Degenerate Bell Polynomials Associated with Umbral Calculus Taekyun Kim1,Daesankim2, Han-Young Kim1, Hyunseok Lee2 and Lee-Chae Jang3* - DocsLib

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Kim et al. Journal of Inequalities and Applications (2020)2020:226 https://doi.org/10.1186/s13660-020-02494-7

R E S E A R C H Open Access Degenerate Bell polynomials associated with umbral calculus Taekyun Kim1,DaeSanKim2, Han-Young Kim1, Hyunseok Lee2 and Lee-Chae Jang3*

*Correspondence: [email protected] Abstract 3Graduate School of Education, Konkuk University, Seoul 143-701, Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials Republic of Korea which is the pioneering work on degenerate versions of special numbers and Full list of author information is polynomials. In recent years, studying degenerate versions regained lively interest of available at the end of the article some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modiﬁed degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa. MSC: 05A19; 05A40; 11B68; 11B73; 11B83 Keywords: Degenerate Bell polynomials; Umbral calculus; Euler polynomials; Degenerate Bernoulli polynomials of the second kind; Degenerate poly-Bernoulli polynomials

1 Introduction and preliminaries In [3, 4], Carlitz studied degenerate Bernoulli and Euler polynomials, which are degenerate versions of the ordinary Bernoulli and Euler polynomials, and investigated some combinatorial results as well as some arithmetical ones. We have witnessed in recent years that, along the same line as Carlitz’s pioneering work, some mathematicians began to ex- plore degenerate versions of quite a few special polynomials and numbers which include the degenerate Bernoulli numbers of the second kind, degenerate Stirling numbers of the first and second kinds, degenerate Cauchy numbers, degenerate Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate gamma function, and so on (see [8, 9, 11, 14–16, 20] and the references therein). They have been studied by various means like combinatorial methods, generating functions, differential equations, umbral calculus techniques, p-adic analysis, and probability theory. Here we would like to point out that the generating functions are also useful in related fields like mathemati- cal physics (see [23, 24]). Moreover, those degenerate versions of some special polynomials found some applications to other areas of mathematics such as differential equations, identities of symmetry, and probability theory [12, 13, 18].

© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 2 of 15

The Bell number Bn countsthenumberofpartitionsofasetwithn elements into disjoint nonempty subsets. The Bell polynomials are natural extensions of Bell numbers and also called Touchard or exponential polynomials (see [2]). As a degenerate version of these Bell polynomials and numbers, the degenerate Bell polynomials and numbers in (18)areintro- duced and studied under the different names of the partially degenerate Bell polynomials and numbers in [16]. The purpose of the present paper is to study the degenerate Bell polynomials and numbers by means of umbral calculus and generating functions. We investigate several properties of those numbers and polynomials which include Dobinski-type formula, recurrence relations, and their derivatives. Moreover, with the help of umbral calculus techniques we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa. Even though the degenerate Bell polynomials were introduced earlier in [16], only some basic properties were studied there by using generating functions. The novelty of this paper is that they are further explored by employing a different method, namely umbral calculus. In particular, this enables us to represent various known families of polynomials in terms of degenerate Bell polynomials and vice versa, which can be viewed as a classical connec- tion problem. This paper is outlined as follows. In Sect. 1, firstly we recall the degenerate exponential functions, the degenerate logarithms, and the degenerate polylogarithm functions which are respectively degenerate versions of exponential functions, logarithms, and polylogarithm functions. Then we go over the definitions of some degenerate special polynomials and numbers, namely the modified degenerate poly-Bernoulli polynomials, the degenerate Bernoulli polynomials of the second kind of order r, the degenerate Stirling numbers of the first and second kinds, and the degenerate Bell polynomials. In addition, we briefly state some basic facts about umbral calculus. For details, we let the reader refer to [21]. In Sect. 2, firstly we derive Dobinski-type formulas, recurrence relations, and the derivatives for the degenerate Bell polynomials. Then we find a formula expressing any polynomial in terms of the degenerate Bell polynomials. We apply this formula to Euler polynomials and to find the inversion formula of (28). By applying the transfer formula we obtain a relation involving the degenerate Bell polynomials, Stirling numbers of the first and second kinds, and the higher-order degenerate Bernoulli polynomials of the second kind. In addition, by using the general formula (26) expressing one Sheffer polynomial in terms of another Sheffer polynomial, we express Euler polynomials and modified degenerate poly-Bernoulli polynomials in terms of the degenerate Bell polynomials and also represent the degenerate Bell polynomials in terms of falling factorials and degenerate Bernoulli polynomials of the second kind. Finally, Sect. 3 is the conclusion of this paper. For any 0 = λ ∈ R, we recall that the degenerate exponential functions are defined by

