Fibonacci numbers -identity

Discussion in 'Recreational Math' started by Alex. Lupas, Aug 11, 2005.

  1. Alex. Lupas

    Alex. Lupas Guest

    Let F_0=F_1=1 and F_{n+2}=F_{n+1}+F_{n} for n=0,1,... .

    If C(m,p):=m(m-1)...(m-p+1)/p! for p=0,1,... ,prove that

    SUM_{k=0 to k=n}C(n,k)C(n+k,k)F_k=

    =SUM_{k=0 to k=n}C(n,k)C(n+k,k)(-1)^{n-k}F_{2k} , n=0,1,... .
    Alex. Lupas, Aug 11, 2005
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