The Leonardo polynomials and their algebraic properties

In the last few years, the Leonardo numbers emerge as a curious integer sequence, which is an inhomogeneous extension of Fibonacci numbers. Here, we introduce the Leonardo polynomials, which generalizes the concept of Leonardo numbers given by Catarino and Borges and also extends the Fibonacci polynomials in some sense. We investigate their numerous algebraic properties such as summations formulas, generating functions, Pascal 2-triangle, interrelations with Fibonacci, Lucas, and Chebyshev polynomials, etc. in closed form. The study shows that the Leonardo polynomials form a class of irreducible polynomials. Moreover, we investigate the derivatives of the Leonardo polynomials and their explicit expressions.