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Kim Advances in Difference Equations (2020) 2020:687 https://doi.org/10.1186/s13662-020-03152-4 RESEARCH Open Access Degenerate Lah–Bell polynomials arising from degenerate Sheffer sequences Hye Kyung Kim 1* * Correspondence: [email protected] 1 Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea Abstract Umbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim–Kim (J. Math. Anal. Appl. 493(1):124521, 2021) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined the λ-linear functionals and λ-differential operators, respectively, instead of the linear functionals and the differential operators of umbral calculus established by Rota. In this paper, the author gives various interesting identities related to the degenerate Lah–Bell polynomials and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derives the inversion formulas of these identities. MSC: 11B73; 11B83; 05A19 Keywords: Degenerate Lah–Bell numbers and polynomials; The degenerate Sheffer sequence; The degenerate Bernoulli (Euler) polynomials; The degenerate Frobenius–Euler polynomials; The degenerate Daehee polynomials; The degenerate Bell polynomials 1 Introduction It is important to note that many academics in the field of mathematics have been re- searching various degenerate versions of special polynomials and numbers not only in some arithmetic and combinatorial aspects but also in applications to differential equa- tions, identities of symmetry and probability theory [9, 12, 14, 1623], beginning with Carlitz’s degenerate Bernoulli polynomials and the degenerate Euler polynomials [2]. Moreover, umbral calculus, established by Rota in the 1970s, was based on modern con- cepts such as linear functionals, linear operators, and adjoints [28]. Umbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials [5, 6, 24, 25, 28]. Recently, Kim–Kim [11] in- troduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined the λ- linear functionals and λ- differential operators, respectively, instead of the linear func- tionals and the differential operators used by Rota [28]. Also, Kim et al. introduced the Lah–Bell polynomials and studied some identities of Lah–Bell polynomials [10, 25]. The two papers mentioned above inspired me. So, I focus on finding the noble identities of degenerate Lah–Bell polynomials in terms of quite a few well-known special polynomials © 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/.
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Page 1: Degenerate Lah–Bell polynomials arising from degenerate ......KimAdvancesinDifferenceEquations20202020:687 Page2of16 andnumbersarisingfromthedegenerateSheffersequence.Inaddition,theauthorderives

Kim Advances in Difference Equations (2020) 2020:687 https://doi.org/10.1186/s13662-020-03152-4

R E S E A R C H Open Access

Degenerate Lah–Bell polynomials arisingfrom degenerate Sheffer sequencesHye Kyung Kim1*

*Correspondence: [email protected] of MathematicsEducation, Daegu CatholicUniversity, Gyeongsan 38430,Republic of Korea

AbstractUmbral calculus is one of the important methods for obtaining the symmetricidentities for the degenerate version of special numbers and polynomials. Recently,Kim–Kim (J. Math. Anal. Appl. 493(1):124521, 2021) introduced the λ-Sheffer sequenceand the degenerate Sheffer sequence. They defined the λ-linear functionals andλ-differential operators, respectively, instead of the linear functionals and thedifferential operators of umbral calculus established by Rota. In this paper, the authorgives various interesting identities related to the degenerate Lah–Bell polynomialsand special polynomials and numbers by using degenerate Sheffer sequences, and atthe same time derives the inversion formulas of these identities.

MSC: 11B73; 11B83; 05A19

Keywords: Degenerate Lah–Bell numbers and polynomials; The degenerate Sheffersequence; The degenerate Bernoulli (Euler) polynomials; The degenerateFrobenius–Euler polynomials; The degenerate Daehee polynomials; The degenerateBell polynomials

1 IntroductionIt is important to note that many academics in the field of mathematics have been re-searching various degenerate versions of special polynomials and numbers not only insome arithmetic and combinatorial aspects but also in applications to differential equa-tions, identities of symmetry and probability theory [9, 12, 14, 16–23], beginning withCarlitz’s degenerate Bernoulli polynomials and the degenerate Euler polynomials [2].

Moreover, umbral calculus, established by Rota in the 1970s, was based on modern con-cepts such as linear functionals, linear operators, and adjoints [28]. Umbral calculus isone of the important methods for obtaining the symmetric identities for the degenerateversion of special numbers and polynomials [5, 6, 24, 25, 28]. Recently, Kim–Kim [11] in-troduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined theλ- linear functionals and λ- differential operators, respectively, instead of the linear func-tionals and the differential operators used by Rota [28]. Also, Kim et al. introduced theLah–Bell polynomials and studied some identities of Lah–Bell polynomials [10, 25]. Thetwo papers mentioned above inspired me. So, I focus on finding the noble identities ofdegenerate Lah–Bell polynomials in terms of quite a few well-known special polynomials

© 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 originalauthor(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or otherthird party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit lineto the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view acopy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Kim Advances in Difference Equations (2020) 2020:687 Page 2 of 16

and numbers arising from the degenerate Sheffer sequence. In addition, the author derivesthe inversion formulas of the identities obtained in this paper. They include the degenerateand other special polynomials and numbers such as Lah numbers, the degenerate fallingfactorial, the degenerate Bernoulli polynomials and numbers, degenerate Frobenius–Eulerpolynomials and numbers of order r, the degenerate Deahee polynomials, the degenerateBell polynomials, and degenerate Stirling numbers of the first and second kinds.

