EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 1-10
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Study on degenerate Stirling numbers
Dae San Kim1, Taekyun Kim2,*, Jongkyum Kwon3,*

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, Gyeongsang National University, Jinju 52828,
Republic of Korea

Abstract. In this paper, we consider various Stirling numbers of both kinds, including the unsigned degen-
erate Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, the unsigned
degenerate r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind.
The aim of this paper is by using generating functions to further study explicit expressions, some identities
and equivalent relations for those Stirling numbers.
2020 Mathematics Subject Classifications: 11B73, 11B83
Key Words and Phrases: unsigned degenerate Stirling numbers of the first kind, unsigned degenerate
r-Stirling numbers of the first kind, degenerate Stirling numbers of the second kind, degenerate r-Stirling
numbers of the second kind

1. Introduction

The Stirling number of the second kind
{n

k

}
enumerates the number of partitions of the set

[n] = {1,2, . . . ,n} into k nonempty disjoint subsets, while the unsigned Stirling number of the
first kind

[n
k

]
counts the number of permutations of the set [n] having k disjoint cycles. Let r be a

positive integer. Then the Stirling numbers of both kinds are generalized as follows. The r-Stirling
number of the second kind

{n
k

}
r enumerates the number of partitions of the set [n] into k nonempty

disjoint subsets in such a way that 1,2, . . . ,r are in distinct subsets, while the unsigned r-Stirling
number of the first kind

[n
k

]
r counts the number of permutations of the set [n] having k disjoint

cycles in such a way that 1,2, . . .r are in distinct cycles.
Carlitz initiated an investigation of degenerate versions of some special numbers and polyno-

mials. Indeed, in [5] he studied degenerate versions of Bernoulli and Euler polynomials, namely
the degenerate Bernoulli and degenerate Euler polynomials. In recent years, this exploration for
degenerate versions have regained interests of some mathematicians and a lot of interesting re-
sults on degenerate versions of many special polynomials and numbers were obtained during the

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5011

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

https://www.ejpam.com 1 © 2024 EJPAM All rights reserved.



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 2

course of this quest (see [12, 14, 15] and the references therein). For instance, the unsigned de-
generate Stirling number of the first kind

[n
k

]
λ

and the degenerate Stirling number of the second
kind

{n
k

}
λ

are respectively degenerate versions of
[n

k

]
and

{n
k

}
. Further, the unsigned degenerate

r-Stirling number of the first kind
[n

k

]
r,λ and the degenerate r-Stirling number of the second kind{n

k

}
r,λ are respectively degenerate versions of

[n
k

]
r and

{n
k

}
r. These investigations about degenerate

versions have been carried out by employing such diverse tools as generating functions, combina-
torial methods, p-adic analysis, p-adic q-analysis, umbral calculus, probability theory, differential
equations, analytic number theory, operator theory, and quantum mechanics.

The aim of this paper is by using generating functions to further study explicit expressions,
some identities and equivalent relations for the aforementioned Stirling numbers of both kinds.
The outline of this paper is as follows. In Section 1, we recall the degenerate exponentials and
degenerate logarithms. We remind the reader of the unsigned Stirling numbers of the first kind
and its generalization the unsigned r-Stirling numbers of the first kind, and the Stirling numbers of
the second kind and its generalization the r-Stirling numbers of the second kind. Then we recall
their degenerate versions, namely the unsigned degenerate Stirling numbers of the first kind and
its generalization the unsigned degenerate r-Stirling numbers of the first kind, and the degenerate
Stirling numbers of the second kind and its generalization the degenerate r-Stirling numbers of
the second kind. Section 2 is the main result of this paper. We find an explicit expression for{n

k

}
λ

as a finite sum involving (l)n,λ and an equivalent inverse relation expressing (k)n,λ in terms
of

{n
j

}
λ

in Theorem 2.1. In Theorem 2.2, we express the finite sum ∑
m
k=1(k)n,λ Hk as a finite

sum involving
{n

k

}
λ

. Here Hk are the usual harmonic numbers. In Theorem 2.3, we find an
explicit expression of

[n+r
k+r

]
r,λ

as a finite sum involving
[n

l

]
λ

. In Theorem 2.4, we derive finite
sum identities involving the unsigned Stirling numbers of the first kind, the degenerate Stirling
numbers of the second kind and the generalized falling factorials. An explicit expression for{n+r

k+r

}
r,λ

is found as a finite sum involving (l + r)n,λ , as a generalization of the corresponding
result. Theorem 2.5 is a generalization of Theorem 2.1, while Theorem 2.6 is that of Theorems
2.2 and 2.4. For the rest of this section, we recall the facts that are needed throughout this paper.

