Some Novel Formulas of Lucas Polynomials via Different Approaches

Some new formulas related to the well-known symmetric Lucas polynomials are the primary focus of this article. Different approaches are used for establishing these formulas. A matrix approach to Lucas polynomials is followed in order to obtain some fundamental properties. Particularly, some recurrence relations and determinant forms are determined by suitable Hessenberg matrices. Conjugate Lucas polynomials and generating functions are derived and examined. Several connection problems between the Lucas polynomials and other celebrated symmetric and non-symmetric orthogonal polynomials such as the first and second kinds of Chebyshev polynomials and their shifted counterparts are solved. We prove that several argument-type hypergeometric functions are involved in the connection coefficients. In addition, we construct new formulas for high-order derivatives of Lucas polynomials in terms of their original polynomials, as well as formulas for repeated integrals of Lucas polynomials.