# Degenerate Poly-Lah-Bell Polynomials and Numbers

Authors: Taekyun Kim; Hye Kyung Kim
Journal: Journal of Mathematics
 (2022)

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## Abstract

Many mathematicians studied “poly” as a generalization of the well-known special polynomials such as Bernoulli polynomials, Euler polynomials, Cauchy polynomials, and Genocchi polynomials. In this paper, we define the degenerate poly-Lah-Bell polynomials arising from the degenerate polyexponential functions which are reduced to degenerate Lah-Bell polynomials whenk=1. In particular, we call these polynomials the “poly-Lah-Bell polynomials” when λ⟶0. We give their explicit expression, Dobinski-like formulas, and recurrence relation. In addition, we obtain various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly-Bell polynomials, the degenerate poly-Bernoulli numbers, and the degenerate poly-Genocchi numbers.


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## Body

## 1. Introduction

Many mathematicians have studied “poly” for the well-known special polynomials such as poly-Bernoulli polynomials, poly-Euler polynomials, poly-Bell polynomials, and poly-Genocchi polynomials [1–10]. Also, a lot of research has been done on degenerate versions of various special polynomials and numbers, such as Stirling numbers, Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Bell polynomials, which has given mathematicians a renewed interest in various degenerate polynomials and numbers [1, 3, 4, 6, 7, 9, 11]. Recently, we introduced degenerate poly-Bell polynomials as one of the generalizations of ordinary Bell polynomials [4]. In relation to this, in this paper, we define the degenerate poly-Lah-Bell polynomials by means of the degenerate polyexponential functions called the “poly-Lah-Bell polynomial”s when λ⟶0. These polynomials are reduced to degenerate Lah-Bell polynomials when k=1. In particular, they are reduced to degenerate Lah-Bell polynomials if k=1. We derive their explicit expression, Dobinski-like formulas, recurrence relation, and various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly-Bell polynomials, the degenerate poly-Bernoulli numbers, and the degenerate poly-Genocchi numbers.The Lah numbers, which was studied by Lah in 1955 [12], have many other interesting applications in analysis and combinatorics (see [13–15]).The unsigned Lah numberLn,k counts the number of ways to partition the set 1,2,…,n into k nonempty linearly ordered subsets, and (see [3, 12, 13])(1)Ln,k=n−1k−1n!k!,k≥0.From (1) (see [13, 14]), the generating function of Ln,k is given by(2)1k!t1−tk=∑n=k∞Ln,ktnn!,k≥0.Kim–Kim [14] introduced the Lah-Bell polynomials BnLx given by(3)ext/1−t=ex1/1−t−1=∑n=0∞BnLxtnn!.Whenx=1, Lah-Bell numbers are given by e1/1−t−1=∑n=0∞BnLtn/n!.It was known that (see [13])(4)1−t−m=∑l=0∞−ml−1ltl=∑l=0∞mltll!,where x0=1 and xn=xx+1⋯x+n−1.Forλ∈ℝ, the degenerate exponential function is defined by (see [1, 3, 4, 6, 7, 11])(5)eλxt=1+λtx/λ,eλt=∑n=0∞1n,λtnn!,where x0,λ=1 and xn,λ=xx−λx−2λ⋯x−n−1λ.In this paper, the degenerate Lah-Bell polynomials are given by (see [14])(6)eλxt1−t=∑n=0∞Bn,λLtnn!.For an integerk, the polylogarithm function is defined by (see [6, 16])(7)Likx=∑m=1∞xmmk.Whenk=1, Li1x=−log1−t.Recently, the polyexponential functions introduced by Kim–Kim are given by (see [8])(8)Eikx=∑n=1∞xnn−1!nk,k∈ℤ.By (8), we see that Ei1x=ex−1.The degenerate polyexponential functions also are given by (see [6, 8])(9)Eik,λx=∑n=1∞1n,λxnn−1!nk,k∈ℤ.We note thatEi1,λx=eλx−1.The degenerate poly-Bernoulli polynomials are given by (see [4])(10)Eik,λlogλ1+teλt−1eλxt=∑n=0∞βn,λkxtnn!.Whenx=0, βn,λk=βn,λk0 are called the degenerate poly-Bernoulli numbers.The degenerate poly-Genocchi polynomials are given by (see [3])(11)2Eik,λlogλ1+teλt+1eλxt=∑n=0∞Gn,λkxtnn!.Whenx=0, Gn,λk=Gn,λk0 are called the degenerate poly-Genocchi numbers.The degenerate Stirling numbers of the second and fist kind are given by (see [3, 4])(12)1k!eλt−1k=∑n=k∞S2,λn,ktnn!,(13)1k!logλ1+tk=∑n=k∞S1,λn,ktnn!,k≥0,respectively (see [4, 9]).where logλeλt=eλlogλt=t and logλt=1/λtλ−1.Recently, fork∈ℤ, we introduced the degenerate poly-Bell polynomials beln,λkx given by (see [4])(14)Eik,λxeλt−1+1=∑n=0∞beln,λkxtnn!,and bel0,λkx=1.Whenx=1, beln,λk=beln,λk1 are called the degenerate poly-Bell numbers.Whenλ⟶0, belnkx are called the poly-Bell polynomials.


