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 SphericalHarmonicY

 http://functions.wolfram.com/05.10.23.0011.01

 Input Form

 Sum[(((n + m - p)! (n - m + p - 1)!)/(n! Sqrt[(2 n + 1) (n - m)! (n + m)!])) SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] w^(n - m), {n, m, Infinity}] == (((2 m - p)! (p - 1)!)/(2^(m + 1) Sqrt[Pi] m!^2)) ((-Sin[\[CurlyTheta]]) E^(I \[CurlyPhi]))^m Hypergeometric2F1[p, 2 m - p + 1, m + 1, (1 - w - Sqrt[1 - 2 w Cos[\[CurlyTheta]] + w^2])/2] Hypergeometric2F1[p, 2 m - p + 1, m + 1, (1 + w - Sqrt[1 - 2 w Cos[\[CurlyTheta]] + w^2])/2] /; m >= 0 && Element[p, Integers] && p >= 0 && p <= 2 m && Element[\[CurlyTheta], Reals] && Element[\[CurlyPhi], Reals] && Abs[w] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "m"]], "\[Infinity]"], " ", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m", "-", "p"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m", "+", "p", "-", "1"]], ")"]], "!"]]]], RowBox[List[RowBox[List["n", "!"]], 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[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "m"]], "-", "p"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], "!"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "!"]], ")"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["2", "m"]], "-", "p", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["1", "-", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", "w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", SuperscriptBox["w", "2"]]]]]], "2"]]], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["2", "m"]], "-", "p", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", FractionBox[RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", "w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", SuperscriptBox["w", "2"]]]]]], "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[LessEqual]", RowBox[List["2", "m"]]]], "\[And]", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]

 MathML Form

 n = m ( n + m - p ) ! ( n - m + p - 1 ) ! n ! ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! Y n m ( ϑ , φ ) w n - m ( 2 m - p ) ! ( p - 1 ) ! 2 m + 1 π m ! 2 ( - sin ( ϑ ) φ ) m 2 F 1 ( p , 2 m - p + 1 ; m + 1 ; 1 - w - 1 - 2 w cos ( ϑ ) + w 2 2 ) 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["p", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "m"]], "-", "p", "+", "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[FractionBox[RowBox[List["1", "-", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", "w", " ", FormBox[RowBox[List["cos", "(", "\[CurlyTheta]", ")"]], TraditionalForm]]], "+", SuperscriptBox["w", "2"]]]]]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 2 F 1 ( p , 2 m - p + 1 ; m + 1 ; 1 + w - 1 - 2 w cos ( ϑ ) + w 2 2 ) 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["p", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "m"]], "-", "p", "+", "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[FractionBox[RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", "w", " ", FormBox[RowBox[List["cos", "(", "\[CurlyTheta]", ")"]], TraditionalForm]]], "+", SuperscriptBox["w", "2"]]]]]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; m 0 p p 2 m ϑ TagBox["\[DoubleStruckCapitalR]", Function[Reals]] φ TagBox["\[DoubleStruckCapitalR]", Function[Reals]] "\[LeftBracketingBar]" w "\[RightBracketingBar]" < 1 Condition n m n m -1 p n -1 m p -1 n 2 n 1 n -1 m n m 1 2 -1 SphericalHarmonicY n m ϑ φ w n -1 m 2 m -1 p p -1 2 m 1 1 2 m 2 -1 -1 ϑ φ m Hypergeometric2F1 p 2 m -1 p 1 m 1 1 -1 w -1 1 -1 2 w ϑ w 2 1 2 2 -1 Hypergeometric2F1 p 2 m -1 p 1 m 1 1 w -1 1 -1 2 w ϑ w 2 1 2 2 -1 m 0 p p 2 m ϑ φ w 1 [/itex]

 Rule Form

 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[RowBox[List["(", RowBox[List["n_", "+", "m_", "-", "p_"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "m_", "+", "p_", "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["SphericalHarmonicY", "[", RowBox[List["n_", ",", "m_", ",", "\[CurlyTheta]_", ",", "\[CurlyPhi]_"]], "]"]], " ", SuperscriptBox["w_", RowBox[List["n_", "-", "m_"]]]]], RowBox[List[RowBox[List["n_", "!"]], " ", 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["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "p"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[CurlyPhi]"]]]]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["2", " ", "m"]], "-", "p", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", SuperscriptBox["w", "2"]]]]]], ")"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["2", " ", "m"]], "-", "p", "+", "1"]], ",", RowBox[List["m", "+", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "w", " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", SuperscriptBox["w", "2"]]]]]], ")"]]]]]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "!"]], ")"]], "2"]]]], "/;", RowBox[List[RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[LessEqual]", RowBox[List["2", " ", "m"]]]], "&&", RowBox[List["\[CurlyTheta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[CurlyPhi]", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29