On type 2 degenerate Bernoulli and Euler polynomials of complex variable

• Published in: Advances in difference equations 2019-11, Vol.2019 (1), p.1-15, Article 490
• Main Authors: Kim, Taekyun, Kim, Dae San, Jang, Lee-Chae, Kim, Han-Young
• Format: Article
• Language: English
• Subjects: Analysis; Complex variables; Difference and Functional Equations; Functional Analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Polynomials; Trigonometric functions; Type 2 degenerate Bernoulli polynomials of complex variable; Type 2 degenerate cosine-Bernoulli polynomials; Type 2 degenerate cosine-Euler polynomials; Type 2 degenerate Euler polynomials of complex variable; Type 2 degenerate sine-Bernoulli polynomials; Type 2 degenerate sine-Euler polynomials
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-019-2419-3

Recently, Masjed-Jamei, Beyki, and Koepf studied the so-called new type Euler polynomials without using Euler polynomials of complex variable. Here we study the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials, which are type 2 degenerate versions of these new type Euler polynomials, by considering the degenerate Euler polynomials of complex variable and by treating the real and imaginary parts separately. In addition, we investigate the corresponding ones for Bernoulli polynomials in the same manner. We derive some explicit expressions for those new polynomials and some identities relating to them. Here we note that the idea of separating the real and imaginary parts separately gives an affirmative answer to the question asked by Hacène Belbachir.