A Study on Dual-Generalized Complex and Hyperbolic-Generalized Complex Numbers

Abstract

Highlights• This paper focuses on the theories of dual-generalized and hyperbolic-generalized complex numbers.• The algebraic structures of dual-generalized and hyperbolic-generalized complex numbers are given.• Dual-generalized complex and hyperbolic-generalized complex valued functions are defined.• The matrix representations of dual-generalized and hyperbolic-generalized complex numbers are stated.• An efficient classification includes complex-generalized complex numbers is examined.

Explicit formulas for exponential of 2×2 split-complex matrices

“…where the split-complex unit j satisfies j 2 = 1 and j ̸ = 1. In the literature, these numbers are also called double, spacetime, hyperbolic or perplex numbers [10], [28], [27], [5], [11], [25], [24], [9], [12]. For any z = z 1 + jz 2 ∈ P we define the real part of z as Re(z) =z 1 and the split-complex part of z as Im( z) = z 2 .…”

Fundamental properties of extended Horadam numbers

“…Table 1. Multiplication scheme of DGC numbers, [12] Moreover, the set of HGC and CGC numbers are introduced as, respectively:…”

“…For the basis elements, we have Jj = jJ and Ji = iJ. The operations for the HGC and CGC numbers can be given similarly (see in [12]).…”

“…where W n is the n-th Horadam number and the set {1, J, i, Ji} has multiplicative properties given in [12].…”

“…In [12], as a new approach, dual-generalized complex (DGC), hyperbolic-generalized complex (HGC) and complex-generalized complex (CGC) numbers are introduced by using the Cayley-Dickson doubling procedure. For the special real values p = −1, p = 0 and p = 1, dual-complex, hyper-dual, dual-hyperbolic, hyperbolic-complex, bihyperbolic and bicomplex numbers are obtained from DGC, HGC and CGC numbers (see detailed classification in [12]). As we move from number systems to elements of Horadam sequences, some papers can be examined in literature, [2,4,7,11,13,14,24,28,33].…”