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

ex
λ
(t) =

∞

∑
k=0

(x)k,λ
tk

k!
, (see [9, 10, 13, 16]), (1)

where the generalized falling factorials are given by

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

For x = 1, for brevity we write eλ (t) = e1
λ
(t) = ∑

∞
k=0(1)k,λ

tk

k! .
As the compositional inverse of eλ (t), the degenerate logarithm is given by

logλ (t) =
1
λ
(tλ −1), (see [10]). (3)

Note that limλ→0 logλ (t) = log t.



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 3

For n ≥ 0, the unsigned Stirling numbers of the first kind are defined by

⟨x⟩n =
n

∑
k=0

[
n
k

]
xk, (see [1, 3, 4, 8, 17, 18]), (4)

where the rising factorials are given by

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

The Stirling numbers of the second kind are given by

xn =
n

∑
k=0

{
n
k

}
(x)k, (n ≥ 0), (see [2, 6, 7]), (5)

where the falling factorials are given by

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

For n,r ≥ 0, the unsigned r-Stirling numbers of the first kind are defined by

⟨x+ r⟩n =
n

∑
k=0

[
n+ r
k+ r

]
r
xk, (see [11, 12, 14]). (6)

The r-Stirling numbers of the second kind are defined by

(x+ r)n =
n

∑
k=0

{
n+ r
k+ r

}
r
(x)k, (see [12, 14]). (7)

For any λ ∈ R, the unsigned degenerate Stirling numbers of the first kind are defined by

⟨x⟩n =
n

∑
k=0

[
n
k

]
λ

⟨x⟩k,λ , (n ≥ 0), (see [10, 14, 15]), (8)

where the generalized rising factorials are given by

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

In view of (5), the degenerate Stirling numbers of the second kind are defined by

(x)n,λ =
n

∑
k=0

{
n
k

}
λ

(x)k, (see [10]). (9)

For n,r ≥ 0, the unsigned degenerate r-Stirling numbers of the first kind are given by

⟨x+ r⟩n =
n

∑
k=0

[
n+ r
k+ r

]
r,λ
⟨x⟩k,λ (n ≥ 0), (see [11, 13−−15]). (10)

In [14], the degenerate r-Stirling numbers of the second kind are defined by

(x+ r)n,λ =
n

∑
k=0

{
n+ r
k+ r

}
r,λ
(x)k, (n ≥ 0). (11)



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 4

2. Some formulas for degenerate Stirling numbers

For (9), we note that

1
k!
(
eλ (t)−1

)k
=

∞

∑
n=k

{
n
k

}
λ

tn

n!
, (k ≥ 0), (see [10, 14]). (12)

By (12), we get
k

∑
l=0

(
k
l

)
(−1)k−l(l)n,λ = k!

{
n
k

}
λ

, (n,k ≥ 0). (13)

Theorem 1. For any nonnegative integers n,k, we have

k!
{

n
k

}
λ

=
k

∑
j=0

(
k
j

)
(−1)k− j( j)n,λ ⇐⇒ (k)n,λ =

k

∑
j=0

(
k
j

)
j!
{

n
j

}
λ

. (14)

Proof. (=⇒) Assume that

k!
{

n
k

}
λ

=
k

∑
j=0

(
k
j

)
(−1)k− j( j)n,λ .

