On Higher-Order Generalized Fibonacci Hybrid Numbers with q-Integer Components: New Properties, Recurrence Relations, and Matrix Representations

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple.