EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1385-1402
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Identities on λ-Analogues of Lah Numbers and
Lah-Bell Polynomials

Dae San Kim1, Taekyun Kim2,∗, Hyekyung Kim3, Jongkyum Kwon4,*
1 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3 Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430,
Republic of Korea
4 Department of Mathematics Education, Gyeongsang National University, Jinju 52828,
Republic of Korea

Abstract. In recent years, some applications of Lah numbers were discovered in the real world
problem of telecommunications and optics. The aim of this paper is to study the λ-analogues of
Lah numbers and Lah-Bell polynomials which are λ-analogues of the Lah numbers and and Lah-
Bell polynomials. Here we note that λ-analogues appear when we replace the falling factorials by
the generalized falling factorials in the defining equations. By using generating function method,
we study some properties, explicit expressions, generating functions and Dobinski-like formulas for
those numbers and polynomials. We also treat the more general λ-analogues of r-Lah numbers
and r-extended λ-Lah-Bell polynomials. In addition, we show that the expectations of two random
variables, both associated with the Poisson random variable with parameter α

λ , are equal to the
λ-analogue of the Lah-Bell polynomial evaluated at α for one and the r-extended λ-Lah-Bell
polynomial evaluated at α for the other.

2020 Mathematics Subject Classifications: 11B73, 11B83

Key Words and Phrases: λ-analogues of Lah numbers, λ-analogues of Lah-Bell polynomials, λ-
analogues of Laguerre polynomials, λ-analogues of r-numbers, r-extended λ-Lah-Bell polynomials

1. Introduction

The unsigned Lah number L(n, k) counts the number of ways that a set of n elements
can be partitioned into k non-empty linearly ordered susbsets, while the Lah-Bell number
BL

n is the number of ways that a set of n elements can be partitioned into nonempty
linearly ordered subsets. In recent years, some practical applications of the Lah numbers
were found in telecommunications and optics. Indeed, Lah numbers have been used in

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5288

Email addresses: dskim@sogang.ac.k (D. S. Kim ), tkkim@kw.ac.kr (T. Kim),
hkkim@cu.ac.kr (H. Kim), mathkjk26@gnu.ac.kr (J. Kwon)

https://www.ejpam.com 1385 © 2024 EJPAM All rights reserved.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1386

steganography for hiding data in images (see [6]). It requires lower complexity of cal-
culation, compared to alternatives DFT (discrete Fourier transform) and DWT (discrete
wavelet transform). In addition, the Lah transform naturally arises in the perturbative
description of the chromatic dispersion in optics (see [22,23]) and can tremendously speeds
up optimization problems.

The study of degenerate versions of many special numbers and polynomials has been
done in recent years by some mathematicians for their regained interests (see [2,9,15,16,21]
and the references therein). A lot of fascinating results have been discovered. For example,
the degenerate Stirling numbers of the first kind and the second kind, which are degenerate
versions of the ordinary Stirling numbers of the first kind and the second kind respectively,
occur very frequently when one studies degenerate versions of many special numbers and
polynomials (see [2,9,15,16,21]). The λ- umbral calculus, which is more convenient when
dealing with degenerate versions of Sheffer polynomials, has been uncovered as a natural
degenerate version of the usual umbral calculus (see [13]). In addition, the degenerate
gamma functions were found as a degenerate version of the ordinary gamma functions
(see [14]).

The aim of this paper is to study the λ-analogues of Lah numbers Lλ(n, k) (see (14))
and Lah-Bell polynomials BL

n,λ(x) (see (21), (23)) which are λ- analogues of the Lah
numbers and and Lah-Bell polynomials. We investigate some properties, explicit expres-
sions, Dobinski-like formulas and generating functions for those numbers and polynomi-
als. We also treat the more general λ-analogues of r-Lah numbers Lr,λ(n, k) (see (39))

and r-extended λ-Lah-Bell polynomials LB
(r)
n,λ(x) (see (45)). In addition, we show the

expectation of one random variable and that of another random variable, both related
to the Poisson random variable with parameter α

λ , are respectively equal to BL
n,λ(α) and

LB
(r)
n,λ(α). Here we note that the degenerate versions arise naturally when we replace the

powers of x by the generalized falling factorial polynomials (x)k,λ (see (1)) in the defining
equations, while the λ-analogues appear when we replace the falling factorials (x)k by the
generalized falling factorials.

