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. [ABSTRACT FROM AUTHOR]