For nonnegative integers n, k, s such that n≥k, the s-Lah number L_s(n, k) counts the number of partitions of a set with n + s elements into k + s ordered blocks such that s distinguished elements have to be in distinct ordered blocks [1–5]. When s = 0, the Lah numbers appears non-crossing partitions, Dyck paths as well as falling and rising factorials [6]. As multivariate forms for ordinary Bell polynomials and Stirling numbers of the second kind, respectively, both the complete and incomplete Bell polynomials play important role in combinatorics and number theory. Recently, many mathematicians have been studying various degenerate versions of special polynomials and numbers as well as enumerative combinatorics, probability theory, number theory, etc. [7–17]. In [7], as an example considering the psychological burden of baseball hitters, it well expresses the starting point of degenerate special polynomials and numbers being studied by many scholars. Also, both the complete and incomplete Bell polynomials are multivariate forms for Bell polynomials and Stirling numbers of the second kind, respectively. Beginning with Bell [18], these polynomials have been studied by many mathematicians [1,16,19,20]. Recently, Kwon et al. [16] introduced the degenerate incomplete and complete s-Bell polynomials and Kim et al. [20] introduced the incomplete and complete s-extended Lah-Bell polynomials, respectively. With this in mind, we want to study the degenerate versions of complete and incomplete Lah-Bell polynomials as multivariate forms for a new type of degenerate s-extended Lah-Bell polynomials and numbers respectively. In Section 2, we consider new types of degenerate incomplete and complete s-Bell polynomials respectively different from those introduced in [16] for our goal. We study several properties and explicit formulas for them. In Section 3, we define both degenerate incomplete and complete s-extended Lah-Bell polynomials associated with a new type of degenerate s-extended Lah-Bell polynomials respectively, and derive relations between these polynomials and degenerate polynomials in the first part. We also investigate explicit formulas for degenerate complete and incomplete s-extended Lah-Bell polynomials, respectively.

First, we introduce some definitions and properties we needed in this paper.

For a nonnegative integer s, the s-Stirling numbers S^(s)₂(n, k) of the second kind are given by the generating function

(1) 1k!est(et−1)k=∑n=k∞Sn(s)(n+s,k+s)tnn!,(see [15,16]).

When s = 0, S⁽⁰⁾₂(n, k) = S₂(n, k) are the Stirling numbers of the second kind which are the number of ways to partition a set with n elements into k non-empty subsets.

From (8), it is to see that [15,16] the generating function of the s-Bell polynomials is

(2) ∑n=0∞beln(s)(x)tnn!=ex(et−1)est.

When x = 1, beln(s)(1)=beln(s)=∑k=0nS2(n+s,k+s) are called the s-bell numbers.

When s = 0, beln(0)(x)=beln(x) are the ordinary Bell polynomials.

Furthermore, the incomplete s-Bell polynomials.

Bn+s,k+s(β1,β2,⋯:ν1,ν2,⋯) are given by the generating function

(3) 1k!(∑l=1∞βltll!)k(∑i=0∞νi+1tii!)s=∑n=k∞Bn+s,k+s(β1,β2,⋯:ν1,ν2,⋯)tnn!,(see [16]).

When s = 0, Bn+0,k+0(β1,β2,⋯:ν1,ν2,⋯)=Bn,k(β1,β2,⋯,βn−k+1) are the incomplete Bell polynomials. From (10), we obtain immediately that [16]

(4) Bn+s,k+s(s)(β1,β2,⋯:ν1,ν2,⋯)=∑Λ(n,k,s)[n!k1!,k2!⋯(β11!)k1(β22!)k2(β33!)k3⋯]×[s!s0!s1!s2!⋯(ν10!)s0(ν21!)s1(ν32!)s3⋯],

where Λ(n,k,s) denotes the set of all nonnegative integers (ki)i≥1 and (si)i≥0 such that ∑i≥1ki=k,∑i≥0si=sand∑i≥1i(ki+si)=n.

The combinatorial meaning of the incomplete s-Bell polynomials is in the reference [18].

The complete s-Bell polynomials Bn(s)(x1,x2,⋯:y1,y2,⋯) are given by the generation function

(5) exp(∑l=1∞βltll!)(∑i=0∞νi+1tii!)s=∑n=k∞Bn(s)(β1,β2,⋯:ν1,ν2,⋯)tnn!,(see [18]),

where exp(t) = e^t.

When s = 0, Bn(0)(β1,β2,⋯:ν1,ν2,⋯)=Bn(β1,β2,⋯,βn) are the complete Bell polynomials.

