On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

Abstract

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (Formula presented.) -Bernoulli–Fibonacci and generalized (Formula presented.) -Bernoulli–Lucas polynomials and numbers by using the (Formula presented.) -Bernoulli numbers, unified (Formula presented.) -Bernoulli polynomials, (Formula presented.) -Fibonacci polynomials, and (Formula presented.) -Lucas polynomials. We also introduce the generalized bivariate (Formula presented.) -Bernoulli–Fibonacci and generalized bivariate (Formula presented.) -Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (Formula presented.) -Bernoulli–Fibonacci polynomials.