Morgan-Voyce transformations of the generalized Horadam polynomials

Abstract

In this study, the small and large Morgan-Voyce transformations are applied to generalized Horadam polynomials. We obtain the new sequences from these transformations. In addition, we provide the generating functions, recurrence relations, and some formulas for these polynomial sequences. Moreover, we give matrix representations of Morgan-Voyce transformations of generalized Horadam polynomials. All provided results are reduced to the small and large Morgan-Voyce transformations of well-known integer sequences including Fibonacci, Lucas, Pell, Pell-Lucas, Jacosthal, and Jacobsthal-Lucas numbers. Finally, we present some applications of special cases of these matrix representations in cryptography.