Generalized Morgan-Voyce Polynomials with Positive and Negative Indices: Expansions and Analytic Properties

This work develops a unified collection of algebraic and analytic identities for generalized Morgan-Voyce polynomials (GMVPs), including an explicit treatment of negative indices. Although negative-subscript Morgan-Voyce-type sequences have been considered previously, a unified treatment of the present one-parameter GMVP family that combines explicit positive- and negative-index connections with moment, inversion, linearization, differentiation, and integral identities does not appear to have been developed. Accordingly, the present work addresses this gap by establishing the structure of the negative-index GMVPs and their explicit connection with the ordinary nonnegative-index family. A moment representation and two inversion formulas are established, and the negative-index GMVPs are explicitly related to the ordinary nonnegative-index family. Linearization formulas and arbitrary-order derivative expansions are then derived, together with converse representations involving several auxiliary polynomial families. Closed-forms for some definite and weighted definite integrals are obtained. The resulting coefficients in several cases are found through terminating generalized hypergeometric series, which can be further simplified for particular choices of the involved parameters. The two classical Morgan-Voyce polynomial families are recovered as special cases of the generalized class, thereby placing their moment, inversion, connection, differentiation, linearization, and integration properties within a single unified framework.