Generalized Humbert polynomials via generalized Fibonacci polynomials

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