Successive Products Approach and its Application for Binomial Numbers Extensions from the Natural to the Integer Domain
Marlo Moesia, José Karam-Filho, Gilson Antonio Giraldi · Research Square · 2022
Abstract This work starts presenting recursive equations in the domain of sequences indexed by integers and their solutions obtained by applying the successive operations introduced in Moesia (2017). Following the original approach, the methodology is developed considering the traditional algebraic group structure. Starting from the dual interpretation of recursions, the notion of successive operations applied on sequences indexed by integers is revisited and some properties highlighted. More important, the successive operations approach is a general way to study previously extended functions from N to Z. As a demonstration of this capability, a class of binomial numbers extensions from the natural domain to the integer one is established here, allowing seeing that different pre-existing extensions of binomial numbers are, in fact, particular cases of just one general extension, developed in this work through the successive product machinery. To exemplify, binomials derived from two known factorial extensions are represented in this framework and their most known recursive expressions are studied. From this study it is also verified that by using the successive product it is possible to avoid extensive demonstrations required by standard methods, resulting in remarkably short and simple proofs.