In more detail, the outline of this paper is as follows. In Section 1, we recall the
generalized falling factorials (x)n,λ and the generalized rising factorials ⟨x⟩n,λ. We remind
the reader of the unsigned Lah numbers L(n, k) and Lah-Bell numbers BL

n . We recall
the λ-analogues of the Stirling numbers of the first kind S1,λ(n, k) and the second kind{
n
k

}
λ
. We remind the reader of the unsigned λ-Stirling numbers of the first kind

[
n
k

]
λ
=

(−1)n−kS1,λ(n, k), the λ-analogues of r-Stirling numbers of the second
{
n+r
k+r

}
r,λ

, and the

λ-Bell polynomials ϕn,λ(x). Section 2 is the main result of this paper. We define the
λ-analogues of Lah numbers Lλ(n, k), and express it as finite sums of products of

[
n
k

]
λ

and
{
n
k

}
λ
in Theorem 2.1. We define the λ-analogues of Lah-Bell polynomials BL

n,λ(x) and

numbers BL
n,λ, and find explicit formulas for Lλ(n, k) in Theorem 2.2 and Dobinski-like

formulas for BL
n,λ(x) in Theorem 2.3. In Theorem 2.4, ϕn,λ(x) is expressed as a finite sum

involving BL
n,λ(x) and

{
n
k

}
λ
. As an inversion of this, we also express BL

n,λ(x) as a finite sum

involving ϕn,λ(x) and
[
n
k

]
λ
. We define the λ-analogues of Laguerre polynomials L

(α)
n,λ(x) of



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1387

order α. Then we show that the convolution of BL
n,λ(x) and L

(α)
n,λ(x) is equal to ⟨α+1⟩n,λ.

We define the λ-analogues of r-Lah numbers Lr,λ(n, k), and find for those numbers an
explicit expression in Theorem 2.6 and a recurrence relation in Theorem 2.7. We define

the r-extended λ-Lah-Bell polynomials LB
(r)
n,λ(x) and find a Dobinski-like formula for those

polynomials in Theorem 2.8. In Theorem 2.9, we express
{
n+r
k+r

}
r,λ

as a finite sum of the

product of Lr,λ(n, k) and
{
n
m

}
−λ

, and conversely Lr,λ(n, k) as a finite sum of the product

of
{
m+r
k+r

}
r,λ

and
[
n
m

]
λ
. Let X be the Poisson distribution with parameter α

λ > 0. Then,

in Section 3, we show that the expectation of the random variable ⟨Xλ⟩n,λ is equal to

BL
n,λ(α) and that of the random variable ⟨Xλ+ r⟩n,λ is equal to LB

(r)
n,λ(α). In the rest of

this section, we recall the facts that are needed throughout this paper.

For any nonzero λ ∈ R, the generalized falling factorial sequence is given by

(x)0,λ = 1, (x)n,λ = x(x−λ)(x−2λ) · · · (x−(n−1)λ), (n ≥ 1), (see [1−5, 7−26]). (1)

Note that

lim
λ→1

(x)n,λ = x(x− 1)(x− 2) · · ·
(
x− (n− 1)

)
= (x)n, (n ≥ 1).

The generalized rising factorial sequence is defined by

⟨x⟩0,λ = 1, ⟨x⟩n,λ = x(x+ λ)(x+ 2λ) · · · (x+ (n− 1)λ), (n ≥ 1).

Note that

lim
λ→1

⟨x⟩n,λ = x(x+ 1)(x+ 2) · · ·
(
x+ (n− 1)

)
= ⟨x⟩n, (n ≥ 1).

For n ≥ 0, the unsigned Lah numbers are defined by

⟨x⟩n =
n∑

k=0

L(n, k)(x)k, (see[5, 12, 13, 26]). (2)

Note that L(n, k) = n!
k!

(
n−1
k−1

)
, (n ≥ k ≥ 1).

