Decomposition of some Hankel matrices generated by the generalized rencontres polynomials

Abstract The aim of this paper is to give new results about factorizations of the Fibonacci numbers F n and the Lucas numbers L n. These numbers are defined by the second order recurrence relation a n+2 = a n+1+a n with the initial terms F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].

Lucas number

Lucas sequence

Recurrence relation

Pisano period

10.2478/s12175-012-0020-2

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Citations (6)

Diagonal recurrence relations for the Stirling numbers of the first kind

arXiv (Cornell University) (2013)

Feng Qi

In the paper, the author presents diagonal recurrence relations for the Stirling numbers of the first kind. As by-products, the author also recovers three explicit formulas for special values of the Bell polynomials of the second kind.

Bell polynomials

Recurrence relation

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Citations (17)

Diagonal recurrence relations for the Stirling numbers of the first kind

arXiv (Cornell University) (2013)

Feng Qi

Recurrence relation

Bell polynomials

10.48550/arxiv.1310.5920

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Citations (15)

A Class of Eigen Polynomials of Symmetry Three Diagonal Matrixes

Journal of Heze Teachers College (1996)

E Teacher

In this article, We have given a recurrence formula on a class of eigenpolynomials of symmetry three diagnomal matrixes, and a recurrence relation of each polynomial cofficient in the eigenpolyomial sequence. In the meantime, this sequence is orthogonal and a series of the divisible properties of the three diagonal matrkes are proved.

Sequence (biology)

Recurrence relation

Main diagonal

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Comments on “A note on a three-term recurrence for a tridiagonal matrix”

Applied Mathematics and Computation (2005)

Davod Khojasteh Salkuyeh

Tridiagonal matrix algorithm

Recurrence relation

Matrix (chemical analysis)

Block matrix

Matrix splitting

Band matrix

10.1016/j.amc.2005.09.033

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Citations (22)

A note on a three-term recurrence for a tridiagonal matrix

Applied Mathematics and Computation (2002)

Moawwad El-Mikkawy

Tridiagonal matrix algorithm

Recurrence relation

Positive definiteness

Matrix splitting

Matrix (chemical analysis)

10.1016/s0096-3003(02)00212-6

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Citations (51)

On the factorization and the inverse of a tridiagonal matrix

Journal of Discrete Mathematical Sciences and Cryptography (2021)

Boutaina Talibi Ahmed Driss Aiat Hadj Driss Sarsri

In this paper we present a new decomposition of the tridiagonal matrices using the CL factorization to calculate the inverse of these matrices. And we suggest, two algorithms to compare the running time for each one in goal to define the efficiency of our new decomposition. As a consequence, we get some interesting results.

LU decomposition

Tridiagonal matrix algorithm

Matrix (chemical analysis)

10.1080/09720529.2021.1873256

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Citations (2)

Appendix: Tridiagonal Matrix Algorithm

A. Chandrasekar

A system of simultaneous algebraic equations with nonzero coefficients only on the main diagonal, the lower diagonal, and the upper diagonal is called a tridiagonal system of equations. Consider a tridiagonal system of N equations with N unknowns, u1, u2, u3,… uN as follows:

Tridiagonal matrix algorithm

Band matrix

Algebraic equation

Main diagonal

Matrix (chemical analysis)

Diagonal matrix

Block matrix

10.1017/9781009119238.019

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