Now, we give some definitions and properties needed in this paper.The unsigned Lah number L(n, k) counts the number of ways of all distributions of

n balls, labeled 1, 2, . . . , n, among k unlabeled, contents-ordered boxes, with no box leftempty and have an explicit formula

L(n, k) =(

n – 1k – 1

)n!k!

(see [10, 25]). (1)

From (1), the generating function of L(n, k) is given by

1k!

(t

1 – t

)k

=∞∑

n=k

L(n, k)tn

n!, (k ≥ 0) (see [10, 25–27]). (2)

Recently, Lah–Bell polynomials were introduced by Kim–Kim to be

ex( 11–t –1) =

∞∑n=0

BLn(x)

tn

n!(see [10]). (3)

When x = 1, BLn = BL

n(1) are called Lah–Bell numbers.For any nonzero λ ∈ R, the degenerate exponential function is defined by

exλ(t) = (1 + λt)

xλ , eλ(t) = (1 + λt)

1λ (see [2, 9, 12, 14, 16, 17, 19–23]). (4)

By Taylor expansion, we get

exλ(t) =

∞∑n=0

(x)n,λtn

n!(see [9, 12, 14, 16, 17, 19–23]), (5)

where (x)0,λ = 1, (x)n,λ = x(x – λ)(x – 2λ) · · · (x – (n – 1)λ) (n ≥ 1).It is known that

(1 – t)–m =∞∑l=0

(–m

l

)(–1)ltl =

∞∑l=0

〈m〉ltl

l!(see [1, 4]). (6)

where 〈x〉0 = 1, 〈x〉n = x(x + 1)(x + 2) · · · (x + n – 1), (n ≥ 1).The degenerate Bernoulli polynomials and degenerate Euler polynomials of order r, re-

spectively, are given by the generating functions

(t

eλ(t) – 1

)r

exλ(t) =

∞∑n=0

β(r)n,λ(x)

tn

n!(see [2, 7, 21]) (7)

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Kim Advances in Difference Equations (2020) 2020:687 Page 3 of 16

and

(2

eλ(t) + 1

)r

exλ(t) =

∞∑n=0

E(r)n,λ(x)

tn

n!(see [2, 7, 15, 18]). (8)

We note that β(r)n,λ = β

(r)n,λ(0) and E(r)

n,λ = E(r)n,λ(0) (n ≥ 0) are called degenerate Bernoulli and

degenerate Euler numbers of order r, respectively.Kim et al. introduced the degenerate Frobenius–Euler polynomials of order r defined by

(1 – u

eλ(t) – u

)r

exλ(t) =

∞∑n=0

h(r)n,λ(x|u)

tn

n!, (u �= 1, u ∈C) (k ≥ 0) (see [15]). (9)

When x = 0, h(r)n,λ(u) = h(r)

n,λ(0|u) are called degenerate Frobenius–Euler numbers of or-der r.

The degenerate Daehee polynomials are defined by

logλ(1 + t)t

(1 + t)x =∞∑

n=0

Dn,λ(x)tn

n!(see [9]). (10)

Here logλ(1 + t) = 1λ

((1 + t)λ – 1) and logλ(eλ(t)) = eλ(logλ(t)) = t.When x = 0, Dn,λ = Dn,λ(0) are called degenerate Daehee numbers.The Bell polynomials are defined by the generating function

ex(et–1) =∞∑

n=0

Beln(x)tn

n!(see [8, 19, 22]). (11)

Kim–Kim introduced the degenerate Bell polynomial given by

exλ

(eλ(t) – 1

)=

∞∑l=0

Bell,λ(x)tl

l!(see [19]). (12)

For n ≥ 0, it is well known that the Stirling numbers of the first and second kind, respec-tively, are given by

(x)n =n∑

l=0

S1(n, l)xl and1k!

(log(1 + t)

)k =∞∑

n=k

S1(n, k)tn

n!(see [2, 3]) (13)

and

xn =n∑

l=0

S2(n, l)(x)l and1k!

(et – 1

)k =∞∑

n=k

S2(n, k)tn

n!(see [2, 3]), (14)

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

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Kim Advances in Difference Equations (2020) 2020:687 Page 4 of 16

Moreover, the degenerate Stirling numbers of the first and second kind, respectively, aregiven by

(x)n =n∑

l=0

S1,λ(n, l)(x)l,λ and

1k!

(logλ(1 + t)

)k =∞∑

n=k

S1,λ(n, k)tn

n!(k ≥ 0) (see [12, 14]),

(15)

and

(x)n,λ =n∑

l=0

S2,λ(n, l)(x)l and

1k!

(eλ(t) – 1

)k =∞∑

n=k

S2,λ(n, k)tn

n!(k ≥ 0) (see [12, 14]).

(16)

For k ∈ Z, Kim–Kim introduced the modified polyexponential function as

Eis(x) =∞∑

n=1

xn

(n – 1)!ns (see [18, 21]). (17)

By (17), we see that Ei1(x) = ex – 1.Kim–Jang considered the type 2 degenerate poly-Euler polynomials which are given by

the generating function to be

Eis(log(1 + 2t))t(eλ(t) + 1)

exλ(t) =

∞∑n=0

E (s)n,λ(x)

tn

n!(see [13]). (18)

When x = 0, E (s)n,λ = E (s)

n,λ(0) are called type 2 degenerate poly-Euler numbers.Since Ei1(log(1 + 2t)) = 2t, we see that E (1)

n,λ(x) = En,λ(x) (n ≥ 0) are the degenerate Eulerpolynomials.