## 2. Degenerate Poly-Lah-Bell Polynomials and Numbers

In this section, we introduce the degenerate poly-Lah-Bell polynomials by using of the degenerate polyexponential functions and give explicit expression and various noble identities involving those polynomials.Fork∈ℤ, we define the degenerate poly-Lah-Bell polynomials Lbn,λkx, which are arising from degenerate polyexponential functions to be(15)Eik,λt1−tx+1=∑n=0∞Lbn,λkxtnn!,and Lb0,λkx=1.Whenx=1, Lbn,λk=Lbn,λk1 are called degenerate poly-Lah-Bell numbers.Whenλ⟶0, Lbnkx are called the poly-Lah-Bell polynomials.Whenk=1, from (6), we have(16)Ei1,λt1−tx+1=1+∑n=1∞1n,λt/1−txnn−1!n=eλt1−tx=∑n=0∞Bn,λLx.Combining (15) and (16), we obtain(17)Lb1,n,λ1x=Bn,λLx.Theorem 1.
Fork∈ℤ,(18)Lb0,λkx=1,Lbn,λkx=∑s=1n1s,λsk−1Ln,sxs,n≥1.Proof.
From (2), (9), and (15),(19)Eik,λt1−tx=∑s=1∞1s,λs−1!skt1−txs=∑s=1∞1s,λsk−1xs1s!t1−ts=∑s=1∞1s,λsk−1xs∑n=s∞Ln,stnn!=∑n=1∞∑s=1n1s,λsk−1Ln,sxstnn!.
By (15) and (19), we have the desired result.Theorem 2 (Dobinski-like formulas).
Fork∈ℤ and n≥1,(20)Lbn,λkx=∑m=1∞∑j=0mmj1m,λ−1m−jjnm−1!mkxm.Proof.
From (4) and (9), we observe that(21)Eik,λt1−tx=∑m=1∞1m,λm−1!mkt1−txm=∑m=1∞1m,λxmm−1!mk∑j=0mmj−1m−j11−tj=∑m=1∞∑j=0mmj1m,λ−1m−jj!xmm−1!mk∑n=0∞jntnn!=∑n=0∞∑m=1∞∑j=0mmj1m,λ−1m−jjnm−1!mkxmtnn!.
Combining (15) and (21), we get(22)∑m=1∞∑j=0mmj1m,λ−1m−jm−1!mkxm=0,Lbn,λkx=∑d=1∞∑j=0ddj1d,λ−1d−jjnd−1!dkxd,n≥1.Theorem 3.
Fork∈ℤ and n≥1, we have(23)Lbn,λkx=∑j=1n∑m=1jLn,jS1,λj,mbelm,λkx,where belm,λkx are poly-Bell polynomials.Proof.
Replacingt by logλ1/1−t in (14), we obtain(24)1+Eik,λt1−tx=1+∑m=1∞belm,λk1m!logλ11−tm=1+∑m=1∞belm,λkx1m!logλ1+t1−tm=1+∑m=1∞belm,λkx∑j=m∞S1,λj,m1j!t1−tj=1+∑j=1∞∑m=1jbelm,λkxS1,λj,m∑n=j∞Ln,jtnn!=1+∑n=1∞∑j=1n∑m=1jS1,λj,mLn,jbelm,λkxtnn!.
Combining (15) and (24), we get the desired result.
The next theorem is the inversion formula of Theorem3.Theorem 4.
Fork∈ℤ and n≥1,(25)beln,λkx=∑j=1n∑m=1jnj−mn−j,λS2,λj,mLbm,λkx,where belm,λkx are the degenerate poly-Bell polynomials.Proof.
Replacingt by 1−eλ−1t in (15), from (5), we observe that(26)Eik,λxeλt−1+1=1+∑m=1∞Lbm,λkx1−eλ−1tmm!=1+∑m=1∞Lbm,λkxeλ−mteλt−1mm!=1+∑m=1∞Lbm,λkx∑h=0∞−mh,λthh!∑j=m∞S2,λj,mtjj!=1+∑j=1∞∑m=1jLbm,λkxS2,λj,mtjj!∑h=0∞−mh,λthh!=1+∑n=1∞∑j=1n∑m=1jnj−mn−j,λS2,λj,mLbm,λkxtnn!.
By (14) and (20), we obtain the desired identity.Theorem 5.
Fork∈ℤ, the recurrence relation for Lbn,λkx is(27)Lbn+1,λkx−nLbn,λkx=1n+1Lbn+1,λk−1x+Lbn,λk−1xLb1,λkx=Lb1,λk−1x.Proof.
