Fibonacci and Lucas Sequences as the Principal Minors of Some Infinite Matrices

Pooya²

2009

ELA

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. Théor. Nombres Bordeaux, 14:19-41, 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 quasi-Pascal 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 non-homogeneous recurrence relations, and the leading principal minors of which form the Fibonacci sequence.

“…Now, it is obvious that our claim implies the validity of Eq. (11). Clearly β0 = 0, β1 = 1 and β2 = 2.…”

Section: Generalized Pascal Triangle Associated To An Arithmetical Or...mentioning

confidence: 98%

Section: Fibonacci and Lucas Numbers As Principal Minors Of A Quasi-pmentioning

confidence: 99%

See 1 more Smart Citation

Generalized Pascal triangles and Toeplitz matrices

Pooya²

2009

ELA

“…As a matter of fact, in many papers one may encounter certain infinite matrices, the leading principal minors of which constitute a Fibonacci (sub)sequence (see for instance [3,4,5,7,10,11]). One of the interesting examples is the infinite matrix given by:…”

Section: Ela 600mentioning

confidence: 99%

New families of integer matrices whose leading principal minors form some well-known sequences

Moghaddamfar²,

Tajbakhsh³

2011

ELA

“…Furthermore, when one is faced with matrices whose entries obey a recursive relation, inversion can be complicated. Such matrices have been studied e.g., in [1], [6], [8], [10], [11], [12] and [14]. The research done in these papers is mostly related to the determinants; rarely are the inverses discussed.…”

Section: Introductionmentioning

confidence: 99%

Certain matrices related to Fibonacci sequence having recursive entries

Salehy²,

Salehy³

2008

ELA

Let φ = (φ i ) i≥1 and ψ = (ψ i ) i≥1 be two arbitrary sequences with φ 1 = ψ 1 . Let A φ,ψ (n) denote the matrix of order n with entries a i,j , 1 ≤ i, j ≤ n, where a 1,j = φ j and a i,1 = ψ i for 1 ≤ i ≤ n, and where, where one of the sequences φ or ψ is the Fibonacci sequence (i.e., 1, 1, 2, 3, 5, 8, . . .) and the other is one of the following sequences:1, 1, . . . , 1, 0, 0, 0, . . .) ,For some sequences of the above type the inverse of A φ,ψ (n) is found. In the final part of this paper, the determinant of a generalized Pascal triangle associated to the Fibonacci sequence is found.