x x λ 1 eλ(t)=(1+λt) , eλ(t)=eλ(t), (see [9, 14]). (1)

Note that

∞ n x t e (t)= (x) λ (see [9]), (2) λ n, n! n=0 Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 3 of 15

where (x)0,λ =1,(x)n,λ = x(x – λ) ···(x –(n –1)λ)), (n ≥ 1).

The degenerate logarithm logλ(1 + t), which is the compositional inverse of the degenerate exponential function eλ(t) and a motivation for the deﬁnition of degenerate polylogarithm function, is deﬁned by

∞ ∞ n n n–1 t t 1 λ log (1 + t)= λ (1) λ = (λ –1) = (1 + t) –1 ,(see[9]). (3) λ n,1/ n! n–1 n! λ n=1 n=1

Note that limλ→0 logλ(1 + t)=log(1 + t), logλ(eλ(t)) = eλ(logλ(t)) = t. We recall from equation (12) of [9]that,fork ∈ Z, the degenerate polylogarithm functions are deﬁned by

∞ n–1 (–λ) (1)n,1/λ n Li λ(x)= x |x| <1 .(4) k, (n –1)!nk n=1

Note that

∞ xn lim Lik,λ(x)= =Lik(x), λ→0 nk n=1

where Lik(x) is called the polylogarithm.

From (4), we note that Li1,λ(x)=–logλ(1 – x). In [3, 4], Carlitz considered the degenerate Bernoulli polynomials which are given by

∞ n t x t e (t)= β λ(x) .(5) e (t)–1 λ n, n! λ n=0

For x =0,βn,λ = βn,λ(0) are called the degenerate Bernoulli numbers.

It is easy to show that limλ→0 βn,λ(x)=Bn(x), (n ≥ 0), where Bn(x) are the ordinary Bernoulli polynomials given by

∞ t tn ext = B (x) (see [1–18, 20–22, 25, 26]). (6) et –1 n n! n=0

For x =0,Bn = Bn(0) are called the Bernoulli numbers. It is well known that the Euler polynomials are given by

∞ 2 tn ext = E (x) (see [21]). (7) et +1 n n! n=0

For x =0,En = En(0) are called the Euler numbers. From (7), we note that

n n E (x)= E xl (n ≥ 0). (8) n l n–l l=0 Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 4 of 15

Recently, the degenerate poly-Bernoulli polynomials have been deﬁned by means of the degenerate polylogarithms as follows:

∞ n Li λ(1 – eλ(–t)) t k, ex (–t)= B(k) (x) (see [9]). (9) 1–e (–t) λ n,λ n! λ n=0

(1) n ≥ When k =1,wenotethatBn,λ(x)=(–1) βn,λ(x), (n 0). Now, we slightly modify the deﬁ- nition of the degenerate poly-Bernoulli polynomials, called the modiﬁed degenerate poly- Bernoulli polynomials, as follows:

∞ Li (1 – e (–t)) tn k,λ λ ex (t)= β(k)(x) (k ∈ Z), (see [17]). (10) e (t)–1 λ n,λ n! λ n=0

(1) ≥ (k) (k) Note that βn,λ(x)=βn,λ(x), (n 0). When x =0,βn,λ = βn,λ(0) are called the modiﬁed degenerate poly-Bernoulli numbers. The degenerate Bernoulli polynomials of the second kind of order r are deﬁned by