Let n, k, s be nonnegative integers with n≥k. It is well known that [2] an explicit formula and the generating function of s-Lah Bell L_s(n, k) are given by, respectively Ls(n,k)=n!k!(n+2s−1k+2s−1), and

(6) 1k!(t1−t)k(11−t)2s=∑n=k∞Ls(n,k)tnn!.

When s = 0, L₀(n, k) = L(n, k) are the unsigned Lah-numbers.

Kim et al. [2] introduced the s-extended Lah-Bell polynomials Lb_{n, s}(x) given by the generating function

(7) ext1−t(11−t)2s=∑n=0∞Lbn,s(x)tnn!.

When x = 1, Lbn,s(1)=Lbn,s=∑k=0nLs(n,k)(n≥0) are called the s-extended Lah-Bell numbers. When s = 0, Lb_{n, 0}(x) = Lb_n(x) are the Lah-Bell polynomials.

For λ∈R, the degenerate exponential function is defined by

(8) eλx(t)=(1+λt)xλandeλ(t)=∑n=0∞(1)n,λtnn!,(see [8−11]).

where (x)0,λ=1 and (x)n,λ=x(x−λ)(x−2λ)⋯(x−(n−1)λ).

The degenerate fully Bell polynomials are given by

(9) eλ(x(eλ(t)−1))=∑n=0∞Beln,λ(x)tnn!,(see [21]).

When λ→0, Beln,λ(x)=beln(x).

In addition, the partially degenerate Bell polynomials are given by

(10) ex(eλ(t)−1))=∑n=0∞Beln,λ(x)tnn!,(see [12]).

When λ→0, Beln,λ(x)=beln(x).

In this section, we introduce new types of degenerate complete s-Bell polynomials and degenerate incomplete s-Bell polynomials different from (9) and (10), respectively. We also give some identities and explicit formulas for these polynomials.

For our goal, we introduce a new type of the degenerate extended s-Bell polynomials defined by

(11) eλ(x(et−1))eλs(t)=∑n=0∞beln,s,λ(x)tnn!.

When x = 1, beln,s,λ=beln,s,λ(1) are called the degenerate extended s-Bell numbers.

When limλ→0eλ(x(et−1))eλs(t)=exp⁡(x(et−1))exp⁡(t)=∑n=0∞beln,s(x)tnn!.

When s = 0, beln,0,λ(x)=beln,λ(x) are the degenerate Bell polynomials.

Theorem 2.1. For n, s∈N∪0, we have beln,s,λ(x)=∑m=0n∑k=0m(nm)(1)k,λ(s)n−m,λS2(m,k)xk. Proof. From (1), (8) and (11), we observe that

(12) ∑n=0∞beln,s,λ(x)tnn!=eλ(x(et−1))eλs(t)=∑k=0∞(1)k,λxk1k!(et−1)keλs(t)=∑k=0∞(1)k,λxk∑m=k∞S2(m,k)tmm!∑j=0∞(s)j,λtjj!=∑m=0∞∑k=0m(1)k,λxkS2(m,k)tmm!∑j=0∞(s)j,λtjj!=∑n=0∞(∑m=0n∑k=0m(nm)(1)k,λ(s)n−m,λS2(m,k)xk)tnn!.

By comparing with the coefficients of both side of (12), we get the desired result.

Theorem 2.2. For n, s∈N∪0, we have beln,s,λ(x)=∑k=0∞∑h=0n∑m=0k(nh)(km)mn−h(s)h,λ(1)k,λ(−1)k−mk!xk. Proof. From (8) and (11), we observe that

(13) ∑n=0∞beln,s,λ(x)tnn!=eλ(x(et−1))eλs(t)=∑k=0∞(1)k,λxk1k!(et−1)keλs(t)=∑k=0∞(1)k,λxk1k!∑m=0k(km)(−1)k−m∑j=0∞mjtjj!∑h=0∞(s)h,λthh!=∑n=0∞(∑k=0∞∑h=0n∑m=0k(nh)(km)mn−h(s)h,λ(1)k,λ(−1)k−mk!xk)tnn!.

By comparing with the coefficients of both side of (13), we get the desired result.

First, we define a new type of the degenerate complete Bell polynomials Bnλ(β1,β2,⋯,βn) associated with the degenerate Bell polynomials beln,λ(x) by

(14) eλ(∑h=1∞βhthh!)=∑n=0∞Wnλ(β1,β2,⋯,βn)tnn!,

and a new type of the degenerate incomplete Bell polynomials Wn,kλ(β1,β2,⋯,βn−k+1) associated with some degenerate Stirling numbers defined by

(15) 1k!(1)k,λ(∑h=1∞βhthh!)k=∑n=k∞Wn,kλ(β1,β2,⋯,βn−k+1)tnn!,(n≥k≥0).

