A note on generalized k-order F&l hybrinomials

Abstract

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the k-order. Hybrid polynomials defined by this generalization are called k-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized k-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized k-order F&L hybrinomials and give some properties of the matrix representations.