Hybrid Numbers with Fibonacci and Lucas Hybrid Number Coefficients

Abstract

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.

On Higher-Order Generalized Fibonacci Hybrinomials: New Properties, Recurrence Relations and Matrix Representations

“…The set of hybrid numbers forms a non-commutative ring under addition and multiplication (please see [8]). After Özdemir's paper, hybrid numbers, whose components are defined by the homogeneous recurrence relation with constant coefficients, have been studied by a large number of researchers since 2018 (please see [9][10][11][12][13][14][15]). In [16], Kızılateş and Kone introduced Fibonacci divisor hybrid numbers that generalize the Fibonacci hybrid numbers defined by Szynal-Liana and Wloch [9].…”