On Two Bivariate Kinds of Poly-Bernoulli and Poly-Genocchi Polynomials

In this paper, we introduce two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials and study their basic properties. Finally, we consider some relationships for Stirling numbers of the second kind related to bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials.

Differentiating generating function (1) with respect z and equating the coefficients of t n n! , we have d dz A n (z) = nA n−1 (z), A 0 (z) = 0, z = x + iy ∈ C, n ∈ N.
In (2015), Kim et al. [10] introduced the poly-Genocchi polynomials are defined by means of the following generating function For k = 1, we have where G n (x) are called the Genocchi polynomials, (see [3,14]).
The Stirling numbers of the first kind are defined by the coefficients in the expansion of (x) n in terms of power of x as follows, (see [1,2,7]) Subsequently, the Stirling numbers of the second kind are defined by, (see [2,4,5]) Recently, Jamei et al. [13,14] introduced and investigated the new type of Bernoulli and Genocchi polynomials defined by means of the following generating function and 2t e t + 1 e xt cos yt = respectively.
They have also considered the two functions e xt cos yt and e xt sin yt as follows (see [12][13][14][15][16]): and where and In (2018), Kim and Ryoo [1] introduced the cosine Bernoulli polynomials of a complex variable, the sine Bernoulli polynomials of a complex variable and the cosine Euler polynomials of a complex variable, the sine Euler polynomials of a complex variable, respectively are defined as follows and From (16) and (17), we get n (x, y) t n n! , and t e t − 1 e xt sin yt = n (x, y) t n n! , This article is organized as follows. In Section 2, we introduce the cosine poly-Bernoulli and sine poly-Bernoulli polynomials and derive some identities of these polynomials. In Section 3, we establish the cosine poly-Genocchi and sine poly-Genocchi polynomials and derive some identities of these polynomials. Finally Section 4, we investigated some relationships for Stirling numbers of the second kind related to poly-Bernoulli and poly-Genocchi polynomials.

Poly-Bernoulli Polynomials of Complex Variable
This section presents sine and cosine variant of poly-Bernoulli polynomials. These variants are processed by separating the real and imaginary parts of the complex poly-Bernoulli polynomials and study on their basic properties are expressed. Now, we consider the poly-Bernoulli polynomials that are given by the generating function On the other hand, we observe that, (see [1]) e (x+iy)t = e xt e iyt = e xt (cos yt + i sin yt), Thus, by (18) and (19), we have and From (20) and (21), we get and

Definition 1. The two bivariate kinds of cosine poly-Bernoulli polynomials B
(k,c) n (x, y) and sine poly-Bernoulli polynomials B (k,s) n (x, y), for non negative integer n are defined by and respectively. (24) and (25), we get new type polynomials as follows and respectively.

It is clear that
From (28) and (29), we can derive the following equations and Therefore, by (30) and (31), we get and Now, we start some basic properties of these polynomials. and Proof. By using (20) and (21), we can easily get. So we omit the proof.

Theorem 2. B
(k,c) n (x, y) and B (k,s) n (x, y) can be represented in terms of poly-Bernoulli numbers as follows Proof. By noting the general identity, we have ∞ ∑ n=0 a n t n n!
n−m C m (x, y) t n n! , which proves (36). The proof of (37) is similar.

Theorem 3.
For every n ∈ Z + , the following formula holds true and B (k,s) n Proof. From (24), we have as well as Similarly Equation (39) can be proved.
and ∂B Proof. Equation (24) yields In particular for k = 2, we have On comparing the coefficients of t n n! on both sides of the above equation, we get the result (46). The proof of (47) is similar.

Poly-Genocchi Polynomials of Complex Variable
This section presents sine and cosine variant of poly-Genocchi polynomials. These variants are processed by separating the real and imaginary parts of the complex poly-Genocchi polynomials and study on their basic properties are expressed. Now, we consider the poly-Genocchi polynomials that are given by the generating function By using (48) and (19), we have and From (49) and (50), we get and and respectively.
From (51)-(54), we have The cosine poly-Genocchi and sine poly-Genocchi polynomials can be determined explicitly. A few of them are Remark 2. For x = 0 in (53) and (54), we get new type polynomials as follows and respectively.
It is clear that G From (55) and (56), we can derive the following equations Therefore, by (57) and (58), we get and and G (k) Proof. By using (50) and (51), we can easily get. So we omit the proof.
n−m C m (x, y) t n n! , which proves (63). The proof of (64) is similar.
Theorem 9. For every n ∈ Z + , the following formula holds true and G (k,s) n Proof. From (53), we have as well as Similarly Equation (66) can be proved. m (x, y)r n−m t n n! , which proves (67). The result (68) can be similarly proved.
Theorem 11. For every n ∈ N, the following formula holds true and ∂G Proof. From Equation (53), we have In particular for k = 2, we have On comparing the coefficients of t n n! on both sides of the above equation, we get the result (73). The proof of (74) is similar.

Relationship between Stirling Numbers of the Second Kind
In this section, we prove some relationships for Stirling numbers of the second kind related to poly-Bernoulli polynomials of complex variable and poly-Genocchi polynomials of complex variable. We start a following theorem. Replacing n by n − p in the above equation and comparing the coefficients of t n n! on either side, we get the result (75). The proof of (76) is similar. By comparing the coefficients of t n on both sides , we get (77). The proof of (78) is similar.