∞ t r tn (1 + t)x = b(r) (x) . (11) log (1 + t) n,λ n! λ n=0

(r) (r) For x =0,bn,λ = bn,λ(0) are called the degenerate Bernoulli numbers of the second kind of order r.Notethat

∞ r r n t x t x (r) t lim (1 + t) = (1 + t) = bn (x) , λ→0 log (1 + t) log(1 + t) n! λ n=0

(r) where bn (x) are called the Bernoulli polynomials of the second kind of order r (see [21]). (1) When r =1,bn(x)=bn (x), (n ≥ 0), are called the Bernoulli polynomials of the second kind. For n ≥ 0, the falling factorial sequence is deﬁned by

(x)0 =1, (x)n = x(x –1)···(x – n +1), (n ≥ 1), (see [21]).

The Stirling numbers of the ﬁrst kind are given by

n l (x)n = S1(n, l)x (n ≥ 0), (see [21]). (12) l=0

As an inversion formula of (12), the Stirling numbers of the second kind are deﬁned as

n n x = S2(n, l)(x)l (n ≥ 0), (see [1, 16, 21]). (13) l=0

In [9], the degenerate Stirling numbers of the ﬁrst kind are deﬁned by Kim and Kim as follows:

n (x)n = S1,λ(n, l)(x)l,λ (n ≥ 0). (14) l=0 Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 5 of 15

Note that limλ→0 S1,λ(n, l)=S1(n, l), (n ≥ 0). As an inversion formula of (14), the Stirling numbers of the second kind are given by

n (x)n,λ = S2,λ(n, l)(x)l (n ≥ 0), (see [11]). (15) l=0

Thus, by (14)and(15), we get

∞ n 1 k t log (1 + t) = S λ(n, k) (k ≥ 0), (see [9]), (16) k! λ 1, n! n=k

and

∞ n 1 k t eλ(t)–1 = S λ(n, k) (k ≥ 0), (see [11]). (17) k! 2, n! n=k

In [16], the degenerate Bell polynomials are deﬁned by

∞ tn x(eλ(t)–1) e = Bel λ(x) . (18) n, n! n=0

When x =1,Beln,λ =Beln,λ(1) are called the degenerate Bell numbers. From (18), we note that n n Bel λ(x + y)= Bel λ(x)Bel λ(y), (n ≥ 0). (19) n, l l, n–l, l=0

For the rest of this section, we will brieﬂy go over some basic facts about umbral calculus. For the details on this fascinating mathematics, the interested reader may refer to [6, 21, 22, 26]. Let C be the ﬁeld of complex numbers, and let F be the set of all formal power series in the variable t over C,givenby ∞ tk F = f (t)= a |a ∈ C . k k! k k=0

Let P = C[x], and let P∗ be the vector space of all linear functionals on P. The action of the linear functional L ∈ P∗ on a polynomial p(x)isdenotedbyL|p(x) , which is linearly extended by the rule

cL + c L |p(x) = c L p(x) + c L p(x) ,

∈ C where c, c . ∞ tk |· P For f (t)= k=0 ak k! , the action f (t) of the linear functional f (t)on is deﬁned by

n f (t)|x = an for all n ≥ 0(see[1, 19, 21, 22]). (20)

Thus, by (20), we get

k n t |x = n!δn,k (n, k ≥ 0), (see [1, 10, 19, 21]), (21) Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 6 of 15

where δn,k is the Kronecker symbol. Let

∞ tk f (t)= L|xk . L k! k=0

Then, by (21), we have

∞ L|xk f (t)|xn = tk|xn = L|xn . L k! k=0

∗ Moreover, the map L → fL(t) is a vector space isomorphism from P onto F. Henceforth, F denotes both the algebra of the formal power series in t and the vector space of all linear functionals on P, and so an element of F will be thought of as both a formal power series and a linear functional. We call F the umbral algebra. The umbral calculus is the study of the umbral algebra. Note that eyt|p(x) = p(y). The order O(f (t)) of a power series f (t)(= 0) is the smallest integer k for which the coeﬃcient of tk does not vanish. An invertible series is a series f (t)withO(f (t))=0,whileadeltaseriesisaseries f (t)withO(f (t)) = 1. Let f (t), g(t) ∈ F with O(f (t)) = 1 and O(g(t))=0.Thenitisknown

that there exists a unique sequence sn(x), (n ≥ 0), of polynomials satisfying the conditions

k g(t) f (t) |sn(x) = n!δn,k (n, k ≥ 0), (see [21, p. 17]). (22)

