-", "m_"]]]]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "+", "m_"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_"]], ")"]], "!"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["p", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "+", "1"]], "2"], ",", RowBox[List[FractionBox["p", "2"], "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["w", " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], RowBox[List["1", "-", RowBox[List["w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]]], ")"]], "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["m", "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]]]], ")"]], RowBox[List["p", "+", "1"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "0"]], "&&", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
