Exploring variable-sensitive $q$-difference equations for $q$-SINE Euler polynomials and $q$-COSINE-Euler polynomials

In this study, we introduced several types of higher-order difference equations involving $q$-SINE Euler (QSE) and $q$-COSINE Euler (QCE) polynomials. Depending on the parameters selected, these higher-order difference equations exhibited properties of trigonometric functions or related Euler numbers. Approximate root construction focused on the QSE polynomial, which was the solution of the $q$-difference equations obtained earlier. We also showed the structure of the approximate roots of higher-order polynomials among the QSE polynomials, understood them, and considered the associated conjectures.