Fibonacci: polynomials , numbers

Discussion in 'General Math' started by Alex.Lupas, Sep 14, 2003.

  1. Alex.Lupas

    Alex.Lupas Guest

    Fibonacci polynomials f_1(x),f_2(x),...,f_n(x),...
    are defined by f_1(x)=1 , f_2(x)=x and
    f_{n+1}(x)=x*f_n(x)+f_{n-1}(x) for n >= 1 .
    Let F_n = f_n(1) be the n-th Fibonacci number.
    I am interested to solve following questions:
    1) To find all positive integers x such that
    F_{nx} = f_n(L_x)*F_x .
    It's true that x may be any odd natural number ?

    2) I have observed that f_n(4)=F_{3n}/2 , f_n(11)=F_{5n}/5 .
    Which is the set of all positive integers p for which there
    exists a pair (m,s) of positive integers such that
    f_n(p)= F_{nm}/s ?
    Alex.Lupas, Sep 14, 2003
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  2. Alex.Lupas

    Alex.Lupas Guest

    NOTE: L_0=2 , L_1=1 , L_{x+1}=L_x + L_{x-1} when x is a positive integer.
    In other words, L_{n-1} is the n-th Lucas number.Alex/Proposer
    Alex.Lupas, Sep 14, 2003
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