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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Polynomials > Fibonacci[n,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/05.12.13.0007.01









  


  








Input Form





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








Standard Form





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MathML Form







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Rule Form





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


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





2007-05-02