Theorem 2.3. For n≥k≥0, we have W0λ(β1,β2,⋯,βn)=1Wnλ(β1,β2,⋯,βn)=∑k=1n(1)k,λBn,k(β1,β2,⋯,βn−k+1)if n≥1. In particular, we have Wnλ(x,x,⋯,x)=beln,λ(x).

Proof. From (3), (4) and (14), we have

(16) ∑n=0∞Wnλ(β1,β2,⋯,βn)tnn!=eλ(∑h=1∞βhthh!)=1+∑k=1∞(1)k,λ1k!(∑h=1∞βhthh!)k=1+∑k=1∞(1)k,λ∑n=k∞Bn,k(β1,β2,⋯,βn−k+1)tnn!=1+∑n=1∞(∑k=1n(1)k,λBn,k(β1,β2,⋯,βn−k+1))tnn!.

Therefore, by comparing with coefficients of both sides of (16), we have the desired result.

In particular, from (16), we have

(17) ∑n=0∞Wnλ(x,x,⋯,x)tnn!=eλ(x∑h=1∞thh!)=eλ(x(et−1))=∑n=0∞beln,λ(x)tnn!.

Thus, by comparing with coefficients of both sides of (17), we have Wnλ(x,x,…,x)=beln,λ(x).

In next theorem, we obtain a new type of degenerate Stirling numbers of second kind (1)k,λS2(n,k).

Theorem 2.4. For n≥k≥0, we have W0λ(β1,β2,…,βn)=1Wnλ(β1,β2,…,βn)=∑k=1nWn,kλ(β1,β2,⋯,βn−k+1)if n≥1. In particular, we get Wn,kλ(1,1,…,1)=(1)k,λS2(n,k).

Proof. From (8), (14) and (15), we observe that

(18) ∑n=0∞Wnλ(β1,β2,⋯,βn)tnn!=eλ(∑h=1∞βhthh!)=∑k=0∞1k!(1)k,λ(∑h=1∞βhthh!)k=1+∑k=1∞∑n=k∞Wn,kλ(β1,β2,⋯,βn−k+1)tnn!=1+∑n=1∞∑k=1nWn,kλ(β1,β2,⋯,βn−k+1)tnn!.

Therefore, by comparing with coefficients of both sides of (18), we have what we want.

In addition, from (18), we get ∑n=k∞Wn,kλ(1,1,⋯,1)=(1)k,λ1k!(et−1)k=(1)k,λ∑n=k∞S2(n,k). Thus, we have Wn,kλ(1,1,⋯,1)=(1)k,λS2(n,k).

Next, for λ∈R, we consider a new type of degenerate incomplete s-Bell polynomials defined by

(19) Wn+s,k+sλ(β1,β2,⋯:ν1,ν2,⋯)=1k!(∑h=1∞(1)k,λβhthh!)k(∑m=0∞(1)m,λνm+1tmm!)s.

From (4) and (19), we have the following explicit formula

For n≥k≥0, we have

(20) Wn+s,k+sλ(β1,β2,⋯:ν1,ν2,⋯)=∑n=k∞Bn+s,k+s((1)k,λβ1,(1)k,λβ2,⋯:(1)0,λν1,(1)1,λν2,⋯)tnn!=∑Λ(n,k,λ)[n!k1!k2!⋯((1)k,λβ11!)k1((1)k,λβ12!)k2⋯]×[r!s0!s1!⋯((1)0,λν10!)s0((1)1,λν21!)s1⋯],

where Λ(n,k,s) denote the set of all nonnegative integers (ki)i≥1 and (si)i≥1 such that ∑i≥1ki=k,∑i≥0si=sand∑i≥1i(ki+si)=n

Naturally, we define a new type of the degenerate complete s-Bell polynomials by

(21) Wn(s),λ(β1,β2,⋯:ν1,ν2,⋯)=eλ(∑h=1∞βhthh!)(∑m=0∞(1)m,λνm+1tmm!)s,

where λ∈R and n, k∈N with n≥k.

We note that limλ→∞Wn(s),λ(β1,β2,⋯:ν1,ν2,⋯)=Bn(s)(β1,β2,⋯:ν1,ν2,⋯).

From (19) and (21), we note that Wn(s),λ(β1,β2,⋯:ν1,ν2,⋯)=∑k=0nWn+s,k+sλ(β1,β2,⋯:ν1,ν2,⋯).