The sequence sn(x) is called the Sheﬀer sequence for (g(t), f (t)), which we denote by sn(x) ∼

(g(t), f (t)). For pn(x) ∼ (1, f (t)), qn(x) ∼ (1, g(t)), we have the transfer formula f (t) n q (x)=x x–1p (x), (n ≥ 1), (see [21, p. 51]). (23) n g(t) n

It is well known that sn(x) ∼ (g(t), f (t)) if and only if

∞ 1 s (x) exf (t) = k tk (see [21, p. 107]) (24) k! g(f (t)) k=0

for all x ∈ C,wheref (t) is the compositional inverse of f (t)satisfyingf (f (t)) = f (f (t)) = t.

Let sn(x) ∼ (g(t), f (t)), and let rn(x) ∼ (h(t), l(t)). Then we have

∞ sn(x)= an,mrm(x), (n ≥ 0), (25) m=0

where 1 h(f (t)) m n an,m = (l f (t) |x (see [21, p. 132]). (26) m! g(f (t)) Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 7 of 15

2 Some identities of degenerate Bell polynomials From (18), we note that

∞ ∞ ∞ tn tn x(eλ(t)–1) l Bel λ(x) = e = x S λ(n, l) n, n! 2, n! n=0 l=0 n=l ∞ n n l t = S λ(n, l)x . (27) 2, n! n=0 l=0

Thus, by (27), we get

n l Beln,λ(x)= S2,λ(n, l)x (n ≥ 0). (28) l=0

On the other hand, by (18), we have

∞ ∞ ∞ 1 xl xl tn x(eλ(t)–1) –x x(eλ(t)) l –x e = e e = e (t)=e (l) λ ex l! λ l! n, n! l=0 l=0 n=0 ∞ ∞ 1 xl tn = (l) λ . (29) ex l! n, n! n=0 l=0

Now, (18)and(29) together yield the following Dobinski-type formula.

Lemma 1 For n ≥ 0, we have

∞ 1 (l)n,λ l Bel λ(x)= x . n, ex l! l=0

For n ∈ N, with the help of Lemma 1,weobtain

n n –1 x Bel λ(x)(1 – λ) λ k –1 k–1, n–k, k=1 n ∞ n –1 –x (l)k–1,λ l = x (1 – λ) λe x k –1 n–k, l! k=1 l=0 ∞ l n–1 –x x n –1 = xe (1 – λ) λ(l) λ l! k n–1–k, k, l=0 k=0 ∞ ∞ l l –x x –x x = xe (l +1–λ) λ = xe (l +1) λ l! n–1, (l +1)! n, l=0 l=0 ∞ l –x x = e (l) λ =Bel λ(x). (30) l! n, n, l=1

Therefore, by (30), we have shown the following proposition. Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 8 of 15

Proposition 2 For n ∈ N, we have n n –1 Bel λ(x)=x Bel λ(x)(1 – λ) λ. n, k –1 k–1, n–k, k=1

By Proposition 2,weget n Beln,λ(x) n –1 = Bel λ(x)(1 – λ) λ x k –1 k–1, n–k, k=1 n–1 n –1 = Bel λ(x)(1 – λ) λ k k, n–k–1, k=0 n–1 n –1 = Bel λ(x)(1) λ (n ∈ N). (31) k k, n–k, k=0

By using (31), we may proceed to deriving the following equation: n Beln+1,λ(x) n = Bel λ(x)(1) λ x k k, n+1–k, k=0 n n = Bel λ(x)(1) λ 1–(n – k)λ k k, n–k, k=0 n n n–1 n = Bel λ(x)(1) λ – λ (n – k)Bel λ(x)(1) λ k k, n–k, k k, n–k, k=0 k=0 n n n–1 n –1 = Bel λ(x)(1) λ – nλ Bel λ(x)(1) λ k k, n–k, k k, n–k, k=0 k=0 n n Beln,λ(x) = Bel λ(x)(1) λ – nλ (n ∈ N). (32) k k, n–k, x k=0

