Explicit Properties ofq-Cosine andq-Sine Euler Polynomials Containing Symmetric Structures

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