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

Abstract

This paper constructs and introduces ( p , q ) -cosine and ( p , q ) -sine Bernoulli polynomials using ( p , q ) -analogues of ( x + a ) n . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using ( p , q ) -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials.

Some Symmetric Properties and Location Conjecture of Approximate Roots for (p,q)-Cosine Euler Polynomials

“…Based on the previous theory, many mathematicians have researched Bernoulli, Euler, and Genocchi polynomials combining (p, q)-numbers. Moreover, they make polynomials of various kinds which have some interesting properties and identities, see [9,12,[14][15][16]. We introduce a few polynomials which are needed in this paper.…”