Some Identities of the Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable

In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the ﬁ rst kind. Moreover, we de ﬁ ne degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.

The following paper is as follows. In Section 2, we define the degenerate multi-poly-Bernoulli polynomials and numbers by using the degenerate multiple polyexponential functions and derive some properties and relations of these polynomials. In Section 3, we consider the degenerate multi-poly-Bernoulli polynomials of a complex variable and then we derive several properties and relations. Also, we examine the results derived in this study [28,29].
Before going to investigate the properties of the degenerate multi-poly-Bernoulli polynomials, we first give the following result.
Proof. Recall Definition 1 that which gives the asserted result (20).
The degenerate Bernoulli polynomials of order r are given by the following series expansion: (cf. [3,6,8,17]). We provide the following theorem.
Proof. Recall from Definition 1 and (10) that which means the claimed result (23).

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Proof. In view of Definition 1, we see that which implies the desired result (25).
Proof. To investigate the derivative property of B which provides the asserted result (27).
We here give a relation including the degenerate multipoly-Bernoulli polynomials with numbers and the degenerate Stirling numbers of the second kind.
Kim [5] introduced the degenerate Whitney numbers are given by Kim also provided several correlations including the degenerate Stirling numbers of the second kind and the degenerate Whitney numbers (see [5]). We now give a correlation as follows.

Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable
In [25], Kim et al. defined the degenerate sine sin λ t and cosine cos λ t functions by and cos where i = ffiffiffiffiffi ffi −1 p . Note that lim λ→0 sin ðxÞ λ ðtÞ = sin xt and lim λ→0 cos ðxÞ λ ðtÞ = cos xt. From (34), it is readily that By these functions in (34), the degenerate sinepolynomials S k,λ ðx, yÞ and degenerate cosine-polynomials C k,λ ðx, yÞ are introduced (cf. [25]) by Several properties of these polynomials in (36) and (37) are studied and investigated in [25]. Also, by means of these functions, Kim et al. [25] introduced the degenerate Euler and Bernoulli polynomials of complex variable and investigate some of their properties. Motivated and inspired by these considerations above, we define type 2 degenerate multi-poly-Bernoulli polynomials of complex variable as follows.
Definition 11. Let k 1 , k 2 , ⋯, k r ∈ ℤ. We define a new form of the degenerate multi-poly-Bernoulli polynomials of complex variable by the following generating function: By (34) and (38), we observe that and In view of (39) and (40), we consider the degenerate multi-poly-sine-Bernoulli polynomials B ðk 1 ,k 2 ,⋯,k r ;SÞ n,λ ðx, yÞ with two parameters and the degenerate multi-poly-cosine-Bernoulli polynomials B ðk 1 ,k 2 ,⋯,k r ;CÞ n,λ ðx, yÞ with two parameters as follows: We now give the two summation formulae by the following theorem.
Proof. The proofs of this theorem can be done by using the same proof methods used in Theorems 5 and 7. So, we omit the proofs.
We here provide the two derivative formulae by the following theorem.
Theorem 13. For k 1 , k 2 , ⋯, k r ∈ ℤ and n ≥ 0, we have Proof. The proofs of this theorem can be done by using the same proof methods used in Theorem 8. So, we omit the proofs.
We give the following theorem.
which complete the proof of the theorem.
We note that (cf. [25]) We give the following theorem.
Theorem 15. For k 1 , k 2 , ⋯, k r ∈ ℤ and n ≥ 0, we have where the notation ½· is Gauss' notation and represents the maximum integer which does not exceed a number in the square bracket.