Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures

Abstract

In this paper, we introduce q-cosine and q-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. We investigate the approximate roots of q-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of q-cosine Euler polynomials in a complex plane using the Newton method.

Keywords

• q-cosine Euler polynomials
• q-sine Euler polynomials
• q-trigonometric function
• q-exponential function