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Study on r-truncated degenerate stirling numbers of the second kind

The degenerate Stirling numbers of the second kind and of the first kind, which are respectively degenerate versions of the Stirling numbers of the second kind and of the first kind, appear frequently when we study various degenerate versions of some special numbers and polynomials. The aim of this paper is to consider the r-truncated degenerate Stirling numbers of the second kind, which reduce to the degenerate Stirling numbers of the second for r = 1, and to investigate their explicit expressions, some properties and related identities, in connection with several other degenerate special numbers and polynomials.


INTRODUCTION
Carlitz [3,4] initiated to study the degenerate Bernoulli and Euler polynomials and numbers, which are degenerate versions of the Bernoulli and Euler polynomials and numbers. In recent years, studying degenerate versions of some special numbers and polynomials have regained interests of some mathematicians and yielded quite a few interesting results (see [8][9][10][11][12][13][14][15]17] and the references therein). It is noteworthy that studying degenerate versions is not only limited to polynomials but also extended to transcendental functions, like gamma functions. It is also remarkable that the degenerate umbral calculus is introduced as a degenerate version of the classical umbral calculus. Degenerate versions of special numbers and polynomials have been explored by various methods, including combinatorial methods, generating functions, umbral calculus techniques, p-adic analysis, differential equations, special functions, probability theory and analytic number theory.
The Stirling number of the second S 2 (n, k) is the number of ways to partition a set of n objects into k nonempty subsets. The (signed) Stirling number of the first kind S 1 (n, k) is defined such that the number of permutations of n elements having exactly k cycles is the nonnegative integer (−1) n−k S 1 (n, k) = |S 1 (n, k)|. The degenerate Stirling numbers of the second kind S 2,λ (n, k) (see (5), (8)) and of the first kind S 1,λ (n, k) (see (4), (7)) were introduced as degenerate versions of the Stirling numbers of the second and of the first kind, respectively. These degenerate Stirling numbers of both kinds appear frequently when we study degenerate versions of some special numbers and polynomials.
Here we consider the r-truncated degenerate Stirling numbers of the second which reduce to the degenerate Stirling numbers of the second for r = 1. The aim of this paper is by using generating functions to study their explicit expressions, some properties and related identities on the r-truncated degenerate Stirling numbers of the second kind, in connection with the higher-order r-truncated degenerate Bernoulli polynomials, the degenerate Bernoulli numbers, the degenerate Stirling numbers of the second kind and the higher-order degenerate Bernoulli numbers.
The outline of this paper is as follows. In Section 1, we recall the degenerate exponentials and logarithms. We remind of the degenrate Stirling numbers of both kinds. Furthermore, we recall the higher-order degenerate Bernoulli polynomials. Section 2 is the main results of this paper. We consider the r-truncated degenerate Stirling numbers of the second kind and find three explicit expressions for those numbers in Theorems 1-3. Then we introduce the higher-order r-truncated degenerate Bernoulli polynomials and obtain some results in connection with the r-truncated degenerate Stirling numbers of the second kind. In Theorem 4, the degenerate Bernoulli numbers are expressed in terms of the Stirling numbers of the second kind. In Theorems 5 and 8, obtained are some identities involving the r-truncated degenerate Stirling numbers of the second kind (for r=2 and r=3, respectively), the degenerate Bernoulli numbers and the degenerate Stirling numbers of the second. In Theorem 6, we find an identity connecting the 2-truncated degenerate Stirling numbers of the second kind and the degenerate Bernoulli numbers. In Theorem 7, we get an identity relating the 2-truncated degenerate Stirling numbers of the second kind and the higher-order degenerate Bernoulli numbers. In the rest of this section, we recall the facts that are needed throughout this paper.
For any nonzero λ ∈ R, the degenerate exponentials are defined by .
n,λ (0) are called the degenerate Bernoulli numbers of order α. In particular, for α = 1, β n,λ (x) = β (1) n,λ (x) are called the degenerate Bernoulli polynomials. For n ≥ 0, the degenerate Stirling numbers of the first kind are defined by where As the inversion formula of (4), the degenerate Stirling numbers of the second kind are defined by The degenerate logarithm log λ (t) is the compositional inverse of e λ (t) satisfying e λ (log λ (t)) = log λ (e λ (t)). Then we have From (4) and (5), we note that [8]).

r-TRUNCATED DEGENERATE STIRLING NUMBERS
For r ∈ N, we consider the r-truncated degenerate Stirling numbers of the second kind given by We agree that S [r] 2,λ (n, kr) = 0, for 0 ≤ n < kr. Note that S Thus, by (9) and (11), we obtain the following theorem.
Theorem 1. For n, k ≥ 0 with n ≥ kr, we have From the binomial expansion, we note that Therefore, by comparing the coefficients on both sides of (12), we obtain the following theorem.

Funding
This work was supported by the Basic Science Research Program, the National Research Foundation of Korea, (NRF-2021R1F1A1050151).