On some identities for the DGC Leonardo sequence

Abstract

In this study, we examine the Leonardo sequence with dual-generalized complex ($\mathcal{DGC}$) coefficients for $\mathfrak{p} \in \mathbb R$. Firstly, we express some summation formulas related to the $\mathcal{DGC}$ Fibonacci, $\mathcal{DGC}$ Lucas, and $\mathcal{DGC}$ Leonardo sequences. Secondly, we present some order-$2$ characteristic relations, involving d’Ocagne's, Catalan's, Cassini's, and Tagiuri's identities. The essential point of the paper is that one can reduce the calculations of the $\mathcal{DGC}$ Leonardo sequence by considering $\mathfrak{p}$. This generalization gives the dual-complex Leonardo sequence for $ \mathfrak p =-1$, hyper-dual Leonardo sequence for $ \mathfrak p =0$, and dual-hyperbolic Leonardo sequence for $ \mathfrak p =1$.