Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences

Abstract

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.

k-Order Fibonacci Polynomials on AES-Like Cryptology

“…There are a lot of generalizations of Fibonacci numbers satisfied and studied by some authors. For more information one can see in [4][5][6][7][8]. The Fibonacci Q-matrix is defined in [9,10] as follows:…”

“…Kizilates et al studied a new generalization of convolved (p, q) −Fibonacci and (p, q) −Lucas polynomials in [4]. Also, Qi et al gave a closed formula for the Horadam polynomials in terms of a tridiagonal determinant in 2019 in [15] and Kizilates et al defined several determinantel expressions of generalized tribonacci polynomials and sequences in [5]. In [6], Kizilates et al introduced new families of three-variable polynomials coupled with well-known polynomials and numbers in 2019.…”

“…Throughout this section, we take m = 5 for next process. Since m = 5, we consider the finite Galois field containing 32 elements in this algorithm and this Galois field is denoted as GF 2 5 . Note that there are exactly 2 5 = 32 such polynomials.…”

“…Since m = 5, we consider the finite Galois field containing 32 elements in this algorithm and this Galois field is denoted as GF 2 5 . Note that there are exactly 2 5 = 32 such polynomials. The set of these 32 polynomials is the finite field GF 2 5 .…”

“…Note that there are exactly 2 5 = 32 such polynomials. The set of these 32 polynomials is the finite field GF 2 5 . Each elements of this polynomials correspond to one letter of the alphabet.…”