Let C be the complex number field and let F be the set of all power series in the variablet over C with

F =

{f (t) =

∞∑k=0

aktk

k!

∣∣∣ak ∈C

}. (19)

Let P = C[x] and P∗ be the vector space all linear functional on P.

Pn ={

P(x) ∈C[x]|deg P(x) ≤ n}

(n ≥ 0). (20)

Then Pn is an (n + 1)-dimensional vector space over C.Recently, Kim–Kim [11] considered the λ-linear functional and λ-differential operator

as follows:For f (t) =

∑∞k=0 ak

tk

k! ∈ F and a fixed nonzero real number λ, each λ gives rise to thelinear functional 〈f (t)|·〉λ on P, called λ-linear functional given by f (t), which is defined

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Kim Advances in Difference Equations (2020) 2020:687 Page 5 of 16

by

⟨f (t)|(x)n,λ

⟩λ

= an, for all n ≥ 0 (see [11]). (21)

and in particular 〈tk|(x)n,λ〉λ = n!δn,k , for all n, k ≥ 0, where δn,k is Kronecker’s symbol.For λ = 0, we observe that the linear functional 〈f (t)|·〉0 agrees with the one in 〈f (t)|xn〉 =

ak , (k ≥ 0).For each λ ∈ R, and each nonnegative integer k, they also defined the differential oper-

ator on P by

(tk)

λ(x)n,λ =

⎧⎨⎩

(n)k(x)n–k,λ, if k ≤ n,

0 if k ≥ n (see [11]).(22)

and for any power series f (t) =∑∞

k=0 aktk

k! ∈F , (f (t))λ(x)n,λ =∑n

k=0(n

k)ak(x)n–k,λ (n ≥ 0).

Note that different λ give rise to different linear functionals on P (see [11] p. 5, p. 8).The order o(f (t)) of a power series f (t)(�= 0) is the smallest integer k for which the coeffi-

cient of tk does not vanish. The series f (t) is called invertible if o(f (t)) = 0 and such serieshas a multiplicative inverse 1/f (t) of f (t). f (t) is called a delta series if o(f (t)) = 1 and it hasa compositional inverse f (t) of f (t) with f (f (t)) = f (f (t)) = t.

Let f (t) and g(t) be a delta series and an invertible series, respectively. Then there existunique sequences sn,λ(x) such that we have the orthogonality conditions

⟨g(t)

(f (t)

)k|sn,λ(x)⟩λ

= n!δn,k (n, k ≥ 0) (see [11]). (23)

The sequences sn,λ(x) are called the λ-Sheffer sequences for (g(t), f (t)), which are denotedby sn,λ(x) ∼ (g(t), f (t))λ.

The sequence sn,λ(x) ∼ (g(t), f (t))λ if and only if

1g(f (t))

exλ

(f (t)

)=

∞∑k=0

sk,λ(x)k!

tk (n, k ≥ 0) (see [11]). (24)

Assume that, for each λ ∈ R∗ of the set of nonzero real numbers, sn,λ(x) is λ-Sheffer for

(gλ(t), fλ(t)). Assume also that limλ→0 fλ(t) = f (t) and limλ→0 gλ(t) = g(t), for some delta se-ries f (t) and an invertible series g(t). Then limλ→0 f λ(t) = f (t), where is the compositionalinverse of f (t) with f (f (t)) = f (f (t)) = t. Let limλ→0 sk,λ(x) = sk(x). In this case, Kim–Kimcalled this the family {sn,λ(x)}λ∈R–{0} of λ-Sheffer sequences sn,λ are the degenerate (Shef-fer) sequences for the Sheffer polynomial sn(x).

Let sn,λ(x) ∼ (g(t), f (t))λ and rn,λ(x) ∼ (h(t), g(t))λ (n ≥ 0). Then

sn,λ(x) =n∑

k=0

μn,krk,λ(x) (n ≥ 0),

where μn,k =1k!

⟨h(f (t))g(f (t))

(l(f (t)

))k∣∣∣(x)n,λ

⟩λ

(n, k ≥ 0) (see [11]).

(25)

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Kim Advances in Difference Equations (2020) 2020:687 Page 6 of 16

2 Degenerate Lah–Bell polynomials arising from degenerate Sheffersequences

In this section, we derive several identities between the degenerate Lah–Bell polynomialsand some other polynomials arising from degenerate Sheffer sequences.

Kim–Kim introduced the degenerate Lah–Bell polynomials given by

exλ

(t

1 – t

)=

∞∑n=0

BLn,λ(x)

tn

n!(n, k ≥ 0) (see [10]). (26)

When x = 1, BLn,λ := BL

n,λ(1) are called the nth degenerate Lah–Bell numbers.When λ → 0, limλ→0 BL

n,λ = BLn are the nth Lah–Bell numbers.

For n ∈N∪ {0} and P(x) =∑n

k=0 ZkBLk,λ(x) ∈ Pn,

by using (23), we observe that

⟨(t

1 + t

)k∣∣∣P(x)⟩λ

=n∑

l=0

Zl

⟨(t

1 + t

)k∣∣∣BLl,λ(x)

⟩λ

=n∑

l=0

Zll!δk,l = k!Zk . (27)

From (27), we have

Zk =1k!

⟨(t

1 + t

)k∣∣∣P(x)⟩λ

. (28)

Thus, we have

P(x) =n∑

k=0

ZkBLk,λ(x) where Zk =

1k!

