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

Abstract

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.

Keywords

• $ q $-derivative
• $ q $-sine euler (qse) polynomials
• $ q $-cosine euler (qce) polynomials
• $ q $-difference equation
• approximate root