The Lah-Bell number BL
n is defined by

BL
n =

n∑
k=0

L(n, k), (n ≥ 0), (see [12, 13]). (3)

The λ-analogues of the Stirling numbers of the first kind are defined by

(x)n,λ =

n∑
k=0

S1,λ(n, k)x
k, (n ≥ 0). (4)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1388

From (4), we get

1

λk

1

k!

(
log(1 + λt)

)k
=

∞∑
n=k

S1,λ(n, k)
tn

n!
, (k ≥ 0). (5)

Note that limλ→1 S1,λ(n, k) = S1(n, k) are the ordinary Stirling numbers of the first kind
given by

(x)n =
n∑

k=0

S1(n, k)x
k, (see [1− 5, 7− 13, 15− 21, 24− 26]). (6)

The unsinged λ-Stirling numbers of the first kind are given by

(−1)n−kS1,λ(n, k) =

[
n

k

]
λ

, (n, k ≥ 0),

and hence we see from (4) and (5) that

⟨x⟩n,λ =
n∑

k=0

[
n

k

]
λ

xk, (n ≥ 0),
1

λk

1

k!

(
− log(1− λt)

)k
=

∞∑
n=k

[
n

k

]
λ

tn

n!
, (k ≥ 0). (7)

The λ-analogues of Stirling numbers of the second kind are defined by

xn =
n∑

k=0

{
n

k

}
λ

(x)k,λ, (n ≥ 0), (see [10]). (8)

From (8), we have

1

λk

1

k!

(
eλt − 1

)k
=

∞∑
n=k

{
n

k

}
λ

tn

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

Note that limλ→1

{
n
k

}
λ
=

{
n
k

}
are the ordinary Stirling numbers of the second kind defined

by

xn =

n∑
k=0

{
n

k

}
(x)k, (n ≥ 0), (see [1− 5, 7− 13, 15− 32]).

For r ∈ N ∪ {0}, the λ-analogues of r-Stirling numbers of the second kind are given by

(x+ r)n =

n∑
k=0

{
n+ r

k + r

}
r,λ

(x)k,λ, (n ≥ 0), (see [10, 11, 15]). (10)

Thus, by (10), we get

1

λk

1

k!

(
eλt − 1

)k
ert =

∞∑
n=k

{
n+ r

k + r

}
r,λ

tn

n!
, (k ≥ 0), (see [10, 11]). (11)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1389

The λ-Bell polynomials are given by

e
x
λ
(eλt−1) =

∞∑
n=0

ϕn,λ(x)
tn

n!
, (see [10]). (12)

Note that

lim
λ→1

ϕn,λ(x) = ϕn(x) =

n∑
k=0

{
n

k

}
xk

are the ordinary Bell polynomials.
Here we note that

ϕn,λ(x) = λne−
x
λ

∞∑
k=0

kn

k!

(x
λ

)k
, ϕn(x) = e−x

∞∑
k=0

kn

k!
xk, ϕn,λ(x) = λnϕn

(x
λ

)
. (13)

From (9) and (12), we have

ϕn,λ(x) =

n∑
k=0

{
n

k

}
λ

xk.

In particular, for x = 1, ϕn,λ = ϕn,λ(1) are called the λ-Bell numbers.

2. Identities on λ-analogues of Lah numbers and Lah-Bell polynomials

In view of (2), we consider the λ-analogues of Lah numbers defined by

⟨x⟩n,λ =
n∑

k=0

Lλ(n, k)(x)k,λ, (n ≥ 0). (14)

From (2), we note that limλ→1 Lλ(n, k) = L(n, k), (n, k ≥ 0).
By (14), we get

n∑
k=0

Lλ(n, k)(x)k,λ = ⟨x⟩n,λ =
n∑

j=0

[
n

j

]
λ

xj =

n∑
j=0

[
n

j

]
λ

j∑
k=0

{
j

k

}
λ

(x)k,λ (15)

=

n∑
k=0

( n∑
j=k

[
n

j

]
λ

{
j

k

}
λ

)
(x)k,λ.

Theorem 1. For n, k ≥ 0, we have

Lλ(n, k) =

n∑
j=k

[
n

j

]
λ

{
j

k

}
λ

.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1390

We note from (14) that

(x)n,λ = (−1)n⟨−x⟩n,λ = (−1)n
n∑

k=0

Lλ(n, k)(−x)k,λ

=
n∑

k=0

(−1)n−kLλ(n, k)⟨x⟩k,λ.

