A matrix approach to generalized Bernoulli–Fibonacci polynomials of order m and applications

We know that the matrices provide a flexible framework to study combinatorial structures. In fact, the generalized Fibonacci matrices allow us to develop the applications to coding theory. In the beginning of this work, a new family of generalized Bernoulli–Fibonacci polynomials of order m is introduced followed by investigating various properties associated with this polynomial class, as well as its relationships with other polynomial families and numbers. These include explicit relations, difference equations, summation formulae, linear and differential recurrence relations. Furthermore, we focus on matrix approach associated with this family by providing the generalized Fibo–Bernoulli polynomials matrix, Fibo–Pascal polynomial matrix and other important matrices. Some product and inverse formulae for the generalized Fibo–Bernoulli polynomials matrix involving other matrices are also derived at the end.