By (32), we get n n Beln+1,λ(x) Beln,λ(x) Bel λ(x)(1) λ = + nλ (n ∈ N). (33) k k, n–k, x x k=0

From (33), we note that

∞ d tn d x(eλ(t)–1) x(eλ(t)–1) Bel λ(x) = e = eλ(t)–1 e dx n, n! dx n=0

x(eλ(t)–1) x(eλ(t)–1) = eλ(t)e – e ∞ ∞ n n tn tn = Bel λ(x)(1) λ – Bel λ(x) m m, n–m, n! n, n! n=0 m=0 n=0 ∞ n Beln+1,λ(x) Beln,λ(x) t = + nλ –Bel λ(x) . (34) x x n, n! n=0 Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 9 of 15

By comparing the coeﬃcients on both sides of (34), we obtain the following theorem.

Theorem 3 For n ≥ 1, we have

d 1 Bel λ(x)= Bel λ(x)+(nλ – x)Bel λ(x) . dx n, x n+1, n,

Let Pn = p(x) ∈ C[x]| deg p(x) ≤ n ,(n ≥ 0).

For p(x) ∈ Pn,let

n p(x)= amBelm,λ(x), (n ≥ 0). (35) m=0

∼ Then, by (22) and noting that Beln,λ(x) (1, logλ(1 + t)), we get

n n m| m| logλ(1 + t) p(x) = al logλ(1 + t) Bell,λ(x) = alδl,ml!=amm!. (36) l=0 l=0

Hence, by (36), we get

1 m a = log (1 + t) |p(x) ,(m ≥ 0). (37) m m! λ

Therefore, by (35)and(37), we obtain the following lemma.

Lemma 4 For p(x) ∈ Pn, we have

n p(x)= amBelm,λ(x), m=0

where 1 m a = log (1 + t) |p(x) . m m! λ

Let us take p(x)=En(x) ∈ Pn.Then,by(8) and Lemma 4,weget

n En(x)= amBelm,λ(x), (n ≥ 0), (38) m=0

where 1 m a = log (1 + t) |E (x) m m! λ n n 1 n m = E log (1 + t) |xl m! l n–l λ l=m Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 10 of 15

n l k n t l = E S λ(k, m) |x l n–l 1, k! l=m k=m n l n 1 k l = E S λ(k, m) t |x l n–l 1, k! l=m k=m n n = E S λ(l, m). (39) l n–l 1, l=m

Thus, (38)and(39) together give the following theorem.

Theorem 5 For n ≥ 0, we have

n n n E (x)= E S λ(l, m) Bel λ(x). n l n–l 1, m, m=0 l=m

n Now, in order to get the inversion formula of (28), we take p(x)=x ∈ Pn.Then,by Lemma 4,weget

n n x = akBelk,λ(x), (40) k=0

where

n 1 k S (m, k) a = log (1 + t) |xn = 1,λ tm|xn k k! λ m! m=k n S1,λ(m, k) = n!δ = S λ(n, k). (41) m! m,n 1, m=k

Now, from (40)and(41), we have the following theorem.

Theorem 6 For n ≥ 0, we have

n n x = S1,λ(n, k)Belk,λ(x). k=0

∼ n ∼ For Beln,λ(x) (1, logλ(1 + t)), x (1, t), by using the transfer formula in (23), we get n t –1 n Beln,λ(x)=x x x logλ(1 + t) t n n–1 = x S (n –1,l)(x) log (1 + t) 2 l λ l=0 n–1 t n = x S (n –1,l) (x) . (42) 2 log (1 + t) l l=0 λ Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 11 of 15

Now, we observe that

∞ t n tk l (x) = b(n) S (l, m)xm log (1 + t) l k,λ k! 1 λ k=0 m=0 l m m = b(n) xm–kS (l, m) k,λ k 1 m=0 k=0 l m m = S (l, m)b(n) xk. (43) k 1 m–k,λ m=0 k=0

