Generalized bivariate conditional Fibonacci and Lucas hybrinomials

Abstract

The Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. In recent years, studies related with hybrid numbers have been increased significantly. In this paper, we introduce the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Also, we present the Binet formula, generating functions, some significant identities, Catalan’s identities and Cassini’s identities of the generalized bivariate conditional Fibonacci and Lucas hybrinomials. Finally, we give more general results compared to the previous works.

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

“…which means that ( 22) is also true for n + 1. Therefore, by the principle of mathematical induction, (22) is true for all n ∈ N. The proof is complete. Lemma 6.…”

“…we know that (22) holds for n = 1. Suppose that ( 22) is true for n. Then, by the inductive hypothesis, we get…”

“…with the initial values h 1 (x) = a and h 2 (x) = bx. For some other papers, please see [20][21][22][23]. Motivated by the above papers, especially articles [17][18][19], in this paper, we define higher-order generalized Fibonacci hybrinomials.…”