Explicit Properties of q-Cosine and q-Sine Euler Polynomials Containing Symmetric Structures

Cheon Seoung Ryoo,Jung Yoog Kang

doi:10.3390/sym12081247

Abstract

In this paper, we introduce q-cosine and q-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. We investigate the approximate roots of q-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of q-cosine Euler polynomials in a complex plane using the Newton method.

Highlights

In 1990, Jackson who published influential papers on the subject introduced the q-number and its notation stems, see [1]
We present some figures of the approximate roots of these polynomials in a complex plane using Newton’s method
By comparing Equations (80) and (81), we investigate a relation between the q-cosine Euler polynomials and q-cosine Bernoulli polynomials and complete the proof of Theorem 14

Summary

Introduction

In 1990, Jackson who published influential papers on the subject introduced the q-number and its notation stems, see [1]. The cosine and sine Euler polynomials in the second row of the diagram contain a motive in this paper. In [16], the definitions and representative properties of cosine and sine Euler polynomials are as follows. The main goal of this paper is to find various properties of q-cosine and q-sine Euler polynomials such as addition theorem, partial q-derivative, basic symmetric properties so on. Euler polynomials such as the identity of q-sine Euler polynomials using q-analogues of subtraction and addition. This is based on the properties of q-trigonometric and q-exponential functions. We present some figures of the approximate roots of these polynomials in a complex plane using Newton’s method

Some Basic Properties of q-cosine and q-sine Euler Polynomials

Conclusions