Various Structures of the Roots and Explicit Properties of q-cosine Bernoulli Polynomials and q-sine Bernoulli Polynomials

Abstract

In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about the polynomials above.

Structure of Approximate Roots Based on Symmetric Properties of (p, q)-Cosine and (p, q)-Sine Bernoulli Polynomials

“…In [11], we confirmed the properties of q-cosine and q-sine Bernoulli polynomials. Their definitions and representative properties are as follows.…”

“…Here, we aim to confirm that changes in the value of the (p, q)-cosine Bernoulli polynomials changes the structure of the approximate roots as the value changes. The structure of the approximate roots in polynomials when p = 1 and q changes, can be found in the q-cosine Bernoulli polynomials (see [11]).…”

“…The different variations of Bernoulli polynomials, q-Bernoulli polynomials and (p, q)-Bernoulli polynomials are illustrated in the diagram below. Kim, Ryoo and many mathematicians have studied the first and second rows of the polynomials in the diagram(see [8][9][10][11][12]). These studies began producing valuable results in areas related to number theory and combinatorics.…”

“…Based on the theorems and definitions, it is significant to study these properties in various ways by the combining with Bernoulli, Euler, and Genocchi polynomials. Mathematicians are studying the extended versions of these polynomials and are researching new polynomials by combining mathematics with other fields, such as physics or engineering (see [9][10][11][12][13][14]). The definition of Bernoulli polynomials combined with (p, q)-numbers follows: Definition 1.…”

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

“…For a long time, the topics of Bernoulli, Euler, and Genocchi polynomials have been extensively researched in many mathematical applications including analytical number theory, combinatorial analysis, p-adic analytic number theory, and other fields. Therefore, many mathematicians have started researching Bernoulli, Euler, and Genocchi polynomials combining q-numbers, see [9,13,[15][16][17][18][19][20][21].…”