A matrix approach to some second-order difference equations with sign-alternating coefficients

Anđelić, Milica; Du, Zhibin; Fonseca, Carlos M. da; Kılıç, Emrah

doi:10.1080/10236198.2019.1709180

Cited by 16 publications

(16 citation statements)

References 15 publications

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“…However, they can be traced back to the foundational work by Stieltjes [13] , [14] . On the other hand, the connection between (1.1) and the determinant of tridiagonal k -Toeplitz matrices

is quite useful, but not always conveniently explored (see, e.g., [15] , [16] , [17] , [18] ).…”

Section: The Main Resultsmentioning

confidence: 99%

On the constant coefficients of a certain recurrence relation: A simple proof

Anđelić

Fonseca

2021

Heliyon

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“…However, they can be traced back to the foundational work by Stieltjes [13] , [14] . On the other hand, the connection between (1.1) and the determinant of tridiagonal k -Toeplitz matrices

is quite useful, but not always conveniently explored (see, e.g., [15] , [16] , [17] , [18] ).…”

Section: The Main Resultsmentioning

confidence: 99%

On the constant coefficients of a certain recurrence relation: A simple proof

Anđelić

Fonseca

2021

Heliyon

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“…The Horadam sequence has been scrutinized for the past decades with many new extensions emerging in the literature. As an example, the reader is referred to [2] and the references therein.…”

Section: * Corresponding Authormentioning

confidence: 99%

“…In the spirit of Section 2 and [2,5], the aim of this section is to discuss the generalization of the Horadam sequence defined by the recurrence relation…”

Section: The Bi-periodic Horadam Sequencementioning

confidence: 99%

The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach

Anđelić

Fonseca

Yılmaz

2022

Heliyon

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In this note we consider the so-called bi-periodic Horadam sequences. Explicit formulas in terms of Chebyshev polynomials of the second kind and the determinant of some perturbed tridiagonal 2-Toeplitz matrices are established. Several illustrative examples are provided as well. The Horadam sequenceIn 1965, Alwyn F. Horadam considered the sequence {𝑤 𝑛 ≡ 𝑤 𝑛 (𝑎, 𝑏; 𝑝, 𝑞)} defined by the second-order homogeneous linear recurrence.1) with initial conditions 𝑤 0 = 𝑎 and 𝑤 1 = 𝑏 , (1.2) for arbitrary integers 𝑎 and 𝑏 [24, Section 3]. This is one of the possible extensions of the Fibonacci numbers, setting 𝑎 = 0 and 𝑏 = 𝑝 = −𝑞 = 1. Horadam studied many of its properties and other instances [21, 22, 23]. If today {𝑤 𝑛 } is familiar, in the 1960's it was a great novelty. According to A.G. Shannon [31, 32], with initial conditions 𝑈 −1 (𝑥) = 0 and 𝑈 0 (𝑥) = 1. One of the most wellknown explicit formulas for these polynomials is 𝑈 𝑛 (𝑥) = sin(𝑛 + 1)𝜃 sin 𝜃 , with 𝑥 = cos 𝜃 (0 ⩽ 𝜃 < 𝜋),for all 𝑛 = 0, 1, 2 ….

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“…Fonseca and Petronilho [22] gave explicit inverses of some tridiagonal matrices. Some of the important works on the number sequences are determinant and permanent representations [23][24][25][26][27][28].…”

Section: P Q P Q P Q P Q M M =mentioning

confidence: 99%

On (p,q)-Rogers-Szegö matrices

Sahin¹

2020

Adıyaman University Journal of Science

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In the present article, we have discussed the (,)-numbers, the Rogers-Szegő polynomial and the (,)-Rogers-Szegő polynomial and have defined the (,)-matrices and the (,)-Rogers-Szegő matrices. We have presented some algebraic properties of these matrices and have proved them. In particular, we have obtained the factorization of these matrices, their inverse matrices, as well as the matrix representations of the (,)-numbers, the Rogers-Szegő polynomials and the (,)-Rogers-Szegő polynomials.

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A matrix approach to some second-order difference equations with sign-alternating coefficients

Cited by 16 publications

References 15 publications

On the constant coefficients of a certain recurrence relation: A simple proof

On the constant coefficients of a certain recurrence relation: A simple proof

The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach

On (p,q)-Rogers-Szegö matrices

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