A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

1 Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 2 Department of Mathematics, Hajjah University, Hajjah, Yemen 3 Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, P.O Box 9004, Postal Code:61413. Abha, Saudi Arabia 4 Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia 5 Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India 6 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab Emirates 7 International Center for Basic and Applied Sciences, Jaipur 302029, India


Introduction
In [1,2], Carlitz initiated study of the degenerate Bernoulli and Euler polynomials and obtained some arithmetic and combinatorial results on them. In recent years, many mathematicians have drawn their attention to various degenerate versions of some old and new polynomials and numbers, namely some degenerate versions of Bernoulli numbers and polynomials of the second kind, Changhee numbers of the second kind, Daehee numbers of the second kind, Bernstein polynomials, central Bell numbers and polynomials, central factorial numbers of the second kind, Cauchy numbers, Eulerian numbers and polynomials, Fubini polynomials, Stirling numbers of the first kind, Stirling polynomials of the second kind, central complete Bell polynomials, Bell numbers and polynomials, type 2 Bernoulli numbers and polynomials, type 2 Bernoulli polynomials of the second kind, poly-Bernoulli numbers and polynomials, poly-Cauchy polynomials, and of Frobenius-Euler polynomials, to name a few [3,14,[16][17][18] and the references therein. They have studied those polynomials and numbers with their interest not only in combinatorial and arithmetic properties but also in differential equations and certain symmetric identities [4,5] and references therein, and found many interesting results related to them [12,[19][20][21][22][23][24][25][26][27][28]. It is remarkable that studying degenerate versions is not only limited to polynomials but also extended to transcendental functions.
It is noteworthy to mention that The degenerate Daehee polynomials are defined by (see [15]) In the case when x = 0, D q,λ = D q,λ (0) denotes the degenerate Daehee numbers. The degenerate Bernoulli polynomials of the second kind which are defined by Kim et al. as follows (see [9]) (1.12) When x = 0, b q,λ = b q,λ (0) are called the degenerate Bernoulli numbers of the second kind. Note here that lim λ→0 b q,λ (x) = b q (x), (q ≥ 0). The degenerate Stirling numbers of the first kind are defined by [11,12] are the Stirling numbers of the first kind presented by [7,17]).
The degenerate Stirling numbers of the second kind are defined by (see [8]) It is clear that lim are the Stirling numbers of the second kind specified by ).
Motivated by the works of Kim et al. [11,14], in this paper, we study the type 2 degenerate poly-Bernoulli polynomials of the second kind arising from modified degenerate polyexponential function and obtain some related identities and explicit expressions. Also, we establish the type 2 degenerate unipoly-Bernoulli polynomials of the second kind attached to an arithmetic function by using modified degenerate polyexponential function and discuss some properties of them.

Type 2 degenerate poly-Bernoulli polynomials of the second kind
Here, the type 2 degenerate poly-Bernoulli polynomials of the second kind are defined by using the modified degenerate polyexponential function which is called the degenerate poly-Bernoulli polynomials of the second kind as j,λ (0) are called the type 2 degenerate poly-Bernoulli numbers of the second kind.
Note that where Pb (k) j (x) are called the type 2 poly-Bernoulli polynomials of the second kind (see [9]). First, we note that By making use of (2.1) and (2.3), we see that Therefore, by (2.3) and (2.4), we obtain the following theorem. Theorem 2.1. For k ∈ Z and j ≥ 0, we have For s ∈ C, the function χ k,λ (s) is given as For any s ∈ C, the second integral is absolutely convergent and thus, the second term on the r.h.s. vanishes at non-positive integers. That is, On the other hand, the first integral in Eq (2.7), for (s) > 0 can be written as which defines an entire function of s. Thus, we may include that χ k,λ (s) can be continued to an entire function of s. Further, from (2.6) and (2.7), we obtain In view of (2.8), we obtain the following theorem.
Therefore, by comparing the coefficients on both sides of (2.19), we obtain the following theorem. Theorem 2.8. For j ≥ 0, we have λ (r, q).

The degenerate unipoly-Bernoulli polynomials of the second kind
Let p be any arithmetic real or complex valued function defined on N. Kim-Kim [7] presented the unipoly function attached to polynomials p(x) as x j j k = Li k (x), (see [10,14]), (3.2) represent the known ordinary polylogarithm function. Dolgy and Khan [3] introduced the degenerate unipoly function attached to polynomials p(x) are considered as follows We see that u k,λ x| 1 Γ = Ei k,λ (x), (see [14]) (3.4) is the modified degenerate polyexponential function. Now, we introduce the degenerate unipoly-Bernoulli polynomials of the second kind attached to polynomials p(x) as When x = 0, Pb (k) j,λ,p = Pb (k) j,λ,p (0) are called the degenerate unipoly-Bernoulli numbers of the second kind attached to p.

Numerical computations
In this section, certain numerical computations are done to calculate certain zeros of the degenerate poly-Bernoulli polynomials of the second kind and show some graphical representations. The first five members of Pb (k) j,λ (x) are calculated and given as: To show the behavior of Pb (k) j,λ (x), we display the graph Pb (k) j,λ (x) for k = 4 and λ = 3, this graph is presented in Figure 1. Next, the approximate solutions of Pb (k) j,λ (x) = 0 when k = 4 and λ = 3, are calculated and listed in Table 1.

Conclusions
In this article, we introduced the type 2 degenerate poly-Bernoulli polynomials of the second kind and derived many related interesting properties. Furthermore, we defined the degenerate unipoly Bernoulli polynomials of the second kind and established some considerable results. Finally, certain related beautiful zeros and graphs are shown.