New Identities and Operational Formulas for Leonardo-Type Polynomial Bases

This paper investigates a class of Leonardo polynomials defined by a nonhomogeneous recurrence relation, whose algebraic structure differs significantly from that of classical Fibonacci-type polynomial families. First, we develop a new connection formula expressing Leonardo polynomials in terms of Fibonacci polynomials, and then we derive the corresponding inverse connection formula. These two identities are subsequently employed to develop an explicit power-form representation of the Leonardo polynomials together with its corresponding monomial inversion formula. Building upon these fundamental representations, the paper derives new derivative and connection formulas involving generalized Fibonacci and generalized Lucas polynomial families in terms of the Leonardo basis. Several consequences for the classical Fibonacci and Lucas polynomials are also established. Furthermore, new product identities involving Leonardo polynomials are developed. The derived representations also reveal parity-dependent symmetry patterns in the Leonardo polynomial coefficients and clarify how these patterns are inherited from the associated Fibonacci- and Lucas-type bases. Many of the resulting coefficients are expressed in terms of terminating hypergeometric functions, yielding compact representations that are expected to be useful in approximation theory and operational methods for differential equations.