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.
Citation: Jung Yoog Kang, Cheon Seoung Ryoo. Exploring variable-sensitive $ q $-difference equations for $ q $-SINE Euler polynomials and $ q $-COSINE-Euler polynomials[J]. AIMS Mathematics, 2024, 9(6): 16753-16772. doi: 10.3934/math.2024812
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.
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Jung Yoog Kang, Cheon Seoung Ryoo. Exploring variable-sensitive $ q $-difference equations for $ q $-SINE Euler polynomials and $ q $-COSINE-Euler polynomials[J]. AIMS Mathematics, 2024, 9(6): 16753-16772. doi: 10.3934/math.2024812
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