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    A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers
    (PERGAMON-ELSEVIER SCIENCE LTD, 2009) Nalli, Ayse; Civciv, Haci
    In this paper, we construct the symmetric tridiagonal family of matrices M(-alpha-beta)(k), k = 1, 2,... whose determinants form any linear subsequence of the Fibonacci numbers. Furthermore, we construct the symmetric tridiagonal family of matrices T(-alpha-beta)(k), k = 1, 2,... whose determinants form any linear subsequence of the Lucas numbers. Thus we give a generalization of the presented in Cahill and Narayan (2004) [Cahill ND, Narayan DA. Fibonacci and Lucas numbers as tridiagonal matrix determinants. Fibonacci Quart 2004;42(3):216-21]. (C) 2007 Elsevier Ltd. All rights reserved.
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    On generalized Fibonacci and Lucas polynomials
    (PERGAMON-ELSEVIER SCIENCE LTD, 2009) Nalli, Ayse; Haukkanen, Pentti
    Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q(h)(x) that generalizes the Q-matrix [GRAPHICS] whose powers generate the Fibonacci numbers. (C) 2009 Elsevier Ltd. All rights reserved.