For any nonzero λ ∈ R, the λ-exponentials are defined by

exλ(t) = (1 + λt)
x
λ =

∞∑
k=0

(x)k,λ
k!

tk, (see [11, 15− 22]). (16)

From (16), we have

e−x
λ (−t) =

∞∑
n=0

(−x)n,λ
(−t)n

n!
=

∞∑
n=0

⟨x⟩n,λ
tn

n!
(17)

=

∞∑
n=0

( n∑
k=0

Lλ(n, k)(x)k,λ

)
tn

n!

=
∞∑
k=0

( ∞∑
n=k

Lk(n, k)
tn

n!

)
(x)k,λ.

On the other hand, by (16), we get

e−x
λ (−t) = (1− λt)−

x
λ =

(
1

1− λt

) x
λ

=

(
1 +

λt

1− λt

) x
λ

(18)

=
∞∑
k=0

1

k!

(
t

1− λt

)k

(x)k,λ.

By (17) and (18), we get

1

k!

(
t

1− λt

)k

=
1

λk

1

k!

(
1

1− λt
− 1

)k

=
∞∑
n=k

Lλ(n, k)
tn

n!
, (k ≥ 0). (19)

The left hand side of (19) can be written as

1

k!

(
t

1− λt

)k

=
1

λk

1

k!

k∑
l=0

(
k

l

)
(−1)k−l(1− λt)−

λl
λ (20)

=

∞∑
n=0

(
1

λk

1

k!

k∑
l=0

(
k

l

)
(−1)k−l⟨λl⟩n,λ

)
tn

n!
.

Therefore, by (19) and (20), we obtain the following theorem.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1391

Theorem 2. For n, k ≥ 0, with n ≥ k, we have

Lλ(n, k) =
1

λk

1

k!

k∑
l=0

(
k

l

)
(−1)k−l⟨λl⟩n,λ.

Note that

L(n, k) = lim
λ→1

Lλ(n, k) =
1

k!

k∑
l=0

(
k

l

)
(−1)k−l⟨l⟩n.

In view of (3), we define the λ-analogues of Lah-Bell numbers as

BL
n,λ =

n∑
k=0

Lλ(n, k), (n ≥ 0). (21)

From (19) and (21), we can easily derive the following equation:

e
t

1−λt =
∞∑
n=0

BL
n,λ

tn

n!
. (22)

Now, we define the λ-analogues of Lah-Bell polynomials as

BL
n,λ(x) =

n∑
k=0

Lλ(n, k)x
k, (n ≥ 0). (23)

Note that BL
n,λ(1) = BL

n,λ, (n ≥ 0). By (22) and (23), we get

e
x
λ
( 1
1−λt

−1) =

∞∑
n=0

BL
n,λ(x)

tn

n!
. (24)

From (24), we have

e
x
λ
( 1
1−λt

−1) = e−
x
λ e

x
λ
( 1
1−λt

) = e−
x
λ

∞∑
k=0

xk

k!

1

λk

(
1

1− λt

)k

(25)

= e−
x
λ

∞∑
k=0

xk

λkk!

∞∑
n=0

⟨λk⟩n,λ
tn

n!

= e−
x
λ

∞∑
n=0

( ∞∑
k=0

1

λkk!
⟨λk⟩n,λxk

)
tn

n!
.

Therefore, by comparing the coefficients on both sides of (25), we obtain the following
theorem.

Theorem 3. For n ≥ 0, we have

BL
n,λ(x) = e−

x
λ

∞∑
k=0

1

λk

⟨λk⟩n,λ
k!

xk.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1392

We observe that

BL
n (x) = lim

λ→1
BL

n,λ(x) = e−x
∞∑
k=0

⟨k⟩n
k!

xk,

where BL
n (x) are the ordinary Lah-Bell polynomials given by

BL
n (x) =

n∑
k=0

L(n, k)xk, (n ≥ 0).

Replacing t by 1
λ(1− e−λt) in (24) and from (9), we get

e
x
λ
(eλt−1) =

∞∑
k=0

BL
k,λ(x)

1

k!

