Properties of <i>q</i>-Differential Equations of Higher Order and Visualization of Fractal Using <i>q</i>-Bernoulli Polynomials

Abstract

We introduce several q-differential equations of higher order which are related to q-Bernoulli polynomials and obtain a symmetric property of q-differential equations of higher order in this paper. By giving q-varying variations, we identify the shape of the approximate roots of q-Bernoulli polynomials, a solution of q-differential equations of higher order, and find several conjectures associated with them. Furthermore, based on q-Bernoulli polynomials, we create a Mandelbrot set and a Julia set to find a variety of related figures.

Keywords

• <i>q</i>-Bernoulli polynomials
• <i>q</i>-difference equation of higher order
• Mandelbrot set
• Julia set