A generalization of Fibonacci and Lucas matrices

Abstract

We define the matrix U n ( a , b , s ) of type s , whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix F n ( a , b , s ) , 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 U n ( a , b , s ) is derived. In partial case we get the inverse of the generalized Fibonacci matrix F n ( 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 r -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 U n ( a , b , s ) , F n ( a , b , s ) and the generalized Pascal matrices are considered. In the case a = 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 = 1 . Some combinatorial identities involving generalized Fibonacci numbers are derived.