Solutions of first and second order power quantum difference equations

Abstract

Abstract Jackson in 1908 introduced the most well–known and used quantum difference operator Dqf(t) = (f(qt)−f(t))/(qt−t) for a fixed 0 < q < 1. Aldwoah in 2009 [3], introduced the power quantum n, q–difference operator Dn,qf(t) = (f(qt n) − f(t))/(qtn − t), where n is an odd natural number and q ∈ (0, 1) are fixed. Dn,q yields Jackson q–difference operator, when n = 1. In this paper, we define the n, q–exponential and n, q–trigonometric (hyperbolic) functions and we establish some of their properties. We prove that they are solutions of power quantum difference equations of first and second order respectively.