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