Thus, by (42)and(43), we get

n–1 l m m (n) k Bel λ(x)=x S (l, m)b S (n –1,l)x n, k 1 m–k,λ 2 l=0 m=0 k=0 n–1 l l m = x S (l, m)b(n) S (n –1,l)xk k 1 m–k,λ 2 l=0 k=0 m=k n–1 n–1 l m = x S (l, m)S (n –1,l)b(n) xk. (44) k 1 2 m–k,λ k=0 l=k m=k

From Proposition 2 and (44), we note that

n–1 n–1 l Bel λ(x) m n, = S (l, m)S (n –1,l)b(n) xk x k 1 2 m–k,λ k=0 l=k m=k n n –1 = Bel λ(x)(1 – λ) λ (n ∈ N). (45) k –1 k–1, n–k, k=1

Therefore, by combining (44)with(45), we obtain the following theorem.

Theorem 7 For n ∈ N, we have

n n–1 n–1 l n –1 m (n) k Bel λ(x)(1 – λ) λ = S (l, m)S (n –1,l)b x . k –1 k–1, n–k, k 1 2 m–k,λ k=1 k=0 l=k m=k

Let us consider the following two Sheﬀer sequences: 1 β(k)(x) ∼ g(t), f (t)= eλt –1 ,Bel(x) ∼ 1, log (1 + t) , n,λ λ n,λ λ

1 Lik,λ(1–eλ(–t)) where g(t) is uniquely determined by ¯ = .From(25)and(26), we have g(f (t)) eλ(t)–1

n (k) βn,λ(x)= an,mBelm,λ(x), (46) m=0 Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 12 of 15

where we have m 1 Lik,λ(1 – eλ(–t)) 1 | n an,m = logλ 1+ log(1 + λt) x m! eλ(t)–1 λ n l S λ(l, m) Li λ(1 – eλ(–t)) 1 = 1, k, log(1 + λt) |xn l! eλ(t)–1 λ l=m n S (l, m) Li (1 – e (–t)) 1 1,λ k,λ λ log λ l| n = l (1 + t) x λ eλ(t)–1 l! l=m n n λr S1,λ(l, m) Lik,λ(1 – eλ(–t)) r| n = l S1,λ(r, l) t x λ r! eλ(t)–1 l=m r=l n n n r–l Lik,λ(1 – eλ(–t)) n–r = S1,λ(l, m)S1,λ(r, l)λ |x r eλ(t)–1 l=m r=l n n ∞ n r–l (k) 1 j n–r = S λ(l, m)S λ(r, l)λ β t |x r 1, 1, j,λ j! l=m r=l j=0 n n n–r n r–l (k) 1 = S λ(l, m)S λ(r, l)λ β (n – r)!δ r 1, 1, j,λ j! j,n–r l=m r=l j=0 n n n r–l (k) = S λ(l, m)S λ(r, l)λ β . (47) r 1, 1, n–r,λ l=m r=l

Therefore, by (46)and(47), we obtain the following theorem.

Theorem 8 For n ≥ 0, we have n n n (k) n r–l (k) β (x)= S λ(l, m)S λ(r, l)λ β Bel λ(x). n,λ r 1, 1, n–r,λ m, m=0 l=m r=l

∼ ∼ t For Beln,λ(x) (1, logλ(1 + t)), (x)n (1, e –1),wehave

n Beln,λ(x)= an,m(x)m, (48) m=0

where we have 1 m a = e(eλ(t)–1) –1 |xn n,m m! 1 m m = (–1)m–l el(eλ(t)–1) |xn m! l l=0 m n 1 m m–l 1 k n = (–1) Bel λ(l) t |x m! l k, k! l=0 k=0 m 1 m m–l = (–1) Bel λ(l). (49) m! l n, l=0

Now, by (48)and(49), we get the following theorem. Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 13 of 15

Theorem 9 For n ≥ 0, we have n m 1 m m–l Bel λ(x)= (–1) Bel λ(l) (x) . n, m! l n, m m=0 l=0

Consider the following two Sheﬀer sequences: t t Bel λ(x) ∼ 1, log (1 + t) , b (x) ∼ , e –1 . n, λ n et –1

