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In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.

In this paper, we introduce the split k-Fibonacci and k-Lucas quaternions. We obtain the Binet formulas, generating functions and exponential generating functions of these quaternions. Moreover, we give the Catalan, Cassini and d'Ocagne identities for the split k-Fibonacci and k-Lucas quaternions.

In this paper, we define a geometric circulant matrix whose entries are the generalized Fibonacci numbers and hyperharmonic Fibonacci numbers. Then we give upper and lower bounds for the spectral norms of these matrices. MSC: Primary 15A60; 11B39; secondary 15B05

In this paper, by applying finite operator to the Horadam sequence, we define the Horadam finite operator sequences. We not only give some special cases of this new sequences but also investigate some properties of our new sequences such as recurrent relation, Binet-like formula, summation formula, and generating function, respectively. We also present closed formula for the Horadam finite operator numbers and their special cases in terms of tridiagonal determinants. Moreover, by using the tridiagonal determinant, we once again verify the recurrence relation of the Horadam finite operator numbers.

In the paper, the authors present several explicit formulas for the $(p,q,r)$-Tribonacci polynomials and generalized Tribonacci sequences in terms of the Hessenberg determinants and, consequently, derive several explicit formulas for the Tribonacci numbers and polynomials, the Tribonacci--Lucas numbers, the Perrin numbers, the Padovan (Cordonnier) numbers, the Van der Laan numbers, the Narayana numbers, the third order Jacobsthal numbers, and the third order Jacobsthal--Lucas numbers in terms of special Hessenberg determinants.

In this paper, we study norms of circulant and r-circulant matrices involving harmonic Fibonacci and hyperharmonic Fibonacci numbers. We obtain inequalities by using matrix norms. Primary 15A60; secondary 15B05; 11B39 MSC: