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.

We construct a new type of Genocchi polynomials using degenerate quantum exponential functions and find various forms of $(q,h)$-difference equations with these polynomials as solutions. This paper includes properties of the symmetric structures of $(q,h)$-difference equations and also presents $(q,h)$-difference equations with other polynomials as coefficients. By understanding the approximate roots structure of degenerate quantum Genocchi polynomials (DQG), which are common solutions to various forms of $(q,h)$-difference equations, we identify the properties of the solutions.