No abstract

In the literature one may encounter certain infinite tridiagonal matrices, the principal minors of which, constitute the Fibonacci or Lucas sequence. The major purpose of this article is to find new infinite matrices with this property. It is interesting to mention that the matrices found are not tridiagonal which have been investigated before. Furthermore, we introduce the sequences composed of Fibonacci and Lucas k-numbers for the positive integer k and we construct the infinite matrices the principal minors of which generate these sequences.

Let λ = (λi)i ≥ 1, μ = (μi)i ≥ 1, ν = (νi)i ≥ 1, ω = (ωi)i ≥ 1 and ψ = (ψi)i ≥ 1 be given sequences, and let (ai,j)i,j ≥ 1 be the doubly indexed sequence given by the recurrence [Formula: see text](i ≥ 3, j ≥ 2), with various choices for the two first rows a1,j, a2,j and first column ai,1. Note that the coefficients depend on the row index only. In this article we study the principal minors of doubly indexed sequences (ai,j)i,j ≥ 1 for certain sequences and certain initial conditions. Moreover, let (bi,j)i,j ≥ 1 be the doubly indexed sequence given by the recurrence [Formula: see text] with various choices for the first row b1,j and first column bi,1. We also study the principal minors of doubly indexed sequence (bi,j)i,j ≥ 1.

This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this type.

Abstract. The purpose of this article is to prove several evaluations of determinants of matrices, the entries of which are given by the recurrence a i,j = a i−1,j−1 +a i−1,j , i, j ≥ 2, with various choices for the first row a 1,j and first column a i,1 .