| Fibonacci numbers, Lucas numbers, Fibonacci polynomials, Lucas polynomials and their various generalizations have important applications in combinatorics, number theory,numerical analysis and other mathematical ?elds, and they have been widely studied all the time. In this thesis, we will study the(p, q)-Fibonacci polynomials, the generalized(p, q)-Fibonacci polynomials, the generalized(p, q)-Lucas polynomials and the generalized Humbert polynomials. The thesis is organized as follows:(1) We study identities, recurrence relation and generating function of the products of two(p, q)-Fibonacci polynomials, establish recurrence relation and explicit expressions of the sums of products of two(p, q)-Fibonacci polynomials. Moreover, we establish many new combinatorial identities by applying the general results on the(p, q)-Fibonacci polynomials to the classical Fibonacci polynomials and Chebyshev polynomials.(2) We de?ne the generalized(p, q)-Fibonacci polynomials un,m(x) and generalized(p, q)-Lucas polynomials vn,m(x), establish the corresponding generating functions, explicit expressions and recurrence relations. Moreover, we obtain some combinatorial identities of polynomials un,m(x) and vn,m(x) by de?ning the generalized Fibonacci-Q matrix.(3) We de?ne the generalized Humbert polynomials u(r)n,m(x), study the corresponding explicit expressions, hypergeometric series representation, recurrence relations, di?erential recurrence relations and expansions. We also establish some identities of the generalized Humbert polynomials, generalized(p, q)-Fibonacci polynomials and generalized(p, q)-Lucas polynomials. Finally, we establish the determinantal representation of the generalized(p, q)-Fibonacci polynomials and give the algebraic interpretation of the generalized Humbert polynomials by the properties of the upper Hessenberg matrix. |