1

λk

(
1− e−λt

)k
(26)

=

∞∑
k=0

BL
k,λ(x)

∞∑
n=k

{
n

k

}
λ

(−1)n−k t
n

n!

=

∞∑
n=0

( n∑
k=0

(−1)n−kBL
k,λ(x)

{
n

k

}
λ

)
tn

n!
.

From (12) and (26), we have

ϕn,λ(x) =

n∑
k=0

(−1)n−k

{
n

k

}
λ

BL
k,λ(x). (27)

Replacing t by 1
λ log

(
1

1−λt

)
in (12) and from (7), we get

e
x
λ
( 1
1−λt

−1) =
∞∑
k=0

ϕk,λ(x)
1

k!

(
1

λ
log

(
1

1− λt

))k

(28)

=
∞∑
k=0

ϕk,λ(x)(−1)k
1

k!

(
log(1− λt)

λ

)k

=
∞∑
k=0

ϕn,λ(x)
∞∑
n=k

[
n

k

]
λ

tn

n!

=
∞∑
n=0

( n∑
k=0

ϕk,λ(x)

[
n

k

]
λ

)
tn

n!
.

Thus, by (24) and (28), we get

BL
n,λ(x) =

n∑
k=0

ϕk,λ(x)

[
n

k

]
λ

, (n ≥ 0). (29)

Therefore, by (27) and (29), we obtain the following theorem.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1393

Theorem 4. For n ≥ 0, we have

ϕn,λ(x) =
n∑

k=0

(−1)n−k

{
n

k

}
λ

BL
n,λ(x), and BL

n,λ(x) =
n∑

k=0

[
n

k

]
λ

ϕn,λ(x).

It is well known that the Laguerre polynomials L
(α)
n (x) of order α, (α > −1), are given

by

(1− t)−α−1ex(
t

t−1
) =

∞∑
n=0

L(α)
n (x)

tn

n!
. (30)

Now, we consider the λ-analogues of Laguerre polynomials L
(α)
n,λ(x) of order α, (α > −1),

which are given by

(1− λt)−
α+1
λ ex(

t
λt−1

) =
∞∑
n=0

L
(α)
n,λ(x)

tn

n!
. (31)

Note that
lim
λ→1

L
(α)
n,λ(x) = L(α)

n (x), (n ≥ 0).

From (31), we have

(1− λt)−
α+1
λ = ex(

t
1−λt

)
∞∑
k=0

L
(α)
k,λ(x)

tk

k!
(32)

=
∞∑

m=0

BL
m,λ(x)

tm

m!

∞∑
k=0

L
(α)
k,λ(x)

tk

k!

=
∞∑
n=0

( n∑
m=0

(
n

m

)
BL

m,λ(x)L
(α)
n−m,λ(x)

)
tn

n!
.

On the other hand, by binomial expansion, we get

(1− λt)−
α+1
λ =

∞∑
n=0

⟨α+ 1⟩n,λ
tn

n!
. (33)

Therefore, by (32) and (33), we obtain the following theorem.

Theorem 5. For n ≥ 0, we have

⟨α+ 1⟩n,λ =

n∑
m=0

(
n

m

)
BL

m,λ(x)L
(α)
n−m,λ(x). (34)

We observe that

⟨α+ 1⟩n,λ =

n∑
k=0

Lλ(n, k)(α+ 1)k,λ =

n∑
k=0

Lλ(n, k)

k∑
j=0

(
k

j

)
(α)j,λ(1)k−j,λ (35)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1394

=

n∑
j=0

n∑
k=j

(
k

j

)
Lλ(n, k)(α)j,λ(1)k−j,λ.

Hence, by (34) and (35), we get

n∑
m=0

n∑
k=m

(
k

m

)
Lλ(n, k)(α)m,λ(1)k−m,λ =

n∑
m=0

(
n

m

)
BL

m,λ(x)L
(α)
n−m,λ(x). (36)

Now, we consider the bivariate λ-Lah-Bell polynomials given by(
1 + y

t

1− λt

)x

=

∞∑
n=0

BL
n,λ(x, y)

tn

n!
. (37)

Thus, by (37) and (19), we get(
1 + y

t

1− λt

)x

=

∞∑
k=0

(
x

k

)
yk
(

t

1− λt

)k

=

∞∑
k=0

(x)ky
k 1

k!

