Generalized Humbert polynomials via generalized Fibonacci polynomials

In this paper, we define the generalized (p, q)-Fibonacci polynomials u(n, m) (x) and generalized (p, q)-Lucas polynomials v(n,m) (x), and further introduce the generalized Humbert polynomials u(n,m)((r)) (x) as the convolutions of u(n, m) (x). We give many expressions, expansions, recurrence relations and differential recurrence relations of u(n,m)((r)) (x), and study the matrices and determinants related to the polynomials u(n,m) (x), v(n,m) (x) and u(n,m)((r)) (x). Finally, we present an algebraic interpretation for the generalized Humbert polynomials u(n,m)((r)) (x). It can be found that various well-known polynomials are special cases of u(n,m) (x), v(n,m) (x) or u(n,m)((r)) (x). Therefore, by introducing the general polynomials u(n,m) (x), v(n,m) (x) and u(n,m)((r)) (x), we have a unified approach to dealing with many special polynomials in the literature. (C) 2017 Elsevier Inc. All rights reserved.