⟨(t

1 + t

)k∣∣∣P(x)⟩λ

. (29)

Theorem 1 For n ∈N∪ {0}, we have

BLn,λ(x) =

n∑k=0

L(n, k)(x)k,λ =n∑

k=0

(1k!

k∑l=0

(kl

)(–1)k–l〈l〉n

)(x)k,λ. (30)

As the inversion formula of (30), we have

(x)n,λ =n∑

k=0

(–1)n–kL(n, k)BLk,λ(x) =

n∑k=0

(1k!

k∑l=0

(kl

)(–1)l+n〈l〉n

)BL

k,λ(x). (31)

Proof From (5), (24) and (26), we consider the following two Sheffer sequences:

BLn,λ(x) ∼

(1,

t1 + t

and (x)n,λ ∼ (1, t)λ. (32)

From (6), (25) and (32), we have

BLn,λ(x) =

n∑k=0

μn,k(x)k,λ, (33)

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Kim Advances in Difference Equations (2020) 2020:687 Page 7 of 16

where

μn,k =1k!

⟨(t

1 – t

)k∣∣∣(x)n,λ

⟩λ

= L(n, k),

or μn,k =1k!

k∑l=0

(kl

)(–1)k–l

⟨(1

1 – t

)l∣∣∣(x)n,λ

⟩λ

=1k!

k∑l=0

(kl

)(–1)k–l〈l〉n.

(34)

Therefore, we have the identity (30).To find the inversion formula of (30), let P(x) = (x)n,λ. From (29),

(x)n,λ =n∑

k=0

ZkBLk,λ(x), (35)

where

Zk =1k!

⟨(t

1 + t

)k∣∣∣(x)n,λ

⟩λ

= (–1)n–kL(n, k),

or Zk =1k!

k∑l=0

(kl

)(–1)l

⟨(1

1 + t

)l∣∣∣(x)n,λ

⟩λ

=1k!

k∑l=0

(kl

)(–1)l+n〈l〉n.

(36)

Therefore, from (35) and (36), we have the identity (31). �

Theorem 2 For n ∈N∪ {0} and r ∈ N, we have

BLn,λ(x) =

n∑k=0

( n∑l=k

n–l∑m=0

(nl

)(1)m+1,λ

(m + 1)L(l, k)L(n – l, m)

)βk,λ(x). (37)

As the inversion formula of (37), we have

βn,λ(x) =n∑

k=0

(1k!

k∑l=0

n∑m=0

(kl

)(l + n – m – 1

n – m

)(nm

)(–1)l+n–m(n – m)!βm,λ

)BL

k,λ(x)

=n∑

k=0

( n∑m=0

(nm

)(–1)n–m–kL(n – m, k)βm,λ

)BL

k,λ(x),

(38)

where βn,λ(x) are the degenerate Bernoulli polynomials.

Proof From (7), (24) and (26), we consider the following two degenerate Sheffer sequences:

BLn,λ(x) ∼

(1,

t1 + t

and βn,λ(x) ∼(

eλ(t) – 1t

, t)

λ

. (39)

From (2), (5) and (25), we have

BLn,λ(x) =

n∑k=0

μn,kβk,λ(x), (40)

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Kim Advances in Difference Equations (2020) 2020:687 Page 8 of 16

where

μn,k =1k!

⟨(eλ( t1–t ) – 1

t1–t

)(t

1 – t

)k∣∣∣(x)n,λ

⟩λ

=⟨(eλ( t

1–t ) – 1t

1–t

)∣∣∣(

1k!

(t

1 – t

)k)λ

(x)n,λ

⟩λ

=n∑

l=k

(nl

)L(l, k)

⟨(eλ( t1–t ) – 1

t1–t

)∣∣∣(x)n–l,λ

⟩λ

=n∑

l=k

(nl

)L(l, k)

⟨ ∞∑m=1

(1)m,λ

m!

(t

1 – t

)m–1∣∣∣(x)n–l,λ

⟩λ

=n∑

l=k

(nl

)L(l, k)

⟨ ∞∑m=0

(1)m+1,λ

(m + 1)!

(t

1 – t

)m∣∣∣(x)n–l,λ

⟩λ

=n∑

l=k

(nl

)L(l, k)

n–l∑m=0

(1)m+1,λ

(m + 1)

⟨1

m!

(t

1 – t

)m∣∣∣(x)n–l,λ

⟩λ

=n∑

l=k

(nl

)L(l, k)

n–l∑m=0

(1)m+1,λ

(m + 1)L(n – l, m).

(41)

To find the inversion formula of (37), let P(x) = βn,λ(x). From (29), we have

βn,λ(x) =n∑

k=0

ZkBLk,λ(x) (n ≥ 0).

where

Zk =1k!

⟨(t

1 + t

)k∣∣∣βn,λ(x)⟩λ

=1k!

k∑l=0

(kl

)(–1)l

⟨(1

1 + t

)l∣∣∣βn,λ(x)⟩λ

=1k!

k∑l=0

(kl

)(–1)l

n∑ν=0

(l + ν – 1

ν

)(–1)ν

⟨tν

∣∣∣n∑

m=0

(nm

)βm,λ(x)n–m,λ

⟩λ

=1k!

k∑l=0

(kl

)(–1)l

n∑ν=0

(l + ν – 1

ν

)(–1)ν

n∑m=0

(nm

)βm,λ

⟨tν |(x)n–m,λ

⟩λ

=1k!

k∑l=0

(kl

)(l + n – m – 1

n – m

)(–1)l+n–m

n∑m=0

(nm

)βm,λ(n – m)!.

