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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/07.06.13.0007.01









  


  










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










Standard Form





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MathML Form







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</mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <mn> 1 </mn> <mtext> </mtext> </mrow> <mroot> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mn> 4 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msubsup> <mo> ( </mo> <semantics> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; 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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["\[Nu]_", "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["\[Nu]", ",", 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"]]], "+", "\[Nu]"]], ",", 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





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