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Paper IPM / M / 11159  


Abstract:  
The purpose of this article is to study determinants of
matrices which are known as generalized Pascal triangles (see R.
Bacher. Determinants of matrices related to the Pascal triangle.
J. Theor. Nombres Bordeaux, 14:1941, 2002). 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 quasiPascal
matrices are obtained whose leading principal minors generate any
arbitrary linear subsequences (F_{nr+s})_{n ≥ 1} or
(L_{nr+s})_{n ≥ 1} of the Fibonacci or Lucas
sequence. New matrices are constructed whose entries are given by
certain linear nonhomogeneous recurrence relations, and the
leading principal minors of which form the Fibonacci sequence.
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