(
t

1− λt

)k

(38)

=
∞∑
k=0

(x)ky
k

∞∑
n=k

Lλ(n, k)
tn

n!
=

∞∑
n=0

( n∑
k=0

Lλ(n, k)(x)ky
k

)
tn

n!
.

By (37) and (38), we get

BL
n,λ(x, y) =

n∑
k=0

Lλ(n, k)(x)ky
k, (n ≥ 0).

Replacing y by y
x and letting x → ∞, we see that BL

n,λ(y) = limx→∞BL
n,λ(x,

y
x). For

r ∈ N ∪ {0}, we define the λ-analogues of r-Lah numbers by

⟨x+ r⟩n,λ =

n∑
k=0

Lr,λ(n, k)(x)k,λ, (n ≥ 0). (39)

From (39), we note that

e
−(x+r)
λ (−t) =

∞∑
n=0

⟨x+ r⟩n,λ
tn

n!
=

∞∑
n=0

( n∑
k=0

Lr,λ(n, k)(x)k,λ

)
tn

n!
(40)

=

∞∑
k=0

( ∞∑
n=k

Lr,λ(n, k)
tn

n!

)
(x)k,λ.

On the other hand, by binomial expansion, we get

e
−(x+r)
λ (−t) =

(
1

1− λt

) r
λ

(1− λt)−
x
λ =

(
1

1− λt

) r
λ
(
1 +

λt

1− λt

) x
λ

(41)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1395

=
∞∑
k=0

1

k!

(
t

1− λt

)k( 1

1− λt

) r
λ

(x)k,λ.

By (40) and (41), we get

1

k!

(
t

1− λt

)k( 1

1− λt

) r
λ

=
∞∑
n=k

Lr,λ(n, k)
tn

n!
, (k ≥ 0). (42)

The left hand side of (42) can be written as

1

k!

(
t

1− λt

)k( 1

1− λt

) r
λ

=
1

λk

1

k!

(
1

1− λt
− 1

)k( 1

1− λt

) r
λ

(43)

=
k∑

l=0

(
k

l

)
(−1)k−l 1

λk

1

k!

(
1

1− λt

) r+lλ
λ

=
k∑

l=0

(
k

l

)
(−1)k−l 1

λkk!

∞∑
n=0

⟨r + lλ⟩n,λ
tn

n!

=
∞∑
n=0

(
1

λkk!

k∑
l=0

(
k

l

)
(−1)k−l⟨r + lλ⟩n,λ

)
tn

n!
.

Therefore, by (42) and (43), we obtain the following theorem.

Theorem 6. For n, k ≥ 0, with n ≥ k, we have

Lr,λ(n, k) =
1

λk

1

k!

k∑
l=0

(
k

l

)
(−1)k−l⟨r + lλ⟩n,λ.

From (39), we note that

n+1∑
k=0

Lr,λ(n+ 1, k)(x)k,λ = ⟨x+ r⟩n+1,λ = ⟨x+ r⟩n,λ(x+ r + nλ) (44)

=
n∑

k=0

Lr,λ(x)k,λ
(
x− kλ+ r + (n+ k)λ

)
=

n∑
k=0

Lr,λ(n, k)(x)k+1,λ +
n∑

k=0

Lr,λ(n, k)
(
r + (n+ k)λ

)
(x)k,λ

=

n+1∑
k=0

(
Lr,λ(n, k − 1) +

(
r + (n+ k)λ

)
Lr,λ(n, k)

)
(x)k,λ.

Comparing the coefficients on both sides of (44), we obtain the following theorem.



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1396

Theorem 7. For n, k ∈ N, with n ≥ k, we have

Lr,λ(n+ 1, k) = Lr,λ(n, k − 1) +
(
r + (n+ k)λ

)
Lr,λ(n, k).