(42)

Stated differently, we get

Zk =1k!

⟨(t

1 + t

)k∣∣∣βn,λ(x)⟩λ

=⟨

1k!

(t

1 + t

)k∣∣∣n∑

m=0

(nm

)βm,λ(x)n–m,λ

⟩λ

(43)

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Kim Advances in Difference Equations (2020) 2020:687 Page 9 of 16

=n∑

m=0

(nm

)βm,λ

⟨1k!

(t

1 + t

)k∣∣∣(x)n–m,λ

⟩λ

=n∑

m=0

(nm

)βm,λ(–1)n–m–kL(n – m, k).

Therefore, from (42) and (43) we have the identity (38). �

Theorem 3 For n ∈N∪ {0}, we have

BLn,λ(x)

=1

(1 – u)r

n∑k=0

( n∑l=k

r∑j=0

n–l∑m=0

(nl

)(rj

)(–u)r–j(j)m,λL(l, k)L(n – l, m)

)h(r)

k,λ(x|u).(44)

As the inversion formula of (44), we have

h(r)n,λ(x|u)

=n∑

k=0

(1k!

k∑l=0

n∑m=0

(kl

)(nm

)(l + n – m – 1

n – m

)(–1)l+n–m(n – m)!h(r)

m,λ(u)

)BL

k,λ(x)

=n∑

k=0

( n∑m=0

(nm

)(–1)n–m–kL(n – m, k)h(r)

m,λ(u)

)BL

k,λ(x),

(45)

where h(r)n,λ(x|u) are the degenerate Frobenius–Euler polynomials of order r.

Proof From (9), (24) and (26), we consider the following two degenerate Sheffer sequences:

BLn,λ(x) ∼

(1,

t1 + t

and h(r)n,λ(x|u) ∼

((eλ(t) – u

1 – u

)r

, t)

λ

. (46)

By using (2), (5), (25) and (46), we have

BLn,λ(x) =

n∑k=0

μn,kh(r)k,λ(x|u), (47)

Here

μn,k =1k!

⟨( (eλ( t1–t ) – u)1 – u

)r( t1 – t

)k∣∣∣(x)n,λ

⟩λ

=⟨( (eλ( t

1–t ) – u)1 – u

)r∣∣∣(

1k!

(t

1 – t

)k)λ

(x)n,λ

⟩λ

=1

(1 – u)r

n∑l=k

(nl

)L(l, k)

⟨(eλ

(t

1 – t

)– u

)r∣∣∣(x)n–l,λ

⟩λ

(48)

=1

(1 – u)r

n∑l=k

(nl

)L(l, k)

r∑j=0

(rj

)(–u)r–j

⟨ejλ

(t

1 – t

)∣∣∣(x)n–l,λ

⟩λ

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Kim Advances in Difference Equations (2020) 2020:687 Page 10 of 16

=1

(1 – u)r

n∑l=k

(nl

)L(l, k)

r∑j=0

(rj

)(–u)r–j

n–l∑m=0

(j)m,λ

⟨1

m!

(t

1 – t

)m∣∣∣(x)n–l,λ

⟩λ

=1

(1 – u)r

n∑l=k

(nl

)L(l, k)

r∑j=0

(rj

)(–u)r–j

n–l∑m=0

(j)m,λL(n – l, m).

Therefore, from (47) and (48), we get the identity (44).To find the inversion formula of (44), by (29), we have

h(r)n,λ(x|u) =

n∑k=0

ZkBLk,λ(x).

In the same way as (42) and (43), we have

Zk =1k!

⟨(t

1 + t

)k∣∣∣h(r)n,λ(x|u)

⟩λ

=1k!

k∑l=0

(kl

)(–1)l

⟨(1

1 + t

)l∣∣∣h(r)n,λ(x|u)

⟩λ

=1k!

k∑l=0

(kl

)(–1)l

n∑ν=0

(l + ν – 1

ν

)(–1)ν

⟨tν

∣∣∣n∑

m=0

(nm

)h(r)

m,λ(u)(x)n–m,λ

⟩λ

=1k!

k∑l=0

(kl

)(–1)l

n∑ν=0

(l + ν – 1

ν

)(–1)ν

n∑m=0

(nm

)h(r)

m,λ(u)⟨tν |(x)n–m,λ〉λ

=1k!

k∑l=0

(kl

)(l + n – m – 1

n – m

)(–1)l+n–m

n∑m=0

(nm

)h(r)

m,λ(u)(n – m)!.

(49)

In another way, we can get

Zk =1k!

⟨(t

1 + t

)k∣∣∣h(r)n,λ(x|u)

⟩λ

=n∑

m=0

(nm

)h(r)

m,λ(u)(–1)n–m–kL(n – m, k). (50)

Therefore, from (49) and (50), we have the identity (45). �

When u = –1 in Theorem 3, we have the following corollary.

Corollary 4 For n ∈N∪ {0} and r ∈N, we have

BLk,λ(x) =

12r

n∑k=0

( n∑l=k

r∑j=0

n–l∑m=0

(nl

)(rj

)(j)m,λL(l, k)L(n – l, m)

)E(r)

k,λ(x). (51)

By the inversion formula of (51), we have

E(r)n,λ(x) =

n∑k=0

(1k!

k∑l=0

n∑m=0

(kl

)(nm

)(l + n – m – 1

n – m

)(–1)l+n–m(n – m)!E(r)

m,λ

)BL

k,λ(x),

where E(r)n,λ(x) are the degenerate Euler polynomials of order r.

