A note on degenerate Euler and Bernoulli polynomials of complex variable

In this paper, we study the degenerate version of the new type Euler polynomials, namely degenerate cosine-Euler polynomials and sime-Euler polynomials and also corresponding ones for Bernoulli polynomials, namely degenerate cosine Bernoulli polynomials and degenerate sine-Bernoulli polynomials by considering the degenerate Euler polynomials of complex variable and the degenerate Bernoulli polynomials of complex variable.

The Stirling numbers of the first kind are defined by the coefficients in the expansion of (x) n in terms of powers of x as follows: S (1) (n, l)x l , (see, [7,11,17]).
In [9], the degenerate stirling numbers of the second kind are defined by the generating function (2) λ (n, k) t n n! , (k ≥ 0).
In [15], the authors deduced many interesting identities and properties for those polynomials.
It is well known that [20]).
From (1) and (2), we note that By (14) and (15), we get n (x, y) t n n! , In view of (4) and (5), we study the degenerate Bernoulli and Euler polynomials with complex variable and investigate some identities and properties for those polynomials. The outline of this paper is as follows. In Section 1, we will beriefly recall the degenerate Bernoulli and Euler polynomials of Carlitz and the degenerate Stirling numbers of the second kind. Then we will introduce so called the new type Euler polynomials, and the cosine-polynomials and sine-polynomials recently introduced in [15]. Then we indicate that the new type Euler polynomials and the corresponding Bernoulli polynomials can be expressed by considering Euler and Bernoulli polynomials of complex variable and treating the real and imaginary parts separately. In Section 2, the degenerate cosine-polynomials and degenerate sine-polynomials were introduced and their explicit expressions were derived. 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. Further, some reflection identities were found for the degenerate cosine-Euler polynomials and degenerate sine-Euler polynomials. In Section 3, the degenerate cosine-Bernoulli polynomials and degenerate sine-Bernoulli polynomials were introduced. 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 degenrate sine-Bernoulli polynomials.

DEGENERATE EULER POLYNOMIALS OF COMPLEX VARIABLE
Here we will consider the degenerate Euler polynomials of complex variable and, by treating the real and imaginary parts separately, introduce the degenerate cosine-Euler polynomials and degenerate sine-Euler polynomials. They are degenerate versions of the new type Euler polynomials studied in [15].
The degenerate sine and cosine functions are defined by (17) .
From (13), we note that lim λ →0 By (5), we get Now, we define the degenerate cosine and degenerate sine function as  (18) and (19), we note that In view of (9) and (10), we define the degenerate cosine-Euler polynomials and degenerate sine-Euler polynomials respectively by n (x, y) and n (x, y) are the new type of Euler polynomials of Masjed-Jamei, Beyki and Koepf (see [15]).
From (22)-(25), we note that We recall here that the generalized falling factorial sequence is defined by We observe that From (20), we can derive the following equation.
By (21), we get sin (y) where [x] denotes the greatest integer ≤ x.
By (29), we get Now, we define the degenerate cosine-polynomials and degenerate sine-polynomials respectively by Note that where C k (x, y) and S k (x, y) are the cosine-polynomials and sine-polynomials of Masijed-Jamei, Beyki and Koepf.
From (24), we note that n m E m,λ C n−m,λ (x, y) t n n! .
Therefore by comparing the coefficients on both sides of (35) and (40), we obtain the following theorem.
Therefore, by comparing the coefficients on both sides of (41), we obtain the following proposition. where r is a fixed real (or complex) number. Now, we consider the reflection symmetric identities for the degenerate cosine-Euler polynomials. By (24), we get n,−λ (x, y) (−1) n t n n! , Therefore, by (42) and (43), we obtain the following theorem Therefore, by (44), we obtain the following theorem.
Theorem 2.7. For n ≥ 0, we have Also, for n ∈ N, we have n−k,λ (y).

DEGENERATE BERNOULLI POLYNOMIALS OF COMPLEX VARIABLE
In this section, we will consider the degenerate Bernoulli polynomials of complex variable and, by treating the real and imaginary parts separately, introduce the degenerate cosine-Bernoulli polynomials and degenerate sine-Bernoulli polynomials.
On the other hand, Therefore, by (53) and (54), we obtain the following theorem. From (49), we have Therefore, by (55), we obtain the following theorem. the imaginary part separately? Our result gives an affirmative answer to the question (see (16)). In this paper, we considered the degenerate Euler and Bernoulli polynomials of complex variable and, by treating the real and imaginary parts separately, were able to introduce degenerate cosine-Euler polynomials, degenerate sine-Euler polynomials, degenerate cosine-Bernoulli polynomials, and degenerate sine-Bernoulli polynomials. They are degenerate versions of the new type Euler polynomials studied by Masjed-Jamei, Beyki and Koepf [15] and of the 'new type Bernoulli polynomials. ' In Section 2, the degenerate cosine-polynomials and degenerate sine-polynomials were introduced and their explicit expressions were derived. 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. Further, some reflection identities were found for the degenerate cosine-Euler polynomials and degenerate sine-Euler polynomials. In Section 3, the degenerate cosine-Bernoulli polynomials and degenerate sine-Bernoulli polynomials were introduced. 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 degenrate sine-Bernoulli polynomials. Further, some expressions involving the degenerate Stirling numbers of the second kind were derived for them.
It was Carlitz [1,2] who initiated the study of degenerate versions of some special polynomials, namely the degenerate Bernoulli and Euler polynomials. Studying degenerate versions of some special polynomials and numbers have turned out to be very fruitful and promising (see [3,[5][6][7][8][9][10][11][13][14]19] and references therein). In fact, this idea of considering degenerate versions of some special polynomials are not limited just to polynomials but can be extended even to transcendental functions like gamma functions [8].