Now, we consider the r-extended λ-Lah-Bell polynomials defined by

LB
(r)
n,λ(x) =

n∑
k=0

Lr,λ(n, k)x
k, (n ≥ 0). (45)

Thus, by (42) and (45), we easily get

e
x
λ
( 1
1−λt

−1)

(
1

1− λt

) r
λ

=

∞∑
n=0

LB
(r)
n,λ(x)

tn

n!
. (46)

In particular, for x = 1, LB
(r)
n,λ = LB

(r)
n,λ(1) are called the r-extended λ-Lah-Bell numbers.

The left hand side of (46) can be written as

e
x
λ
( 1
1−λt

−1)

(
1

1− λt

) r
λ

= e−
x
λ

∞∑
k=0

xk

λk!k!

(
1

1− λt

)λk+r
λ

(47)

= e−
x
λ

∞∑
k=0

xk

λkk!

∞∑
n=0

⟨λk + r⟩n,λ
tn

n!

=

∞∑
n=0

(
e−

x
λ

∞∑
k=0

⟨λk + r⟩n,λ
λkk!

xk
)
tn

n!
.

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

Theorem 8. For n ≥ 0, we have

LB
(r)
n,λ(x) = e−

x
λ

∞∑
k=0

⟨λk + r⟩n,λ
k!λk

xk.

For r ≥ 0, the λ-analogues of r-Stirling numbers of the second kind are defined by

∞∑
n=k

{
n+ r

k + r

}
r,λ

tn

n!
=

1

λk

1

k!

(
eλt − 1

)k
ert, (k ≥ 0), (see [13]). (48)

Replacing t by 1
λ(1− e−λt) in (42) and using (9), we get

1

k!

1

λk

(
eλt − 1

)k
ert =

∞∑
m=k

Lr,λ(m, k)
1

m!

(
e−λt − 1

−λ

)m

(49)

=

∞∑
m=k

Lr,λ(m, k)

∞∑
n=m

{
n

m

}
−λ

tn

n!



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1397

=

∞∑
n=k

( n∑
m=k

Lr,λ(m, k)

{
n

m

}
−λ

)
tn

n!
.

Thus, by (48) and (49), we have{
n+ r

k + r

}
r,λ

=

n∑
m=k

Lr,λ(m, k)

{
n

m

}
−λ

, (k ≥ 0). (50)

Replacing t by 1
λ log

(
1

1−λt

)
in (48) and using (7), we see that

1

λk

1

k!

(
1

1− λt
− 1

)k( 1

1− λt

) r
λ

=
∞∑

m=k

{
m+ r

k + r

}
r,λ

1

m!

(
1

λ
log

(
1

1− λt

))m

(51)

=
∞∑

m=k

{
m+ r

k + r

}
r,λ

∞∑
n=m

[
n

m

]
λ

tn

n!

=

∞∑
n=k

( n∑
m=k

{
m+ r

k + r

}
r,λ

[
n

m

]
λ

)
tn

n!
.

By (42) and (51), we get

Lr,λ(n, k) =
n∑

m=k

{
m+ r

k + r

}
r,λ

[
n

m

]
λ

, (k ≥ 0). (52)

Therefore, by (50) and (52), we obtain the following theorem.

Theorem 9. For n, k ≥ 0 with n ≥ k, we have{
n+ r

k + r

}
r,λ

=
n∑

m=k

Lr,λ(n, k)

{
n

m

}
−λ

,

and

Lr,λ(n, k) =
n∑

m=k

{
m+ r

k + r

}
r,λ

[
n

m

]
λ

.

3. Further Remarks

A Poisson random variable indicates how many events occured within a given period
of time. A random variable X, taking on one of the variables 0, 1, 2, . . . , is said to be the
Poisson random variable with parameter α > 0 if the probability mass function of X is
given by

p(i) = P{X = i} = e−αα
i

i!
, (see [16, 25]). (53)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1398

Let f be a real valued function, and let X be a random variable. Then we define

E[f(X)] =

∞∑
i=0

f(i)p(i), (see [25]). (54)

For λ ∈ R with 0 < λ < 1, assume that X is the Poisson random variable with parameter
α
λ (> 0). Then we note from (54) that

E

[(
1

1− λt

)X]
=

∞∑
i=0

(
1

1− λt

)i

p(i) (55)

=

∞∑
i=0

(
1

1− λt

)i 1

i!