Then we have

k

∑
j=0

(
k
j

)
j!
{

n
j

}
λ

=
k

∑
j=0

(
k
j

) j

∑
l=0

(
j
l

)
(−1) j−l(l)n,λ

=
k

∑
l=0

(l)n,λ

k

∑
j=l

(
k
j

)(
j
l

)
(−1) j−l

=
k

∑
l=0

(l)n,λ

(
k
l

) k−l

∑
j=0

(−1)k−l− j
(

k− l
j

)

=
k

∑
l=0

(l)n,λ

(
k
l

)
(1−1)k−l = (k)n,λ .

(⇐=) Suppose that

(k)n,λ =
k

∑
j=0

(
k
j

)
j!
{

n
j

}
λ

.

Then we have

k

∑
j=0

(
k
j

)
(−1)k− j( j)n,λ =

k

∑
j=0

(
k
j

)
(−1)k− j

j

∑
l=0

(
j
l

)
l!
{

n
l

}
λ

=
k

∑
l=0

l!
{

n
l

}
λ

k

∑
j=l

(
k
j

)(
j
l

)
(−1)k− j



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 5

=
k

∑
l=0

l!
{

n
l

}
λ

(
k
l

) k−l

∑
j=0

(−1)k−l− j
(

k− l
j

)

=
k

∑
l=0

{
n
l

}
λ

(
k
l

)
l!(1−1)k−l = k!

{
n
k

}
λ

.

The harmonic numbers are defined by

H0 = 0, Hk = 1+
1
2
+

1
3
+ · · ·+ 1

k
, (k ∈ N). (15)

Theorem 2. For n,m ∈ N, we have

m

∑
k=1

(k)n,λ Hk =
m

∑
j=1

j!
{

n
j

}
λ

(
m+1
j+1

)(
Hm+1 −

1
j+1

)
.

Proof. From Theorem 2.1 and noting that
{n

0

}
λ
= 0, for n ≥ 1, we see that

m

∑
k=1

(k)n,λ Hk =
m

∑
k=1

Hk

k

∑
j=1

(
k
j

)
j!
{

n
j

}
λ

(16)

=
m

∑
j=1

j!
{

n
j

}
λ

m

∑
k= j

Hk

(
k
j

)
.

The forward difference operator △ is defined by △ f (x) = f (x+1)− f (x). From the definition of
the forward difference operator, we get

f (x)△g(x) =△
(

f (x)g(x)
)
−
(
△ f (x)

)
g(x+1). (17)

Thus, by (17), we get

m−1

∑
k=0

f (k)
(
△g(k)

)
=

m−1

∑
k=0

△
(

f (k)g(k)
)
−

m−1

∑
k=0

(
△ f (k)

)
g(k+1) (18)

=
m−1

∑
k=0

(
f (k+1)g(k+1)− f (k)g(k)

)
−

m−1

∑
k=0

(
△ f (k)

)
g(k+1)

= f (m)g(m)− f (0)g(0)−
m−1

∑
k=0

(
△ f (k)

)
g(k+1).

Let f (k) = Hk and g(k) =
( k

j+1

)
. Then we have

△g(k) = g(k+1)−g(k) =
(

k+1
j+1

)
−
(

k
j+1

)
=

(
k
j

)
,

△ f (x) = f (k+1)− f (k) = Hk+1 −Hk =
1

k+1
.

(19)



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 6

From (18) and (19), we note taht

m−1

∑
k=0

(
k
j

)
Hk =

m−1

∑
k=0

(
△g(k)

)
f (k) = f (m)g(m)−

m−1

∑
k=0

(
△ f (k)

)
g(k+1) (20)

= Hm

(
m

j+1

)
−

m−1

∑
k=0

1
k+1

(
k+1
j+1

)
= Hm

(
m

j+1

)
− 1

j+1

m−1

∑
k=0

(
k
j

)
= Hm

(
m

j+1

)
− 1

j+1

m−1

∑
k=0

[(
k+1
j+1

)
−
(

k
j+1

)]
= Hm

(
m

j+1

)
− 1

j+1

(
m

j+1

)
=

(
m

j+1

)(
Hm − 1

j+1

)
.

