A Generalized Quaternionic Sequence with Vietoris' Number Components

Abstract

In this investigation, the aim is to determine a generalized quaternionic sequence with Vietoris'  number components depending on $2$-parameters $\alpha$ and $\beta$. Considering specific real values $\alpha$ and $\beta$, various types of classical quaternionic sequence with  Vietoris' number components can be obtained as real, split, split-semi and so on. The fundamental algebraic structures, several classical expressions, a two and three term recurrence relations are identified, as well as Catalan-like, generating function and Binet-like formulas. Furthermore,  a determinantal approach is used to generate the generalized quaternionic sequence with Vietoris' number components.