p-Adic integral on Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Z}_{p}$\end{document} associated with degenerate Bernoulli polynomials of the second kind

In this paper, by means of p-adic Volkenborn integrals we introduce and study two different degenerate versions of Bernoulli polynomials of the second kind, namely partially and fully degenerate Bernoulli polynomials of the second kind, and also their higher-order versions. We derive several explicit expressions of those polynomials and various identities involving them.


Introduction and preliminaries
In [1,2], Carlitz studied degenerate versions of Bernoulli and Euler polynomials, namely the degenerate Bernoulli and Euler polynomials, and obtained some interesting arithmetic and combinatorial results. In recent years, various degenerate versions of many special polynomials and numbers regained interest of some mathematicians, and quite a few results have been discovered. These include the degenerate Stirling numbers of the first and second kinds, degenerate central factorial numbers of the second kind, degenerate Bernoulli numbers of the second kind, degenerate Bernstein polynomials, degenerate Bell numbers and polynomials, degenerate central Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate Cauchy numbers, and so on (see [3,10,13,16,18,19] and the references therein). Here we would like to mention that the study of degenerate versions can be done not only for polynomials but also for transcendental functions like gamma functions. For this, we let the reader refer to the paper [14].
The aim of this paper is to study two degenerate versions of Bernoulli polynomials of the second kind, namely the partially and fully degenerate Bernoulli polynomials of the second kind, and their higher-order versions by using p-adic Volkenborn integrals. We derive several explicit expressions for those polynomials and identities involving them and some other special numbers and polynomials. The possible applications of our results are discussed in the last section.
The paper is organized as follows. In this section, we recall what is needed in the rest of the paper, which includes the p-adic Volkenborn integrals, the ordinary and higher-order Bernoulli polynomials, the Bernoulli polynomials of the second kind, the degenerate exponential functions, the Daehee numbers, the Stirling numbers of both kinds, the degenerate Stirling numbers of both kinds, and the degenerate Bernoulli polynomials. In Sect. 2, we define the partially degenerate Bernoulli polynomials of the second kind and their higherorder versions by using p-adic Volkenborn integrals. We derive several explicit expressions for those polynomials. Further, we obtain identities involving those polynomials and some other polynomials including the higher-order Bernoulli polynomials, the Daehee numbers, and the usual and degenerate Stirling numbers of both kinds. In Sect. 3, we define the fully degenerate Bernoulli polynomials of the second kind and their higher-order versions by using p-adic Volkenborn integrals. We deduce several explicit expressions for those polynomials. Moreover, we obtain identities involving those polynomials and some other special numbers and polynomials. Here we observe that, for x = 0, both partial degenerate Bernoulli polynomials of the second kind and fully degenerate Bernoulli polynomials of the second kind become the same degenerate Bernoulli numbers of the second kind.
Throughout this paper, Z p , Q p , and C p denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of an algebraic closure of Q p .
As an inversion formula of (11), the Stirling numbers of the second kind are defined by (see [18,21]) Recently, Kim considered the degenerate Stirling numbers of the second kind given by (see [10]) In light of (11), the degenerate Stirling numbers of the first kind are defined as In [1,2], Carlitz considered the degenerate Bernoulli polynomials given by When x = 0, β n,λ = β n,λ (0) are called the degenerate Bernoulli numbers.

Partially degenerate Bernoulli polynomials of the second kind
In this and next section, we assume that 0 = λ ∈ Z p and t ∈ C p with |t| p < p By (16), we easily see that lim λ→0 log λ (t) = log(t).
From (2) and (16), we can derive the following equation: Let us define the partially degenerate Bernoulli polynomials of the second kind as follows: Then, from (17), we see that Bernoulli numbers of the second kind.
First, from (18) we note that Thus we get the next result by (20).
Therefore, we obtain the following theorem.
In particular, we have From (17), we note that On the other hand, Therefore, by (25) and (26), we obtain the following theorem.
By replacing t by log λ (1 + t) in (15), we get We observe that From (30), we obtain Therefore, by (29) and (31), we obtain the following theorem. In particular, we have From (21), we note that By comparing the coefficients on both sides of (32), we obtain the following theorem.
For r ∈ N, we define the partially degenerate Bernoulli polynomials of the second kind of order r by the following multiple p-adic integrals on Z p : For x = 0, b (r) n,λ = b (r) n,λ (0) are called the degenerate Bernoulli numbers of the second kind of order r.
On the other hand, (33) is also equal to Therefore, by (33) and (34), we obtain the following theorem. By replacing t by e λ (t) -1 in (33), we get In particular, we have S 2,λ (n + r, r) = n + r r n m=0 b (r) m,λ S 2,λ (n, m).
From (13), we note that Thus, by (35) By (14), we get Thus, by (40), we have As is well known, the degenerate Bernoulli polynomials of order r are defined by t n n! (see [1,2]).
From (33), we note that Thus, by (44), we obtain the following theorem.

Fully degenerate Bernoulli polynomials of the second kind
Let us define the fully degenerate Bernoulli polynomials of the second kind as follows: Then, from (17), we see that Note that lim λ→0 b n,λ (x) = b n (x) (n ≥ 0). We note that b n,λ = b n,λ (0) are the degenerate Bernoulli numbers of the second kind. We note here that Here, recalling (14), one should compare (53) with the following: From (51) and (53), we note that From (17), we note that On the other hand, from (53) we have Therefore, by (59) and (60), we obtain the following theorem.
of the degenerate Bernoulli numbers of the second kind as a solution, were derived. As a result, it was possible to derive an identity involving the ordinary and higher-order degenerate Bernoulli numbers of the second kind and generalized harmonic numbers (see also [4]). The third one is their possible application to identities of symmetry. For instance, in [13] we obtained many symmetric identities in three variables related to degenerate Euler polynomials and alternating generalized falling factorial sums. Each of these possible applications of the special polynomials considered in this paper requires considerable amount of work and hence needs to appear in the form of separate papers. Finally, as one of our future projects, we will continue to study various degenerate versions of special polynomials and numbers and investigate their possible applications to physics, science, and engineering as well as to mathematics.