Then, by (25)and(26), we get

n Beln,λ(x)= an,mbm(x), (50) m=0

where we have ∞ eλ(t)–1 n Bl l n an,0 = |x = eλ(t)–1 |x eeλ(t)–1 –1 l! l=0 ∞ m Bl m n = S λ(m, l) t |x m! 2, m=0 l=0 n m n Bl = S λ(m, l)n!δ = S λ(n, l)B , (51) m! 2, n,m 2, l m=0 l=0 l=0

and, for m ≥ 1, 1 e (t)–1 λ eλ(t)–1 m n an,m = e –1 |x m! eeλ(t)–1 –1

1 1 eλ(t)–1 m–1 n eλ(t)–1 m–1 n = e –1 eλ(t) x – e –1 x m! m!

n 1 (1)k,λ m–1 1 m–1 = eeλ(t)–1 –1 tk xn – eeλ(t)–1 –1 xn m! k! m! k=0 1 n n 1 eλ(t)–1 m–1 n–k eλ(t)–1 m–1 n = (1)k,λ e –1 x – e –1 x m! k m! k=0 1 n–1 n eλ(t)–1 m–1 k = (1) λ e –1 |x m! k n–k, k=0 1 n–1 n m–1 m –1 m–1–l l(eλ(t)–1) k = (1) λ (–1) e |x m! k n–k, l k=0 l=0 n–1 m–1 1 n m –1 m–1–l = (1) λ (–1) Bel λ(l). (52) m! k n–k, l k, k=0 l=0

Now, (50)–(52) altogether yield the following result. Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 14 of 15

Theorem 10 For n ≥ 0, we have

n Beln,λ(x)= S2,λ(n, l)Bl l=0 n n–1 m–1 1 n m –1 m–1–l + (1) λ (–1) Bel λ(l) b (x). m! k n–k, l k, n m=1 k=0 l=0

3Conclusion We derived Dobinski-type formulas, recurrence relations, and the derivatives for the degenerate Bell polynomials. After finding a formula expressing any polynomial in terms of the degenerate Bell polynomials, we applied this formula to Euler polynomials and powers of x. We obtained a relation involving the degenerate Bell polynomials, Stirling numbers of the first and second kinds, and the higher-order degenerate Bernoulli polynomials of the second kind. In addition, by using the general formula of expressing one Sheffer polynomial in terms of another Sheffer polynomial, we expressed Euler polynomials and modified degenerate poly-Bernoulli polynomials in terms of the degenerate Bell polynomials and also represented the degenerate Bell polynomials in terms of degenerate Bernoulli polynomials of the second kind and falling factorials. As we briefly mentioned in Sect. 1, degenerate versions of many special polynomials and numbers have applications to identities of symmetry, differential equations, and probability theory as well as to number theory and combinatorics. In more detail, some new combinatorial identities were found from infinite families of ordinary differential equations which are satisfied by the generating functions of some degenerate special polynomials (see [12]). Many identities of symmetry have been discovered for various degenerate versions of quite a few special polynomials in recent years (see [13]). The generating functions of the moments of certain random variables were used in order to derive some identities connecting some special numbers and moments of random variables (see [18]). As one of our future projects, we would like to continue to study degenerate versions of certain special polynomials and numbers and their applications to physics, science, and engineering as well as to mathematics.

Acknowledgements We would like to thank the referees for their useful comments and suggestions. Also, we thank Friday mathematics seminar team of the Kwangwoon University for suggesting the present research subject.

Funding This research received no external funding.

Availability of data and materials Not applicable.

Competing interests The authors declare that they have no competing interests.

Authors’ contributions TK and DSK conceived of the framework and structured the whole paper; TK and DSK wrote the paper; LCJ, HYK, and HL checked the results of the paper; DSK and TK completed the revision of the article. All authors have read and agreed to the published version of the manuscript.

Author details 1Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. 2Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea. 3Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea. Kim et al. Journal of Inequalities and Applications (2020)2020:226 Page 15 of 15

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 4 August 2020 Accepted: 21 September 2020

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