Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.

This article aims to present type 2 degenerate poly-Bernoulli numbers and polynomials arising from moderated degenerate polyexponential functions. Certain explicit expressions for these numbers and polynomials are derived. Also, we introduce type 2 degenerate unipoly-Bernoulli numbers and polynomials by utilizing unipoly functions and show some basic properties of them.
Theorem 2.1 For κ ∈ Z and υ ≥ 0, we have Remark 2.1 Letting λ to 0 in Theorem 2.1 leads to Then, from (2.1) and (2.6), we have Therefore, by (2.7), we arrive at the following theorem.
By making use of (1.7) and (2.1), we note that which on comparing the coefficients on both sides of the above equation yields the following theorem.
Let κ ≥ 1 be an integer. For s ∈ C, the function η κ,λ (s) can be defined as Ei κ,λ log λ (1 + z) dz Here, for any s ∈ C, the second integral converges absolutely; hence, the second term on the r.h.s. vanishes at nonpositive integers, i.e., On the other hand, for (s) > 0, we can write the first integral in (2.10) as which defines an entire function of s. Therefore, we may say that η κ,λ (s) can be continued to an entire function of s.
By making use of (1.2), we note that On the other hand, (2.14) Therefore, by (2.13) and (2.14), we arrive at the following theorem.
From (2.1), we note that On replacing z with e λ (z) -1 in (2.15), we get On the other hand, Therefore, by (2.16) and (2.17), we arrive at the following theorem.
On the other hand, Therefore, by (2.19) and (2.20), we arrive at the following theorem.

Theorem 2.8
For κ ∈ Z and υ ∈ N, we have

Type 2 degenerate unipoly-Bernoulli polynomials and numbers
The unipoly function attached to polynomials p(ω) was defined by Kim and Kim [14] as where p denotes any arithmetic real or complex-valued function defined on N. Moreover, is the ordinary polylogarithm function. Dolgy and Khan [4] introduced the degenerate unipoly function attached to polynomials p(ω) as follows: It is worthy to see that is the moderated degenerate polyexponential function. Now, we define type 2 degenerate unipoly-Bernoulli polynomials which are given by the generating function as follows: For ω = 0, β (κ) r,λ,p = β (κ) r,λ,p (0) denotes type 2 degenerate unipoly-Bernoulli numbers attached to p.
Thus, by (3.6), we have By making use of (3.5), we see that which yields the following theorem.
By Eq. (3.9), we get the following theorem.
Theorem 3.2 Let j be a nonnegative integer and κ ∈ Z. Then which yields the following theorem.

Conclusions
In this paper, we have studied and introduced degenerate versions of type 2 Bernoulli numbers and polynomials and derived some properties of these polynomials. We have given some relationships between higher-order Bernoulli polynomials, degenerate type 2 Bernoulli polynomials, degenerate central Bell polynomials, degenerate Stirling numbers of the first and second kind, degenerate central factorials numbers. Besides, we have introduced degenerate type 2 unipoly-Bernoulli polynomials by using degenerate unipoly polynomials and derived some identities of these polynomials. We have derived some relationship between degenerate type 2 Bernoulli polynomials and degenerate Daehee polynomials.