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 Fibonacci

 http://functions.wolfram.com/05.12.13.0009.01

 Input Form

 Derivative[2][w][z] + (1/(4 + a^2 r^(2 z))) (-2 a^2 r^(2 z) (-Log[r] + Log[s]) - 4 (Log[r] + 2 Log[s])) Derivative[1][w][z] + (1/(4 + a^2 r^(2 z))) (4 Log[s] (Log[r] + Log[s]) - a^2 r^(2 z) ((-1 + n^2) Log[r]^2 + 2 Log[r] Log[s] - Log[s]^2)) w[z] == 0 /; w[z] == Subscript[c, 1] s^z Fibonacci[n, a r^z] + Subscript[c, 2] (s^z/(4 + a^2 r^(2 z))^(1/4)) LegendreP[-(1/2) + n, 1/2, 2, (I a r^z)/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "+", RowBox[List[FractionBox["1", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r", "]"]]]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r", "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s", "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "s", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r", "]"]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["n", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r", "]"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "r", "]"]], " ", RowBox[List["Log", "[", "s", "]"]]]], "-", SuperscriptBox[RowBox[List["Log", "[", "s", "]"]], "2"]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["s", "z"], RowBox[List["Fibonacci", "[", RowBox[List["n", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", FractionBox[SuperscriptBox["s", "z"], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], SuperscriptBox["r", RowBox[List["2", "z"]]]]]]], ")"]], RowBox[List["1", "/", "4"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], ",", FractionBox["1", "2"], ",", "2", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["r", "z"]]], "2"]]], "]"]]]]]]]]]]]]

 MathML Form

 w ′′ ( z ) + - 2 a 2 ( log ( s ) - log ( r ) ) r 2 z - 4 ( log ( r ) + 2 log ( s ) ) a 2 r 2 z + 4 w ( z ) + 1 a 2 r 2 z + 4 ( 4 log ( s ) ( log ( r ) + log ( s ) ) - a 2 r 2 z ( ( n 2 - 1 ) log 2 ( r ) + 2 log ( s ) log ( r ) - log 2 ( s ) ) ) w ( z ) 0 /; w ( z ) c 1 s z F TagBox["F", Fibonacci] n ( a r z ) + c 2 s z a 2 r 2 z + 4 4 P TagBox["P", LegendreP] n - 1 2 1 2 ( 1 2 a r z TagBox[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["r", "z"]]], HoldComplete[LegendreP, 2]] ) Condition z 2 w z -2 a 2 s -1 r r 2 z -1 4 r 2 s a 2 r 2 z 4 -1 z w z 1 a 2 r 2 z 4 -1 4 s r s -1 a 2 r 2 z n 2 -1 r 2 2 s r -1 s 2 w z 0 w z Subscript c 1 s z Fibonacci n a r z Subscript c 2 s z a 2 r 2 z 4 1 4 -1 LegendreP n -1 1 2 1 2 2 1 2 a r z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r_", "]"]]]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], RowBox[List["4", "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "s_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["n_", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "r_", "]"]], " ", RowBox[List["Log", "[", "s_", "]"]]]], "-", SuperscriptBox[RowBox[List["Log", "[", "s_", "]"]], "2"]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]], RowBox[List["4", "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", FractionBox[RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], ",", FractionBox["1", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], RowBox[List["1", "/", "4"]]]]]]]]]]]]]]

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

 2007-05-02