Degenerate poly-Bell polynomials and numbers

*Correspondence: hkkim@cu.ac.kr 2Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea Full list of author information is available at the end of the article Abstract Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when λ → 0. Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if k = 1. We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

With this in mind, in this paper, we define the degenerate poly-Bell polynomials through their degenerate polyexponential functions, reducing them to the degenerate Bell polynomials if k = 1. Hence, we define the poly-Bell polynomials when λ → 0, providing explicit expressions and identities involving those polynomials.
In recent years, much research has been done for various degenerate versions of many special polynomials and numbers. Moreover, various special polynomials and numbers regained interest of mathematicians, and quite a few results have been discovered [3,9,11,12,[14][15][16][17][18]. The polyexponential functions were reconsidered by Kim [8] in view of an inverse to the polylogarithm functions which were first studied by Hardy [5]. In this paper, we define the degenerate poly-Bell polynomials by means of the degenerate polyexponential functions, and they are reduced to the degenerate Bell polynomials if k = 1. In particular, when λ → 0, we call them the poly-Bell polynomials. We also provide explicit representations and combinatorial identities for these polynomials, including Dobinskilike formulas, recurrence relationships, etc.
The Bell polynomials B n (x) = n k=0 S 2 (n, k)x n are natural extensions of the Bell numbers which are a number of ways to partition a set with n elements into nonempty subsets. It is well known that the generating function of the Bell polynomials is given by Bel n (x) t n n! (see [3,11,15,19]).

Degenerate poly-Bell polynomials and numbers
In this section, we define the degenerate poly-Bell polynomials by using of the degenerate polyexponential functions and give explicit expressions and identities involving these polynomials.
Theorem 1 For k ∈ Z and n ≥ 1, we have Proof From (6) and (11), we observe that Combining with (14) and (16), we have the desired result.
Proof By replacing t with log λ (1 + t) in (14), the left-hand side is On the other hand, from (12), the right-hand side is Combining with coefficients of (18) and (19), we get what we want.
Theorem 3 (Dobinski-like formulas) For k ∈ Z and n ≥ 1, we have Proof From (1) and (6), we observe that By comparing with coefficients on both sides of (20), we get Theorem 4 For k ∈ Z and n ≥ 1, we have Proof Differentiating with respect to t in (14), the left-hand side of (14) is On the other hand, the right-hand side of (14) is Combining with (21) and (22), we get From (24), we get By comparing with coefficients on both sides of (25), we have Theorem 5 For k ∈ Z and n ≥ 1, we have Proof From (6) and (12), we observe that By replacing t with e λ (x(e λ (t) -1)) -1 in (27), we get (1) m,λ m k-1 S 1,λ (h, m)S 2,λ (n, j)S 2,λ (j, h)x j t n n! . (28) Combining with (14) and (28), we get what we want.
For the next theorem, we observe that By comparing with coefficients on both sides of (29), we get Theorem 6 For n ≥ 1, we have where β (k) n,λ are the degenerate poly-Bernoulli numbers.
Theorem 7 For k ∈ Z and n ≥ 1, we have where β (k) n,λ are the degenerate poly-Bernoulli numbers.
Theorem 8 For k ∈ Z and n ≥ 1, we have where G (k) n,λ are the degenerate poly-Genocchi numbers.
Let f (x) be a real variable function on X λ . From (17) of Theorem 2, we observe that In addition, for n ∈ N, we also obtain the moments of X λ as follows: E X n λ = ∞ j=0 j n P λ (j) = e -1 λ (α) Thus, we have the following theorem.