Diverse Properties and Approximate Roots for a Novel Kinds of the (<i>p</i>,<i>q</i>)-Cosine and (<i>p</i>,<i>q</i>)-Sine Geometric Polynomials

Abstract

Utilizing p,q-numbers and p,q-concepts, in 2016, Duran et al. considered p,q-Genocchi numbers and polynomials, p,q-Bernoulli numbers and polynomials and p,q-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced (p,q)-special polynomials and numbers and have described some of their properties and applications. In this paper, using the (p,q)-cosine polynomials and (p,q)-sine polynomials, we consider a novel kinds of (p,q)-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the p,q-integral representations and p,q-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures.

Keywords

• (<i>p</i>, <i>q</i>)-trigonometric functions
• (<i>p</i>, <i>q</i>)-calculus, cosine polynomials
• sine polynomials
• geometric polynomials
• (<i>p</i>, <i>q</i>)-geometric polynomials