Theorem 2.5. For n, k, s≥0 with n≥k, we have Wn+s,k+sλ(1,1,⋯:1,1,⋯)=(1)k,λS2(n,k)eλs(t).

Proof. From (1), (8) and (19), we have

Therefore, by comparing with coefficients of both side of (22), we obtain the desired result.

In Theorem 2.5, we obtain a new type of degenerate s-extended Stirling number of second.

Theorem 2.6. For n≥k≥0, we have

Proof. By using (8), we observe that

and

From (23) and (24), we get

On the other hand, from (21), we have

(26) eλ(∑h=1∞βhthh!)(∑m=0∞(1)m,λνm+1tmm!)s=∑n=0∞Wn(s),λ(β1,β2,⋯:ν1,ν2,⋯)tnn!.

Thus, by comparing with coefficients of (25) and (26), we have what we want.

Next, we consider the extended degenerate complete s-Bell polynomials defined by the generating function

(27) eλ(z∑h=1∞βhthh!)(∑m=0∞(1)m,λνm+1tmm!)s=∑n=0∞Wn(s),λ(z|β1,β2,⋯:ν1,ν2,⋯)tnn!.

Theorem 2.7. For n≥k≥0, we have Wn(s),λ(z |β1,β2,⋯:ν1,ν2,⋯)=∑k=0nzkWn+s,k+sλ(β1,β2,⋯:ν1,ν2,⋯). When z = 1, we get Wn(s),λ(1|β1,β2,⋯:ν1,ν2,⋯)=∑k=0nWn+s,k+s(s),λ(β1,β2,⋯:ν1,ν2,⋯).

Proof. From (19) and (27), we have

(28) ∑n=0∞Wn(s),λ(z |β1,β2,⋯:ν1,ν2,⋯)tnn!=∑k=0∞zk(1)k,λ1k!(∑h=1∞βhthh!)k(∑m=0∞(1)m,λνm+1tmm!)s=∑k=0∞zk∑n=k∞Wn+s,k+sλ(β1,β2,⋯:ν1,ν2,⋯)tnn!=∑n=0∞(∑k=0nzkWn+s,k+s(s),λ(β1,β2,⋯:ν1,ν2,⋯))tnn!.

Thus, by comparing with coefficients of both sides of (28), we have what we want.

Theorem 2.8. For n, k, s≥0 with n≥k, we have

Wn(s),λ(z | 1,1,⋯:1,1,⋯)=beln,s,λ(z).

Proof. From (27), we observe that

(29) eλ(z∑l=1∞thl!)(∑m=0∞(1)m,λtmm!)s =∑n=0∞∑k=0nzkWn+s,k+s(s)((1)k,λ,(1)k,λ,⋯:(1)0,λ,(1)1,λ,⋯)tnn!. 

On the other hand, from (11), we get

(30) eλ(z ∑l=1∞tll!)(∑m=0∞(1)m,λtmm!)s=eλ(z(et−1))eλs(t)=∑n=0∞beln,s,λ(z)tnn!.

Thus, from (29) and (30), we get the desired result.

In this section, we introduce a new type of the degenerate Lah-Bell polynomials different from Kim-Kim’s in [8] and define both the s-extended complete and incomplete degenerate Lah-Bell polynomials associated with a new type of the degenerate Lah-Bell polynomials. We also demonstrate some interesting properties related to these polynomials and explicit formulas for them.

We consider a new type of the degenerate Lah-Bell polynomials Lbn,λ(x) given by the generating function

(31) eλ(xt1−t)=∑n=0∞Lbn,λ(x)tnn!.

When x = 1, Lbn,λ(1)=Lbn,λ  (n≥0) are called the degenerate Lah-Bell numbers (see Figs. 1 and 2).

When λ→0, Lbn,λ(x)=Lbn(x).

In view of the ordinary Bell polynomials, the degenerate 2s-extended Lah-Bell polynomials are defined by the generating function

(32) ∑n=k∞Lbn,2s,λ(x)=eλx(t1−t)eλ2s(t).

When x = 1, Lbn,2s,λ=Lbn,2s,λ(1) are called the degenerate extended 2s-extended Lah-Bell numbers numbers. When s = 0, the degenerate complete s-extended Lah-Bell polynomials are the degenerate Lah-Bell polynomials.

Next, we introduce the degenerate complete Lah-Bell polynomials LWnλ(x1,x2,⋯,xn) defined by the generating function

(33) eλ(∑h=1∞βhth)=∑n=0∞LWnλ(β1,β2,⋯,βn)tnn!.