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Kim Advances in Difference Equations (2020) 2020:687 Page 11 of 16

Theorem 5 For n ∈N∪ {0} and r ∈ N, we have

BLn,λ(x) =

n∑k=0

( n∑l=k

n∑m=0

n–m∑j=0

(nm

)(1)j+1,λ

j + 1S2,λ(l, k)L(m, l)L(n – m, j)

)Dk,λ(x). (52)

As the inversion formula of (52), we have

Dn,λ(x) =n∑

k=0

( n∑m=0

n–m∑j=0

(nm

)(–1)n–m–kS1,λ(n – m, j)L(n – m, k)Dm,λ

)BL

k,λ(x), (53)

where Dn,λ(x) are the degenerate Daehee polynomials.

Proof From (10) and (24), we consider the following two degenerate Sheffer sequences:

BLn,λ(x) ∼

(1,

t1 + t

and Dn,λ(x) ∼(

eλ(t) – 1t

, eλ(t) – 1)

λ

. (54)

From (2), (16), (25) and (54), we have

BLn,λ(x) =

n∑k=0

μn,kDk,λ(x), (55)

where

μn,k =1k!

⟨eλ( t1–t ) – 1

t1–t

(eλ

(t

1 – t

)– 1

)k∣∣∣(x)n,λ

⟩λ

=⟨

1 – tt

(eλ

(t

1 – t

)– 1

)1k!

(eλ

(t

1 – t

)– 1

)k∣∣∣(x)n,λ

⟩λ

=n∑

l=k

S2,λ(l, k)⟨

1 – tt

(eλ

(t

1 – t

)– 1

)∣∣∣(

1l!

(t

1 – t

)l)λ

(x)n,λ

⟩λ

=n∑

l=k

S2,λ(l, k)n∑

m=0

(nm

)L(m, l)

⟨1 – t

t

(eλ

(t

1 – t

)– 1

)∣∣∣(x)n–m,λ

⟩λ

=n∑

l=k

S2,λ(l, k)n∑

m=0

(nm

)L(m, l)

⟨ ∞∑j=1

(1)j,λ1j!

(t

1 – t

)j–1∣∣∣(x)n–m,λ

⟩λ

=n∑

l=k

S2,λ(l, k)n∑

m=0

(nm

)L(m, l)

⟨ ∞∑j=0

(1)j+1,λ1

(j + 1)!

(t

1 – t

)j∣∣∣(x)n–m,λ

⟩λ

=n∑

l=k

S2,λ(l, k)n∑

m=0

(nm

)L(m, l)

n–m∑j=0

(1)j+1,λ

j + 1

⟨1j!

(t

1 – t

)j∣∣∣(x)n–m,λ

⟩λ

=n∑

l=k

S2,λ(l, k)n∑

m=0

(nm

)L(m, l)

n–m∑j=0

(1)j+1,λ

j + 1L(n – m, j).

(56)

Therefore, from (55) and (56), we get the identity (52).

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Kim Advances in Difference Equations (2020) 2020:687 Page 12 of 16

To find the inversion formula of (52), from (29), we have

Dn,λ(x) =n∑

k=0

ZkBLk,λ(x). (57)

By using (1 + t)x =∑∞

n=0(x)ntn

n! and the first equation of (15), we have

Zk =⟨

1k!

(t

1 + t

)k∣∣∣Dn,λ(x)⟩λ

=⟨

1k!

(t

1 + t

)k∣∣∣n∑

m=0

(nm

)Dm,λ(x)n–m

⟩λ

=n∑

m=0

(nm

)Dm,λ

⟨1k!

(t

1 + t

)k∣∣∣n–m∑j=0

S1,λ(n – m, j)(x)n–m,λ

⟩λ

=n∑

m=0

(nm

)Dm,λ

n–m∑j=0

S1,λ(n – m, j)⟨

1k!

(t

1 + t

)k∣∣∣(x)n–m,λ

⟩λ

=n∑

m=0

(nm

)Dm,λ

n–m∑j=0

S1,λ(n – m, j)(–1)n–m–kL(n – m, k).

(58)

Therefore, from (57) and (58), we have the identity (53). �

Theorem 6 For n ∈N∪ {0}, we have

BLn,λ(x) =

n∑k=0

( n∑l=k

S1,λ(l, k)L(n, k)

)Belk,λ(x). (59)

As the inversion formula of (59), we have

Beln,λ(x) =n∑

k=0

( n∑l=0

(–1)l–kS2,λ(n, l)L(l, k)

)BL

k,λ(x), (60)

where Beln,λ(x) are the degenerate Bell polynomials.

Proof From (12), (24) and (26), we consider two degenerate Sheffer sequences as follows:

BLn,λ(x) ∼

(1,

t1 + t

and Beln,λ(x) ∼ (1, logλ(1 + t)

)λ. (61)

By using (2), (15), (25) and (61), we have

BLn,λ(x) =

n∑k=0

μn,kBelk,λ(x), (62)

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Kim Advances in Difference Equations (2020) 2020:687 Page 13 of 16

where

μn,k =1k!

