Diverse Properties and Approximate Roots for a Novel Kinds of the (p, q)-Cosine and (p, q)-Sine Geometric Polynomials
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CitationSharma, S.K., Khan, W.A., Ryoo, C.-S., Duran, U. (2022). Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials. Mathematics, 10 (15), art. no. 2709. https://doi.org/10.3390/math10152709
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.
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