Generalized Pascal triangles and Toeplitz matrices. (English) Zbl 1190.15006
Summary: The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles [see R. Bacher, J. Théor. Nombres Bordeaux 14, No. 1, 19–41 (2002; Zbl 1023.11010)]. This article presents a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix, and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a Toeplitz matrix. This equality allows for the evaluation of a few determinants of generalized Pascal matrices associated with certain sequences. In particular, families of quasi-Pascal matrices are obtained whose leading principal minors generate any arbitrary linear subsequences \(({\mathcal F}_{nr+s})_{n\geq 1}\) or \(({\mathcal L}_{nr+s})_{n\geq 1}\) of the Fibonacci or Lucas sequence. New matrices are constructed whose entries are given by certain linear non-homogeneous recurrence relations, and the leading principal minors of which form the Fibonacci sequence.
MSC:
15A15 | Determinants, permanents, traces, other special matrix functions |
15B36 | Matrices of integers |
15B05 | Toeplitz, Cauchy, and related matrices |
11C20 | Matrices, determinants in number theory |
11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
15A23 | Factorization of matrices |