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Math Help - Lucas Sequences

  1. #1
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    Lucas Sequences

    I read some facts about Lucas sequences, but they were not proven, can you help me?

    Let P and Q be integers. Define u_{n+1} = Pu_n - Qu_{n-1} with u_0 = 0, u_1 = 1.
    Similar define v_{n+1} = Pv_n - Qv_{n-1} with v_0 = 2, v_1 = P.

    Show:
    (1) 2u_n = Pu_{n-1} + v_{n-1} (Managed to prove by induction myself)
    (2) 2v_n = (P^2-4Q)u_{n-1} + Pv_{n-1}
    (3) u_{2n} = u_{n}v_{n}
    (4) v_{2n} = v_{n}^2 - 2Q^{n}
    (5) u_{n+m} = u_{n}u_{m+1} -Qu_{m}u_{n-1} = (u_{n}v_{m} + u_{m}v_{n})/2
    (6) v_{n+m} = v_{n}v_{m} -Q^{m}v_{n-m}

    I would be thankful for any proven statement .
    Last edited by EinStone; May 20th 2010 at 04:55 PM.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    If you assume  n \leq m you could prove (5) and (6) by induction where your base cases are when  n=m , which are (3) and (4) respectively.
    Last edited by chiph588@; May 21st 2010 at 06:47 AM.
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