Holditch-Type Theorem for Non-Linear Points in Generalized Complex Plane $\mathbb{C}_{p}$

Abstract

The generalized complex number system and generalized complex plane were studied by Yaglom [1, 2] and Harkin [3]. Moreover, Holditch-type theorem for linear points in C p were given by Erişir et al. [4]. The aim of this paper is to find the answers of the questions "How is the polar moments of inertia calculated for trajectories drawn by non-linear points in C p ?", "How is Holditch-type theorem expressed for these points in C p ?" and finally "Is this paper a new generalization of [4]?".

Bridging the p-Special Functions between the Generalized Hyperbolic and Trigonometric Families

“…This angle is called p-argument of ξ. On the generalized complex numbers and elliptical complex numbers in the literature, we invite the interest of the readers to some interesting studies, namely [3][4][5][6][7][8] and the reference therein. The authors of [1] introduced in C p the p-trigonometric functions p-cosine, p-sine and p-tangent as follows: cosp(θ p ) = cos |p|θ p , (1) sinp…”

A New Construction of Holditch Theorem for Homothetic Motions in C p

“…Erişir et al obtained the Steiner area formula, the polar moment of inertia, and Holditch-type theorem in C p [35,36]. In addition to that, Erişir and Güngör gave the Holditch-type theorem for nonlinear points in a generalized complex plane C p , [37,38]. Moreover, Gürses et al gave the one-parameter planar homothetic motion in C j = fx + Jy :…”

On the Trigonometric and p-Trigonometric Functions of Elliptical Complex Variables

“…This angle is called p-argument of z. There can be found some interesting studies [3,4,5,6,7,8,9,10,11,12] on the generalized complex numbers and elliptical complex numbers in the literature. Recently,Özen and Tosun have extended the trigonometric functions cosine, sine and p-trigonometric functions pcosine, p-sine to the elliptical complex variables [3].…”