⟨(logλ

(1 +

t1 – t

))k∣∣∣(x)n,λ

⟩λ

=n∑

l=k

S1,λ(l, k)⟨

1l!

(t

1 – t

)l∣∣∣(x)n,λ

⟩λ

=n∑

l=k

S1,λ(l, k)L(n, l).(63)

Therefore from (62) and (63), we get the identity (59).To find inversion formula of (59), from (29), we have

Beln,λ(x) =n∑

k=0

ZkBLk,λ(x). (64)

From (5) and (16), we observe that

∞∑n=0

Beln,λ(x)tn

n!= ex

λ

(eλ(t) – 1

)=

∞∑n=0

( n∑l=0

S2,λ(n, l)(x)l,λ

)tn

n!.

Thus, by using Beln,λ(x) =∑n

l=0 S2,λ(n, l)(x)l,λ and (6), we have

Zk =1k!

⟨(t

1 + t

)k∣∣∣Beln,λ(x)⟩λ

=⟨

1k!

(t

1 + t

)k∣∣∣n∑

l=0

S2,λ(n, l)(x)l,λ

⟩λ

=n∑

l=0

S2,λ(n, l)⟨

1k!

(t

1 + t

)k∣∣∣(x)l,λ

⟩λ

=n∑

l=0

S2,λ(n, l)(–1)l–kL(l, k).

(65)

Therefore, from (64) and (65), we have the identity (60). �

Theorem 7 For n ∈N∪ {0}, we have

BLn,λ(x) =

n∑k=0

( n∑l=k

S2,λ(l, k)L(n, l)

)(x)n (n ≥ 0). (66)

Proof Since exλ(log(1 + t)) = (1 + t)x =

∑∞n=0(x)n

tn

n! , we have (x)n ∼ (1, eλ(t) – 1)λ.Therefore, we consider the two degenerate Sheffer sequences as follows:

BLn,λ(x) ∼

(1,

t1 + t

and (x)n ∼ (1, eλ(t) – 1

)λ. (67)

Thus, from (2), (16) and (67), we have

BLn,λ(x) =

n∑k=0

μn,k(x)k (n ≥ 0), (68)

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Kim Advances in Difference Equations (2020) 2020:687 Page 14 of 16

where

μn,k =1k!

⟨(eλ

(t

1 – t

)– 1

)k∣∣∣(x)n,λ

⟩λ

=n∑

l=k

S2,λ(l, k)⟨

1l!

(t

1 – t

)l∣∣∣(x)n,λ

⟩λ

=n∑

l=k

S2,λ(l, k)L(n, l).(69)

Therefore, from (68) and (69), we have the identity (66). �

Theorem 8 For n ∈N∪ {0} and s ∈N, we have

E (s)n,λ(x) =

n∑k=0

( n∑l=k

(nl

)(–1)l–kL(l, k)E (s)

n–l,λ

)BL

k,λ(x), (70)

where E (s)n,λ(x) are type 2 degenerate poly-Euler polynomials.

Proof From (18), (24) and (26), we consider the following two degenerate Sheffer se-quences as follows:

BLn,λ(x) ∼

(1,

t1 + t

and E (s)n,λ(x) ∼

(t(eλ(t) + 1)

Eis(log(1 + 2t)), t

. (71)

By using (2) and (25), we have

E (s)n,λ(x) =

n∑k=0

μn,kBLkλ(x), (72)

where

μn,k =1k!

⟨Eis(log(1 + 2t))

t(eλ(t) + 1)

(t

1 + t

)k∣∣∣(x)n,λ

⟩λ

=⟨

Eis(log(1 + 2t))t(eλ(t) + 1)

∣∣∣(

1k!

(t

1 + t

)k)λ

(x)n,λ

⟩λ

=n∑

l=k

(nl

)(–1)l–kL(l, k)

⟨Eis(log(1 + 2t))

t(eλ(t) + 1)

∣∣∣(x)n–l,λ

⟩λ

=n∑

l=k

(nl

)(–1)l–kL(l, k)E (s)

n–l,λ.

(73)

Therefore, from (72) and (73), we get the identity (70). �

3 ConclusionThe author represented the degenerate Lah–Bell polynomials in terms of quite a few well-known special polynomials and at the same time derived the inversion formulas of thoseidentities by using the degenerate Sheffer sequences. We addressed the special polyno-mials and numbers: the degenerate falling factorial, the Lah numbers and the degenerate

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Kim Advances in Difference Equations (2020) 2020:687 Page 15 of 16

Bernoulli polynomials; the Lah numbers and the degenerate Frobenius–Euler polynomi-als of order r; the Lah numbers and the degenerate Deahee polynomials; the Lah numbersand the degenerate Bell polynomials; the Lah numbers and the type 2 degenerate polyEuler polynomials. Therefore, the paper demonstrates that degenerate versions are notonly applicable for number theory and combinatorics but also to symmetric identities,differential equations and probability theory. Building upon this, the author would like tofurther study into degenerate versions of certain special polynomials and numbers andtheir applications to physics, economics and engineering as well as mathematics.

AcknowledgementsThe author would like to thank the referees for the detailed and valuable comments, which helped improve the originalmanuscript in its present form.

FundingThis work was supported by the Basic Science Research Program, the National Research Foundation of Korea, the Ministryof Education (NRF-2018R1D1A1B07049584).

Availability of data and materialsNot applicable.

Ethics approval and consent to participateThe author declares that there is no ethical problem in the publication of this paper.

