Exploring variable-sensitive $ q $-difference equations for $ q $-SINE Euler polynomials and $ q $-COSINE-Euler polynomials-Reference-Cited by-同舟云学术

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

Published:2024 Issue:6 Volume:9 Page:16753-16772
ISSN:2473-6988
Container-title:AIMS Mathematics
language:
Short-container-title:MATH

Author:

Kang Jung Yoog¹,Ryoo Cheon Seoung²

Affiliation:

1. Department of Mathematics Education, Silla University, Busan 46958, Republic of Korea

2. Department of Mathematics, Hannam University, Daejeon 34430, Republic of Korea

Abstract

<abstract><p>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.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Reference23 articles.

1. W. H. Abdi, On $q$-Laplace transforms, Proc. Natl. Acad. Sci. India, 29 (1961), 389–408.

2. G. Bangerezako, An Introduction to $q$-Difference Equations, Preprint, Bujumbura: University of Burundi, 2007.

3. L. Carlitz, $q$-Bernoulli numbers and polynomials, Duke Math. J., 15 (1948), 987–1000.

4. L. Carlitz, $q$-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76 (1954), 332–350.

5. T. Ernst, A Comprehensive Treatment of $q$-Calculus, New York: Springer Science, Business Media, 2012.