On type 2 degenerate Bernoulli and Euler polynomials of complex variable

Abstract

Abstract 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.

Keywords

• Type 2 degenerate Bernoulli polynomials of complex variable
• Type 2 degenerate Euler polynomials of complex variable
• Type 2 degenerate cosine-Bernoulli polynomials
• Type 2 degenerate sine-Bernoulli polynomials
• Type 2 degenerate cosine-Euler polynomials
• Type 2 degenerate sine-Euler polynomials