Competing interestsThe author declares that they have no competing interests.

Consent for publicationThe author want to publish this paper in this journal.

Authors’ contributionsThe author was the only one contributing to the manuscript. The author read and approved the final manuscript.

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

Received: 24 October 2020 Accepted: 29 November 2020

References1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.

National Bureau of Standards Applied Mathematics Series, vol. 55. U.S. Government Printing Office, Washington(1964) xiv+1046 pp., For sale by the Superintendent of Documents

2. Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51–88 (1979)3. Carlitz, L.: Weighted Stirling numbers of the first and second kind. Fibonacci Q. 18, 147–162 (1980)4. Comtet, L.: Advanced Combinatorics. The Art of Finite and Infinite Expansions. Reidel, Dordrecht (1974). Revised and

enlarged edn., xi+343 pp. ISBN: 90-277-0441-4 05-025. Dere, R., Simsek, Y.: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math.

(Kyungshang) 22(3), 433–438 (2012)6. Ernst, T.: Examples of a q-umbral calculus. Adv. Stud. Contemp. Math. (Kyungshang) 16(1), 1–22 (2008)7. Kim, D.S., Kim, T.: Some identities of Bernoulli and Euler polynomials arising from umbral calculus. Adv. Stud.

Contemp. Math. (Kyungshang) 23(1), 159–171 (2013)8. Kim, D.S., Kim, T.: Some identities of Bell polynomials. Sci. China Math. 58(10), 2095–2104 (2015)9. Kim, D.S., Kim, T.: Some identities of degenerate Daehee numbers arising from certain differential equations. J.

Nonlinear Sci. Appl. 10, 744–751 (2017)10. Kim, D.S., Kim, T.: Lah–Bell numbers and polynomials. Proc. Jangjeon Math. Soc. 23(4), 577–586 (2020)11. Kim, D.S., Kim, T.: Degenerate Sheffer sequence and λ-Sheffer sequence. J. Math. Anal. Appl. 493(1), 124521 (2021)12. Kim, D.S., Kim, T.: A note on a new type of degenerate Bernoulli numbers. Russ. J. Math. Phys. 27(2), 227–235 (2020)13. Kim, H.K., Jang, L.-C.: Type 2 degenerate poly-Euler polynomials. Symmetry 12, 1011 (2020).

https://doi.org/10.3390/sym1206101114. Kim, T.: A note on degenerate Stirling polynomials of the second kind. Proc. Jangjeon Math. Soc. 20(3), 319–331

(2017)15. Kim, T., Kim, D.S.: An identity of symmetry for the degenerate Frobenius–Euler polynomials. Math. Slovaca 68(1),

239–243 (2018)16. Kim, T., Kim, D.S.: Some identities of extended degenerate r-central Bell polynomials arising from umbral calculus.

Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 1, 19 pp. (2020)17. Kim, T., Kim, D.S.: Note on the degenerate gamma function. Russ. J. Math. Phys. 27(3), 352–358 (2020)

Page 16: Degenerate Lah–Bell polynomials arising from degenerate ......KimAdvancesinDifferenceEquations20202020:687 Page2of16 andnumbersarisingfromthedegenerateSheffersequence.Inaddition,theauthorderives

Kim Advances in Difference Equations (2020) 2020:687 Page 16 of 16

18. Kim, T., Kim, D.S.: Degenerate polyexponential functions and degenerate Bell polynomials. J. Math. Anal. Appl. 487(2),124017, 15 pp. (2020)

19. Kim, T., Kim, D.S., Kim, H.Y., Kwon, J.: Some identities of degenerate Bell polynomials. Mathematics 8, 40, 8 pp. (2020)20. Kim, T., Kim, D.S., Kim, H.Y., Lee, H., Jang, L.-C.: Degenerate Bell polynomials associated with umbral calculus. J. Inequal.

Appl. 2020, 226, 15 pp. (2020)21. Kim, T., Kim, D.S., Kwon, J., Lee, H.: Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli

numbers and polynomials. Adv. Differ. Equ. 2020, 168, 12 pp. (2020)22. Kim, T., Kim, D.S., Lee, H., Kwon, J.: A note no some identities of new type degenerate Bell polynomials. Mathematics

7, 1086, 12 pp. (2019)23. Kim, T., Kim, D.S., Lee, H., Park, J.W.: A note on degenerate r-Stirling numbers. J. Inequal. Appl. 2020, 225, 12 pp. (2020)24. Kwasniewski, A.K.: On ψ -umbral extensions of Stirling numbers and Dobinski-like formulas. Adv. Stud. Contemp.

Math. (Kyungshang) 12(1), 73–100 (2006)25. Ma, Y., Kim, D.S., Kim, T., Kim, H.Y., Lee, H.S.: Some identities of Lah–Bell polynomials. Adv. Differ. Equ. 2020, 510, 10 pp.

(2020)26. Nyul, G., Racz, G.: The r-Lah numbers. Discrete Math. 338, 1660–1666 (2015).

https://doi.org/10.1016/j.disc.2014.03.02927. Nyul, G., Racz, G.: Sums of r-Lah numbers and r-Lah polynomials. Ars Math. Contemp. 18(2), 211–222 (2020).

https://doi.org/10.26493/1855-3974.1793.c4d28. Roman, S.: The Umbral Calculus. Pure and Applied Mathematics, vol. 111, x+193 pp. Academic Press, New York (1984)


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