Differentiating with respect tot at (15), we observe that(28)∂∂tEik,λt1−tx=∂∂t∑n=1∞1n,λt/1−txnn−1!nk=x1−t2∑n=0∞1n,λt/1−txn−1n−1!nk−1=1t1−t∑n=1∞1n,λt/1−txnn−1!nk−1=1tEik−1,λt1−tx+11−tEik−1,λt1−tx=∑n=0∞Lbn+1,λk−1n+1tnn!+11−t∑n=1∞Lbn,λk−1xtnn!.
On the other hand, from (15), we get(29)∂∂tEik,λt1−tx=∂∂t∑n=1∞Lbn,λkxtnn!=∑n=1∞Lbn,λkxtn−1n−1!=∑n=0∞Lbn+1,λkxtnn!.
From (28) and (29), we get(30)1−t∑n=0∞Lbn+1,λkxtnn!=1−t∑n=0∞1n+1Lbn+1,λk−1xtnn!+∑n=1∞Lbn,λk−1xtnn!.
From (30), we have(31)∑n=0∞Lbn+1,λkxtnn!−∑n=0∞Lbn+1,λkxtn+1n!=∑n=0∞1n+1Lbn+1,λk−1xtnn!−∑n=0∞1n+1Lbn+1,λk−1xtn+1n!+∑n=1∞Lbn,λk−1xtnn!.
From (31), we have(32)∑n=0∞Lbn+1,λkxtnn!−∑n=1∞nLbn,λkxtnn!=∑n=0∞1n+1Lbn+1,λk−1xtnn!−∑n=1∞nn+1Lbn,λk−1xtnn!+∑n=1∞Lbn,λk−1xtnn!.
By comparing the coefficients of both sides of (32), we get the desired result.Remark 1.
By (9), we get(33)∂∂xEikt1−tx=∂∂x∑n=1∞1n,λt/1−txnn−1!nk=t1−t∑n=1∞1n,λt/1−txn−1n−1!nk−1=t1−tt1−tx−1∑n=1∞1n,λt/1−txnn−1!nk−1=1xEik−1,λt1−tx.
Therefore, fork≥0, by (33), we have(34)Eik,λt1−tx=∫0x1y∫0y1y⋯∫0y1y︸k−2−timesEi1,λt1−tydy⋯dy=∫0x1y∫0y1y⋯∫0y1y︸k−2−timeseλt1−ty−1dy⋯dy=∫0x1y∫0y1y⋯∫0y1y︸k−2−times∑s=1∞1s,λs!t1−tsysdy⋯dy=∑s=1∞1s,λ1s!t1−ts1sk−1xs=∑n=1∞∑s=1n1s,λ1sk−1Ln,sxs.
From (34), we have the same identity of Theorem 1.Theorem 6.
Fork∈ℤ and n≥1,(35)Lbn,λkx=∑j=1n∑ν=1j∑h=1ν1h,λνk−1S1,λν,hS2,λj,νLn,jxj.Proof.
From (9) and (13), we obtain(36)Eik,λlogλ1+t=∑ν=1∞1ν,λlogλ1+tνν−1!νk=∑ν=1∞1ν,λνk−11ν!logλ1+tν=∑n=1∞∑ν=1n1ν,λS1,λn,ννk−1tnn!.
Replacingt by eλt/1−tx−1 in (36), from (2) and (12), we get(37)Eik,λt1−tx=∑ν=1∞∑h=1ν1h,λS1,λν,hhk−11ν!eλt1−tx−1ν=∑ν=1∞∑h=1ν1h,λS1,λν,hhk−1∑j=ν∞xjS2,λj,ν1j!t1−tj=∑j=1∞∑ν=1j∑h=1ν1h,λS1,λν,hhk−1S2,λj,νxj∑n=j∞Ln,jtnn!=∑n=1∞∑j=1n∑ν=1j∑h=1ν1h,λS1,λν,hνk−1S2,λj,νLn,jxjtnn!.
Combining (15) and (37), we have(38)∑n=1∞Lbn,λkxtnn!=∑n=1∞∑j=1n∑ν=1j∑h=1ν1h,λνk−1S1,λν,hS2,λj,νLn,jxjtnn!.
By comparing the coefficients on both sides of (38), the desired identity is obtained.Theorem 7.
Fork∈ℤ and n≥1,(39)Lbn,λkx=∑ω=1n∑h=1ωβh,λk1−βh,λkS2,λω,hLn,ωxω,where βn,λkx and βn,λk are the degenerate poly-Bernoulli polynomials and numbers, respectively.Proof.
By (5) and (10), we obtain easily(40)βn,λkx=∑υ=0nnlxυ,λβn−υ,λk.
Then, by (40), we obtain(41)Eik,λlogλ1+t=eλt−1∑j=0∞βj,λktjj!=∑d=0∞1d,λtdd!−1∑j=0∞βj,λktjj!=∑n=0∞∑d=0n1d,λβn−d,λktnn!−∑n=0∞βn,λktnn!=∑n=0∞βn,λk1−βn,λktnn!.
By replacingt with eλt/1−tx−1 in (41), we get(42)Eik,λt1−tx=∑h=1∞βh,λk1−βh,λkeλt/1−tx−1hh!=∑h=1∞βh,λk1−βh,λk∑ω=h∞S2,λω,h1ω!t1−txω=∑j=1∞∑h=1ωβh,λk1−βh,λkS2,λω,hxω∑n=ω∞Ln,ωtnn!=∑n=1∞∑ω=1n∑h=1ωβh,λk1−βh,λkS2,λω,hLn,ωxωtnn!.
