The Fibonacci Identities of Orthogonality

Abstract In even dimensions, the orthogonal projection onto the two dimensional space of second order recurrence sequences is particularly nice: it is a scaled Hankel matrix whose entries consist of the classical Fibonacci sequence. A new proof is given of this result, and new Fibonacci identities are derived from it. Examples are given showing that familiar Fibonacci identities can be viewed as special cases. We show that the projection in odd dimensions can be written as a rank one Lucas perturbation of a scaled Lucas Hankel matrix, from which more Fibonacci identities are derived. The Fibonacci projection Let R n denote the n dimensional vector space of all real n-tuples, and let R n (k, l) denote the vector subspace consisting of the (s i ) ∈ R n for which s i = ks i−1 + ls i−2 , for i = 3, . . . , n. (Se