(
α

λ

)i

e−
α
λ = e

1
λ

α
1−λt e−

α
λ

= e
α
λ
( 1
1−λt

−1) =

∞∑
n=0

BL
n,λ(α)

tn

n!
.

On the other hand, by binomial expansion, we get

E

[(
1

1− λt

)X]
= E

[(
1

1− λt

)λX
λ
]
=

∞∑
n=0

E
[
⟨Xλ⟩n,λ

] tn
n!
. (56)

Hence, by (55) and (56), we get

E
[
⟨Xλ⟩n,λ

]
= BL

n,λ(α), (n ≥ 0). (57)

For r ≥ 0, from (46) and (55), we observe that

E

[(
1

1− λt

)X+ r
λ
]
= E

[(
1

1− λt

)X](
1

1− λt

) r
λ

(58)

= e
α
λ
( 1
1−λt

−1)

(
1

1− λt

) r
λ

=
∞∑
n=0

LB
(r)
n,λ(α)

tn

n!
.

On the other hand, by binomial expansion, we get

E

[(
1

1− λt

)X+ r
λ
]
= E

[(
1

1− λt

)λX+r
λ

]
=

∞∑
n=0

E[⟨λX + r⟩n,λ]
tn

n!
. (59)

Thus, by (58) and (59), we get

LB
(r)
n,λ(α) = E

[
⟨λX + r⟩n,λ

]
, (n ≥ 0). (60)

From (4), (39), and (60), we note that

LB
(r)
n,λ(α) = E

[
⟨Xλ+ r⟩n,λ

]
=

n∑
k=0

Lr,λ(n, k)E[(λX)k,λ]. (61)



J. Kwon et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1385-1402 1399

=

n∑
k=0

Lr,λ(n, k)

k∑
j=0

S1,λ(k, j)λ
jE[Xj ].

From (13), we have

E[Xj ] =
∞∑
k=0

kjp(k) =
∞∑
k=0

(αλ )
k

k!
e−

α
λ kj (62)

= e−
α
λ

∞∑
k=0

kj

k!
(
α

λ
)k = ϕj(

α

λ
).

Hence, by (13), (61), and (62), we get

LB
(r)
n,λ(α) =

n∑
j=0

n∑
k=j

Lr,λ(n, k)S1,λ(k, j)ϕj,λ(α), (n ≥ 0). (63)

We obtain the following theorem from (57), (60), and (63).

Theorem 10. Assume that X is the Poisson random variable with parameter α
λ (> 0), for

λ with 0 < λ < 1.

E
[
⟨Xλ⟩n,λ

]
= BL

n,λ(α), LB
(r)
n,λ(α) = E

[
⟨λX + r⟩n,λ

]
=

n∑
j=0

n∑
k=j

Lr,λ(n, k)S1,λ(k, j)ϕj,λ(α), (n ≥ 0).

4. Conclusion

The degenerate versions arise when we replace the powers of x by the generalized
falling factorial polynomials (x)k,λ in the defining equations, whereas the λ-analogues
appear when we replace the falling factorials (x)k by the generalized falling factorials.

In this paper, as λ- analogues of the Lah numbers and Lah-Bell polynomials, we studied
the λ-analogues of Lah numbers Lλ(n, k) and Lah-Bell polynomials BL

n,λ(x). For those
numbers and polynomials, we investigated some properties, explicit expressions, generating
functions and Dobinski-like formulas. We also considered the more general λ-analogues

of r-Lah numbers Lr,λ(n, k) and r-extended λ-Lah-Bell polynomials LB
(r)
n,λ(x) and similar

results to Lλ(n, k) and BL
n,λ(x) were derived. In addition, we showed the expectation of

one random variable and that of another random variable, both related to the Poisson

random variable with parameter α
λ , are respectively equal to BL

n,λ(α) and LB
(r)
n,λ(α).

As one of our future research projects, we would like to continue to explore λ-analogues
of some special numbers and polynomials and their applications to physics, science and
engineering as well as to mathematics.



REFERENCES 1400

Acknowledgements

We would like to thank Jangjeon Institute for Mathematical Science for its support
during the preparation of this paper.

Conflict of interest

The authors declare that they have no competing interests in this paper.

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