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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.
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    On Unitary Analogs of GCD Reciprocal LCM Matrices
    (Taylor & Francis Ltd, 2010) Haukkanen, Pentti; Ilmonen, Pauliina; Nallı, Ayşe; Sillanpaa, Juha
    A divisor d is an element of Z(+) of n is an element of Z(+) is said to be a unitary divisor of n if (d, n/d)=1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM.