We establish a new sequence of polynomials that combines the Fibonacci and Lucas polynomials. We will refer to these polynomials as merged Fibonacci-Lucas polynomials (MFLPs). We will show that we can represent these polynomials by combining two certain Fibonacci polynomials. This formula will be essential for determining the power form representation of these polynomials. This representation and its inversion formula for these polynomials are crucial to derive new formulas about the MFLPs. New derivative expressions for these polynomials are given as combinations of several symmetric and non-symmetric polynomials. We also provide the inverse formulas for these formulas. Some new product formulas involving the MFLPs have also been derived. We also provide some definite integral formulas that apply to the derived formulas.

This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical Hermite polynomials. Derivative expressions of these polynomials that involve one parameter are found in terms of other parameter polynomials. Some other important formulas, such as the linearization and connection formulas between these polynomials and some other polynomials, are also given. Most of the coefficients are represented in terms of hypergeometric functions that can be reduced in some specific cases using some standard formulas. Two applications of the developed formulas in this paper are given. The first application is concerned with introducing some weighted definite integrals involving the GHPs. In contrast, the second is concerned with establishing the operational matrix of the integer derivatives of the GHPs.