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A generalization of Fibonacci and Lucas matrices. (English) Zbl 1188.15024

We define the matrix \({\mathcal U}^{(a,b,s)}_n\) of type \(s\), whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix \({\mathcal F}^{(a,b,s)}_n\), whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices (\(s=0\) and \(s=1\)). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix \({\mathcal U}^{(a,b,s)}_n\) is derived. In partial case we get the inverse of the generalized Fibonacci matrix \({\mathcal F}^{(a,b,0)}_n\) and later known results from G.-Y. Lee, J.-S. Kim and S.-G. Lee [Fibonacci Q. 40, No. 3, 203–211 (2002; Zbl 1079.11012)]; P. Stǎnicǎ [Integers 5, No. 2, Paper A16, 11 p., electronic only (2005; Zbl 1139.11307)], and Z. Zhang and Y. Zhang [Indian J. Pure Appl. Math. 38, No. 5, 457–465 (2007; Zbl 1149.15014)].
Correlations between the matrices \({\mathcal U}^{(a,b,s)}_n\), \({\mathcal F}^{(a,b,s)}_n\) and the generalized Pascal matrices are considered. In the case \(a=0\), \(b=1\) we get known result for Fibonacci matrices [cf. G.-Y. Lee, J.-S. Kim and S.-H. Cho, Discrete Appl. Math. 130, No. 3, 527–534 (2003; Zbl 1020.05016)]. Analogous result for Lucas matrices, originated in [Z. Zhang and Y. Zhang, loc.cit.], can be derived in the partial case \(a=2\), \(b=1\). Some combinatorial identities involving generalized Fibonacci numbers are derived.

MSC:

15B36 Matrices of integers
11C20 Matrices, determinants in number theory
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A19 Combinatorial identities, bijective combinatorics
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