Combining (15) and (42), we have(43)∑n=1∞Lbn,λkxtnn!=∑n=1∞∑ω=1n∑h=1ωβh,λk1−βh,λkS2,λω,hLn,ωxωtnn!.
By comparing the coefficients on both sides of (43), we attain the desired result.
The next theorem is the inversion formula of Theorem7.Theorem 8.
Fork∈ℤ, we have(44)βn,λk=∑ν=0n∑m=hν+1∑h=1mnν1ν+1Lm,hS1,λν+1,mLbh,λkβn−ν,λ,where βn,λk are the degenerate poly-Bernoulli numbers.Proof.
Replacingt by 1−1/1+logλ1+t in (15), from (2) and (13), we get(45)Eik,λlogλ1+t=∑h=1∞Lbh,λk1h!1−1logλ1+t+1h=∑h=1∞Lbh,λk∑m=h∞Lm,h1m!logλ1+tm=∑m=1∞∑h=1mLbh,λkLm,h∑n=m∞S1,λn,mtnn!=∑n=1∞∑m=1n∑h=1mLbh,λkLm,hS1,λn,mtnn!.
From (10) and (45), we get(46)∑n=0∞βn,λktnn!=teλt−11tEik,λlogλ1+t=∑i=0∞βi,λtii!∑ν=0∞∑m=1ν+1∑h=1mLbh,λkLm,hS1,λν+1,mj+1tνν!=∑n=0∞∑ν=0n∑m=hν+1∑h=1mnν1ν+1Lm,hS1,λν+1,mLbh,λkβn−ν,λtnn!.
By comparing with the coefficients on both sides of (46), we attain the desired result.Theorem 9.
Fork∈ℤ and n≥1, we have(47)Lbn,λkx=12∑j=1n∑l=1j∑w=1llw1w,λGl−w,λk+2Gl,λkS2,λj,lLn,jxj,where Gn,λk are called the degenerate poly-Genocchi numbers.Proof.
From (11) and G0,λk=0, we observe that(48)2Eik,λlogλ1+t=eλt+1∑i=0∞Gi,λktii!=∑w=0∞1w,λtww!∑i=0∞Gi,λktii!+∑i=1∞Gi,λktii!=∑w=1∞1w,λtww!∑i=0∞Gi,λktii!+2∑i=1∞Gi,λktii!=∑l=1∞∑w=1llw1w,λGl−w,λk1+2Gl,λktll!.
On the one hand, replacingt by eλt/1−tx−1 in (48), from (2) and (12), we get(49)2Eik,λt1−tx=∑l=1∞∑w=1llw1w,λGl−w,λk+2Gl,λk1l!eλt1−tx−1l=∑l=1∞∑w=1llw1w,λGl−w,λk+2Gl,λk∑j=l∞S2,λj,l1j!t1−tjxj=∑n=1∞∑j=1n∑l=1j∑w=1llw1w,λGl−w,λk+2Gl,λkS2,λj,lLn,jxjtnn!.
On the other hand, by (15) and (49), we have(50)∑n=1∞Lbk,n,λxtnn!=Eik,λt1−tx=12∑n=1∞∑j=1n∑l=1j∑w=1llw1w,λGl−w,λk+2Gl,λkS2,λj,lLn,jxjtnn!.
By comparing the coefficients on both sides of (50), we get the desired identity.


## 3. Conclusion

In this paper, we considered the degenerate poly-Lah-Bell polynomials and numbers by using of the degenerate polyexponential functions and obtained some combinatorial identities between these polynomials and numbers and special polynomials and numbers, involving the explicit formula, Dobinski-like formula, and recurrence relation. In addition, we derived the relations between these polynomials and the degenerate poly-Bell polynomials which we recently introduced [4]. Finally, we would like to continue researching “poly” about the generalized Stirling polynomials and numbers.



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