ar X iv : 0 90 1 . 25 97 v 1 [ m at h . R A ] 1 9 Ja n 20 09 GENERALIZED PASCAL TRIANGLES AND TÖEPLITZ MATRICES ∗

The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Töeplitz matrix and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a Töeplitz matrix. This equality allows us to evaluate a few determinants of generalized Pascal matrices associated to certain sequences. In particular, we obtain families of quasi-Pascal matrices whose principal minors generate any arbitrary linear subsequences F (nr + s) or L(nr + s), (n = 1, 2, 3,. . .) of Fibonacci or Lucas sequence.