We note that

(34) ∑n=0∞LWnλ(x,x,⋯,x)tnn!=eλ(x∑h=1∞th)=eλ(xt1−t)=∑n=0∞Lbn,λ(x)tnn!.

From (31), we have LWnλ(x,⋯,x)=Lbn,λ(x) and LWnλ(1,1,⋯,1)=Lbn,λ.

From (20) and (34), we get

(35) ∑n=0∞LWnλ(β1,β2,⋯,βn)tnn!=eλ(∑h=1∞h!βhthh!)=∑n=0∞Wnλ(1!β1,2!β2,⋯,n!βn)tnn!.

By (14), (15), (35) and Theorem 2.3, we obtain the following theorem.

Theorem 3.1. For n≥k≥0, we have LW0λ(β1,β2,⋯,xn)=1 and LWnλ(β1,β2,⋯,βn)=Wnλ(1!β1,2!β2,⋯,n!βn)=∑k=1n(1)k,λBn,k(1!β1,2!β2,⋯,(n−k+1)!βn−k+1).

Naturally, we can define a new type of the degenerate incomplete Lah-Bell polynomials

LWn,kλ(β1,β2,⋯,βn−k+1) are defined by the generating function

(36) 1k!(1)k,λ(∑h=1∞βhth)k=∑n=k∞LWn,kλ(β1,β2,⋯,βn−k+1)tnn!,(n≥k≥0).

Note that when λ→0, LWn,kλ(1,1,⋯,1)=Lbn are the Lah-bell numbers.

From (15) and (36), we observe that

(37) LWn,kλ(β1,β2,⋯,βn−k+1)=Wn,kλ(1!β1,2!β2,⋯,(n−k+1)!βn−k+1).

Theorem 3.2. For n≥k≥0, we have LW0λ(β1,β2,⋯,βn)=1,LWnλ(β1,β2,⋯,βn)=∑k=1nLWn,kλ(β1,β2,⋯,βn−k+1)if  n≥1.

Proof.

From (8), (36) and (37), we observe that

(38) 1+∑n=1∞LWnλ(β1,β2,⋯,βn)tnn!=eλ(∑h=1∞βhth)=1+∑k=1∞1k!(1)k,λ(∑h=1∞βhth)k=1+∑k=1∞∑n=k∞LWn,kλ(β1,β2,⋯,βn−k+1)tnn!=1+∑n=1∞(∑k=1nLWn,kλ(β1,β2,⋯,βn−k+1)tnn!).

Therefore, by comparing with coefficients of both side of (38), we get the desired identity.

We define the degenerate s-extended incomplete Lah-Bell polynomials

LWn+2s,k+2sλ(β1,β2,⋯:ν1,ν2,⋯) by the generating function

(39) 1k!(1)k,λ(∑h=1∞βhth)k(∑m=0∞(1)m,λνm+1tm)2s=∑n=k∞LWn+2s,k+2sλ(β1,β2,⋯:ν1,ν2,⋯)tnn!.

When s = 0, the degenerate incomplete s-extended Lah-Bell polynomials are the degenerate incomplete Lah-Bell polynomials.

From (19) and Theorem 2.5, we get easily the following explicit formula.

Theorem 3.3. For n≥k≥0, we have LWn+2s,k+2sλ(β1,β2,⋯:ν1,ν2,⋯)=Wn+2s,k+2sλ(1!β1,2!β2,⋯:0!ν1,1!ν2,⋯)=∑Λ(n,k,2r)[n!k1!,k2!⋯β1k1β2k2⋯][(2s)!s0!s1!⋯ν1s0ν2s1⋯],

where Λ(n,k,2s) denote the set of all nonnegative integers {ki}i≥1 and {si}i≥0 such that ∑i≥1ki=k,∑i≥0si=2sand∑i≥1i(ki+si)=n.

We also define the degenerate s-extended complete Lah-Bell polynomials

LWn(2s),λ(z|β1,β2,⋯:ν1,ν2,⋯) by the generating function

(40) eλ(z∑h=1∞βhth)(∑m=0∞(1)m,λνm+1tm)2s=∑n=0∞LWn(2s),λ(z|β1,β2,⋯:ν1,ν2,⋯)tnn!.

Theorem 3.4. For n≥k≥0, we have LWn+2s,k+2sλ(x,x,x,⋯:10!11!12!⋯)=Lbn,2s,λ(x). When x = 1, we have LWn+2s,k+2sλ(1,1,1,⋯:10!11!12!⋯)=Lbn,2s,λ.