A special approach to derive new formulas for some special numbers and polynomials


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Kilar N., ŞİMŞEK Y.

TURKISH JOURNAL OF MATHEMATICS, vol.44, no.6, pp.2217-2240, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 44 Issue: 6
  • Publication Date: 2020
  • Doi Number: 10.3906/mat-2005-116
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.2217-2240
  • Keywords: Bernoulli and Euler type numbers and polynomials, Chebyshev polynomials, Fibonacci and Lucas polynomials, Generating functions, Harmonic and trigonometric functions, Laplace operator
  • Akdeniz University Affiliated: Yes

Abstract

By applying Laplace differential operator to harmonic conjugate components of the analytic functions and using Wirtinger derivatives, some identities and relations including Bernoulli and Euler polynomials and numbers are obtained. Next, using the Legendre identity, trigonometric functions and the Dirichlet kernel, some formulas and relations involving Bernoulli and Euler numbers, cosine-type Bernoulli and Euler polynomials, and sine-type Bernoulli and Euler polynomials are driven. Then, by using the generating functions method and the well-known Euler identity, many new identities, formulas, and combinatorial sums among the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, the Chebyshev polynomials, and Bernoulli and Euler type polynomials are given. Finally, some infinite series representations for these special numbers and polynomials and their numerical examples are presented.