A generalization of Fibonacci and Lucas matrices

We define the matrix Un(a,b,s) of type ss, whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix Fn(a,b,s), whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices (s=0s=0 and s=1s=1). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix Un(a,b,s) is derived. In partial case we get the inverse of the generalized Fibonacci matrix Fn(a,b,0) and later known results from [Gwang-Yeon Lee, Jin-Soo Kim, Sang-Gu Lee, Factorizations and eigenvalues of Fibonaci and symmetric Fibonaci matrices, Fibonacci Quart. 40 (2002) 203–211; P. Staˇnicaˇ, Cholesky factorizations of matrices associated with rr-order recurrent sequences, Electron. J. Combin. Number Theory 5 (2) (2005) #A16] and [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)]. Correlations between the matrices Un(a,b,s), Fn(a,b,s) and the generalized Pascal matrices are considered. In the case a=0,b=1a=0,b=1 we get known result for Fibonacci matrices [Gwang-Yeon Lee, Jin-Soo Kim, Seong-Hoon Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math. 130 (2003) 527–534]. Analogous result for Lucas matrices, originated in [Z. Zhang, Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. (in press)], can be derived in the partial case a=2,b=1a=2,b=1. Some combinatorial identities involving generalized Fibonacci numbers are derived.