$${\varvec{p}}$$ p -Trigonometric Approach to Elliptic Biquaternions

Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications

“…In this section, firstly, we recall the fundamental concepts of elliptic biquaternions (see [14] and [16] for more details.) The set of elliptic biquaternions is given in the cartesian form as in the following…”

“…In addition, an elliptic biquaternion which has a unit real semi norm is called unit elliptic biquaternion, [14]. The set of all unit elliptic biquaternions is denoted by η and is given as in the following η = {Q ∈ HC p : N Q = 1} ⊂ HC p , [16].…”

“…in which case, the angle ϕ p = x + Iy ∈ C p is called as elliptical complex angle. For any two elliptical complex angles ϕ p and ψ p , the following addition formulas hold [16]: Let Q = A 0 + A 1 i + A 2 j + A 3 k be any elliptic biquaternion with a non-zero modulus while…”

“…Also, some properties of elliptic biquaternions relating to seminorm and the three types of conjugate have been given and the idempotent elements in the elliptic biquaternion algebra have been discussed in [15]. Lastly, some algebraic relations between elliptic biquaternions and extended p-trigonometric functions have been discussed in [16]. In this paper, firstly, we present an introduction to elliptic biquaternions and recall some information on elliptic matrices for the necessary background.…”

“…2I k in the space HC −1/2 , we know that it can be written in the p-trigonometric form as (see[16] for details). Then, we can writeH + (Q) = 2 cosp ϕ p I 4 + sinp ϕ p H + (w Q )Every power of this matrix can be found from the theorem 4.1.…”