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Abstract
Recently, the so-called new type Euler polynomials have been studied without considering Euler polynomials of a complex variable. Here we study degenerate versions of these new type Euler polynomials. This has been done by considering the degenerate Euler polynomials of a complex variable. We also investigate corresponding ones for Bernoulli polynomials in the same manner. We derive some properties and identities for those new polynomials. Here we note that our result gives an affirmative answer to the question raised by the reviewer of the paper.
Highlights
The ordinary Bernoulli polynomials Bn ( x ) and Euler polynomials En ( x ) are respectively defined by ∞tn t e xt = ∑ Bn ( x ), (1) t e −1 n!n =0 and et tn e xt = ∑ En ( x ), +1 n! n =0 (2).For any nonzero λ ∈ R, the degenerate exponential function is defined by x eλx (t) = (1 + λt) λ, eλ (t) = e1λ (t)
The degenerate cosine-Euler polynomials and degenerate sine-Euler polynomials were expressed in terms of degenerate cosine-polynomials and degenerate sine-polynomials and vice versa
Some reflection identities were found for the degenerate cosine-Euler polynomials and degenerate sine-Euler polynomials
Summary
The ordinary Bernoulli polynomials Bn ( x ) and Euler polynomials En ( x ) are respectively defined by. In [1,2], Carlitz considered the degenerate Bernoulli and Euler polynomials which are given by and x t t tn eλx (t) =. They were expressed in terms of degenerate cosine-polynomials and degenerate sine-polynomials and vice versa. Reflection symmetries were deduced for the degenerate cosine-Bernoulli polynomials and degenerate sine-Bernoulli polynomials
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