Thus, by (16) and (20), we get

m

∑
k=1

(k)n,λ Hk =
m

∑
j=1

j!
{

n
j

}
λ

m

∑
k= j

Hk

(
k
j

)
=

m

∑
j=1

j!
{

n
j

}
λ

(
m+1
j+1

)(
Hm+1 −

1
j+1

)
.

From (10), we note that(
1

1− t

)r( 1
1− t

)x

=
∞

∑
n=0

⟨x+ r⟩n
tn

n!
=

∞

∑
n=0

n

∑
k=0

[
n+ r
k+ r

]
r,λ
⟨x⟩k,λ

tn

n!
(21)

=
∞

∑
k=0

(
∞

∑
n=k

[
n+ r
k+ r

]
r,λ

tn

n!

)
⟨x⟩k,λ .

On the other hand, by (1) and (3), we get(
1

1− t

)r( 1
1− t

)x

= e−x
λ

(
logλ (1− t)

)( 1
1− t

)r

=
∞

∑
k=0

(− logλ (1− t))k

k!

(
1

1− t

)r

⟨x⟩k,λ (22)

=
∞

∑
k=0

1
k!

(
log−λ

(
1

1− t

))k( 1
1− t

)r

⟨x⟩k,λ .

By (21) and (22), we get

1
k!

(
log−λ

(
1

1− t

))k( 1
1− t

)r

=
∞

∑
n=k

[
n+ r
k+ r

]
r,λ

tn

n!
, (k ≥ 0). (23)

From (8) or (23), we see that

1
k!

(
log−λ

(
1

1− t

))k

=
∞

∑
n=k

[
n
k

]
λ

tn

n!
. (24)

Theorem 3. For any nonnegative integers n,k, we have[
n+1
k+1

]
1,λ

=
n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨1⟩l−k,λ .



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 7

In general, for r ≥ 0, we have [
n+ r
k+ r

]
r,λ

=
n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨r⟩l−k,λ .

Proof. From (23), we note that

1
1− t

1
k!

(
log−λ

(
1

1− t

))k

=
∞

∑
n=k

[
n+1
k+1

]
1,λ

tn

n!
. (25)

On the other hand, by (3) and (24), we get

1
k!

(
log−λ

(
1

1− t

))k 1
1− t

=
1
k!

(
log−λ

(
1

1− t

))k ∞

∑
l=0

⟨1⟩k,λ

l!

(
log−λ

(
1

1− t

))l

(26)

=
∞

∑
l=0

⟨1⟩l,λ (k+1)!
l!k!

1
(k+1)!

(
log−λ

(
1

1− t

))k+l

=
∞

∑
l=0

⟨1⟩k,λ (k+ l)!
k!l!

∞

∑
n=k+l

[
n

k+ l

]
λ

tn

n!

=
∞

∑
l=0

(
k+ l

k

)
⟨1⟩l,λ

∞

∑
n=k+l

[
n

k+ l

]
λ

tn

n!
=

∞

∑
l=k

(
l
k

)
⟨1⟩l−k,λ

∞

∑
n=l

[
n
l

]
λ

tn

n!

=
∞

∑
n=k

( n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨1⟩l−k,λ

)
tn

n!
.

Thus, by (25) and (26), we get [
n+1
k+1

]
λ

=
n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨1⟩l−k,λ . (27)

More generally, for any r ≥ 0, we have

∞

∑
n=k

[
n+ r
k+ r

]
r,λ

tn

n!
= e−r

λ

(
logλ (1− t)

) 1
k!

(
log−λ

(
1

1− t

))k

(28)

=
∞

∑
l=0

⟨r⟩l,λ
1
l!

(
log−λ

(
1

1− t

))l 1
k!

(
log−λ

(
1

1− t

))k

=
∞

∑
l=0

⟨r⟩l,λ

(
k+ l

l

)
1

(k+ l)!

(
log−λ

(
1

1− t

))k+l

=
∞

∑
l=k

⟨r⟩l−k,λ

(
l
k

)
1
l!

(
log−λ

(
1

1− t

))l

=
∞

∑
l=k

⟨r⟩l−k,λ

(
l
k

)
∞

∑
n=l

[
n
l

]
λ

tn

n!

