Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants

By Nathan D. Cahill, Darren A. Narayan. The Fibonacci quarterly, 2004

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described in [2] and [3] (also in [5], but with 1 and -1 on the off-diagonals, instead of i). The determinants |H(k)| are all the Fibonacci numbers Fk, starting with k = 2. In a similar family of matrices [1], the (1,1) element of H(n) is replaced with a 3. The determinants now generate the Lucas sequence Lk, starting with k = 2 (the Lucas sequence is defined by the second order recurrence L1 = 1, L2 = 3, Lk+1 = Lk + Lk−1, k ≥ 2). In this article, we extend these results to construct families of tridiagonal matrices whose determinants generate any arbitrary linear subsequence Fαk+β or Lαk+β , k = 1, 2, . . . of the Fibonacci or Lucas numbers. We then choose a specific linear subsequence of the Fibonacci numbers and use it to derive the following factorization:

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

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