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.