New Properties and Determinantal Representations of Leonardo Finite Operator Polynomials

The aim of this paper is to introduce Leonardo finite operator polynomials and obtain some of their new properties. We first present the recurrence relation provided by Leonardo finite operator polynomials. Then, we give a Binet-like formula, generating function, exponential generating function, and a finite sum formula for Leonardo finite operator polynomials. We present a determinant representation for the nth term of Leonardo finite operator polynomials. Ultimately, by utilizing the generating function of the proposed polynomials, we establish generating relations for specific bilinear and bilateral polynomial families. This approach thus broadens the applicability of the finite operator framework to encompass a wider range of special functions.