On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers


Creative Commons License

Bayad A., ŞİMŞEK Y.

Symmetry, vol.14, no.4, 2022 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 14 Issue: 4
  • Publication Date: 2022
  • Doi Number: 10.3390/sym14040654
  • Journal Name: Symmetry
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, INSPEC, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Keywords: Bernoulli and Euler numbers and polynomials, cosine-type Bernoulli and Euler polynomials, generating functions, sine-type Bernoulli and Euler polynomials, special numbers and polynomials, Stirling numbers
  • Akdeniz University Affiliated: Yes

Abstract

© 2022 by the authors. Licensee MDPI, Basel, Switzerland.The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials.