On some identities for the DGC Leonardo sequence

In this study, we examine the Leonardo sequence with dual-generalized complex (DGC) coefficients for p ∈ R. Firstly, we express some summation formulas related to the DGC Fibonacci, DGC Lucas, and 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 DGC Leonardo sequence by considering p. This generalization gives the dual-complex Leonardo sequence for p =-1, hyper-dual Leonardo sequence for p =0, and dual-hyperbolic Leonardo sequence for p =1.