Some Novel Formulas of Lucas Polynomials via Different Approaches

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