=
∞

∑
n=k

( n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨r⟩l−k,λ

)
tn

n!
.



D. Kim, T. Kim, J. Kwon / Eur. J. Pure Appl. Math, 17 (1) (2024), 1-10 8

By comparing the coefficients on both sides of (28), we get[
n+ r
k+ r

]
r,λ

=
n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨r⟩l−k,λ , (n,k ≥ 0).

Note that[
n+1
k+1

]
1
= lim

λ→0

[
n+1
k+1

]
1,λ

= lim
λ→0

n

∑
l=k

(
l
k

)[
n
l

]
λ

⟨1⟩l−k,λ =
n

∑
l=k

(
l
k

)[
n
l

]
,

and [
n+ r
k+ r

]
r
= lim

λ→0

[
n+ r
k+ r

]
r,λ

=
n

∑
l=k

(
l
k

)[
n
l

]
rl−k.

Theorem 4. For n,m ∈ N, we have

m

∑
k=1

[
m
k

]
(k)n,λ =

m

∑
j=1

j!
{

n
j

}
λ

[
m+1
j+1

]
1
.

Proof. By (14), we get

m

∑
k=1

[
m
k

]
(k)n,λ =

m

∑
k=1

[
m
k

] k

∑
j=1

(
k
j

)
j!
{

n
j

}
λ

(29)

=
m

∑
j=1

j!
{

n
j

}
λ

m

∑
k= j

(
k
j

)[
m
k

]
=

m

∑
j=1

j!
{

n
j

}
λ

[
m+1
j+1

]
1
. (30)

From (11), we note that

1
k!
(
eλ (t)−1

)ker
λ
(t) =

∞

∑
n=k

{
n+ r
k+ r

}
r,λ

tn

n!
, (k,r ≥ 0). (31)

By (31), we get

∞

∑
n=k

{
n+ r
k+ r

}
r,λ

tn

n!
=

1
k!
(
eλ (t)−1

)ker
λ
(t) =

1
k!

k

∑
l=0

(
k
l

)
(−1)k−lel+r

λ
(t) (32)

=
∞

∑
n=0

1
k!

k

∑
l=0

(
k
l

)
(−1)k−l(l + r)n,λ

tn

n!
.

Thus, by comparing the coefficients on both sides of (32), we get

k!
{

n+ r
k+ r

}
r,λ

=
k

∑
l=0

(
k
l

)
(−1)k−l(l + r)n,λ , (n ≥ k). (33)

Theorem 5. For n,r ≥ 0, we have

k!
{

n+ r
k+ r

}
r,λ

=
k

∑
l=0

(
k
l

)
(−1)k−l(l + r)n,λ ⇐⇒ (k+ r)n,λ =

k

∑
l=0

(
k
l

)
l!
{

n+ r
l + r

}
r,λ
.



REFERENCES 9

Proof. This can be shown just as the proof of Theorem 2.1.

Theorem 6. For n,m ∈ N, we have

m

∑
k=1

(k+ r)n,λ Hk =
m

∑
j=1

j!
{

n+ r
j+ r

}
r,λ

(
m+1
j+1

)(
Hm+1 −

1
j+1

)
,

and
m

∑
k=1

[
m
k

]
(k+ r)n,λ =

m

∑
j=1

j!
{

n+ r
j+ r

}
r,λ

[
m+1
j+1

]
1
.

Proof. By using (33), this can be proved just as the proofs of Theorem 2.2 and Theorem 2.4.
The details are left to the reader.

3. Conclusion

Various degenerate Stirling numbers of both kinds appear very frequently when we study de-
generate versions of many special numbers and polynomials. In this paper, we investigated several
degenerate Stirling numbers like the unsigned degenerate Stirling numbers of the first kind, the de-
generate Stirling numbers of the second kind, the unsigned degenerate r-Stirling numbers of the
first kind and the degenerate r-Stirling numbers of the second kind. We found some identities,
explicit expressions and some equivalent relations among them. It is one of our future projects to
continue to explore degenerate versions of some special numbers and polynomials and their appli-
cations to statistics, physics, science, engineering, and social sciences as well as to mathematics.

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