EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6069
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Properties of the Non-central Stirling Numbers of
the Second Kind with Complex Parameters

Erwin N. Arlan1∗, Maribeth B. Montero2

1 Mathematics Department, Caraga State University, Butuan City, Philippines
2 Mathematics Department, Mindanao State University, Marawi City, Philippines

Abstract. This work extends the non-central Stirling numbers of the second kind to complex
arguments. This extension is achieved through an integral representation employing a Hankel
contour. Furthermore, we investigate the extent to which key properties, including the recurrence
relations, of the classical Stirling numbers are preserved under this complex generalization.

2020 Mathematics Subject Classifications: 05A15, 05A18, 30C15

Key Words and Phrases: Classical Stirling numbers, Non-central Stirling numbers, Complex
numbers

1. Introduction

Stirling numbers have long been a fundamental concept in combinatorics, originally
defined for non-negative integers. These numbers play a crucial role in counting permu-
tations, partitions, and other combinatorial structures. However, their application has
typically been limited to discrete contexts. Extending Stirling numbers to real and com-
plex arguments not only generalizes many well-known combinatorial identities but also
significantly broadens their applicability across a wide array of mathematical fields.

In [1], Koutras introduced the non-central Stirling numbers of the second kind by a
natural extension of the definition of the classical Stirling numbers of the second kind,
S(n, k). The non-central Stirling numbers of the second kind, denoted by Sa(n, k) are
defined as the coefficients of the following expansion with parameter a,

(t− a)n =
n∑

k=0

Sa(n, k)(t)n

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6069

Email addresses: enarlan@carsu.edu.ph (E. N. Arlan), maribeth.montero@msumain.edu.ph (M. B.
Montero)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 2 of 10

where (t)n is the falling factorial of t of order n given by

(t)n =
n∏

i=1

(t− i+ 1) = t(t− 1)(t− 2) · · · (t− n+ 1),

for n ≥ 1; (t)0 = 1 and Sa(0, 0) = 1, Sa(n, 0) = (−a)n and Sa(0, k) = 0 , n, k ̸= 0.

Note that if a = 0, the classical Stirling numbers, S(n, k), are obtained. This definition
leads to an alternative definition in terms of exponential generating
functions which allows for further extensions to generalized non-central Stirling numbers.

Motivated by the problem of Graham, Knuth, and Patashnik, in [2], that is, to
generalize stirling numbers of the second kind S(n, k) by extending the range values of
parameters n and k to complex numbers, Flajolet and Prodinger, in [3], defined an
extension of the classical Stirling numbers S(n, k) of the second kind to complex arguments.
They established an alternative and more natural extension of Stirling numbers of complex
arguments for which most classical identities are still satisfied. Some of these properties
are listed here.

i) The Stirling numbers S(x, y) of the second kind for complex arguments satisfy the
following recurrence relation

S(x, y) = S(x− 1, y − 1) + yS(x− 1, y).

ii) For any complex x and n ∈ N, we have the following formula

S(x, k) =
1

k!

k∑
j=1

(
k

j

)
(−1)k−jjx.

Thus, following the work of Flajolet and Prodinger, it is interesting if we could extend
non-central Stirling numbers of the second kind to complex arguments.

2. Preliminaries

This section covers fundamental concepts and results in complex analysis and combi-
natorics, including the binomial theorem, which will be applied in the later section. These
results can be found in [4], [5], [6], [7], and [8].

Lemma 1. [4] For any integer n ≥ 0, we have

(x+ y)n =
n∑

r=0

(
n

r

)
xn−ryr.



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 3 of 10

The number

(
n

r

)
is called a binomial coefficient and satisfies the following recurrence

relation (
n

r

)
=

(
n− 1

r

)
+

(
n− 1

r − 1

)
.

The following is the generalized binomial theorem due to Newton.

Lemma 2. [4] For complex numbers x, y, and s, we have

(x+ y)s =

∞∑
r=0

(
s

r

)
xs−ryr,

where (
s

r

)
=

 1, r = 0
s(s− 1)(s− 2) · · · (s− r + 1)

r!
, r > 0

.

Moreover, the Gamma function is being considered. This function is known to be
related to Stirling-type numbers. The Gamma function [7], denoted by Γ, is an extension
of the factorial function to real and complex numbers defined via improper integral that
converges only for complex numbers with a positive real part. That is,

Γ(z) =

∫ ∞

0
tz−1e−tdt,

where Re(z) > 0. When z = 1,

Γ(1) =

∫ ∞

0
e−tdt = 1.

When z = n ∈ Z+, we use integraton by parts and get

Γ(n) = (n− 1)

∫ ∞

0
tn−2e−tdt.

Consequently, after applying integration by parts (n− 1) times, we obtain

Γ(n) = (n− 1)(n− 2) · · · (1)
∫ ∞

0
e−tdt = (n− 1)!.

This shows that Γ(z) is indeed an extension of the factorial function with its argument
shifted down by 1.

The following theorem is a property of the Gamma function.



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 4 of 10

Lemma 3. [9] The Gamma function Γ(z) has an integral representation

1

Γ(z)
=

1

2πi

∫
γ
ett−zdt,

where γ is a Hankel contour.

The preceding formula is known as Hankel’s contour integral. The path of integration
γ starts at −∞ − i0 on the real axis, goes to −ϵ − i0, circles the origin in the counter-
clockwise direction with the radius ϵ to the point −ϵ+i0 and returns to the point −∞+i0.

Before presenting the main result, we introduce the following lemma, an original
contribution by the author, which plays an important role in establishing several
fundamental properties of non-central Stirling numbers with complex indices.

Lemma 4. For complex numbers x and a with Re(a) < 0, we have

x!

2πi

∫
H
e(j−a)u du

ux+1
= (j − a)x.

Proof. Let s = x+ 1 and t = (j − a)u. Then u = (j − a)−1t and du = (j − a)−1dt.

Hence,

x!

2πi

∫
H
e(j−a)u du

ux+1
= (j − a)s−1 (s− 1)!

2πi

∫
H
t−setdt

= (j − a)s−1Γ(s)

2πi

∫
H
t−setdt.

However, by Lemma 3,
Γ(s)

2πi

∫
H
t−setdt = 1.

Thus,
x!

2πi

∫
H
e(j−a)u du

ux+1
= (j − a)x.

In the following section, we begin by presenting the definition of the non-central Stirling
numbers of the second kind for complex arguments and establish several key properties.
This section presents the central result of our work, which we hope will serve as a foun-
dation for further analysis and potential applications.



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 5 of 10

3. The Non-central Stirling Numbers with Complex Indices

The exponential generating funcion for Sa(n, k) is given by

∞∑
n=k

Sa(n, k)
un

n!
= e−au 1

k!
(eu − 1)k.

That is, ∑
n≥0

Sa(n, k)
k!

n!
un = e−au(eu − 1)k.

Hence, by Cauchy Integral Formula, we have

Sa(n, k) =
1

2πi

n!

k!

∫
γ

e−au(eu − 1)kdu

un+1
,

where γ is a small contour encircling the origin.

We can deform γ into a Hankel contour which starts from −∞ below the negative
x − axis surrounding the origin couterclockwise and returns to −∞ above the negative
x − axis. This suggest the following definition. Here, we assume that H is at a distance
≤ 1 from the real axis.

Definition 3.1. The non-central Stirling numbers of the second kind with complex
arguments x and y, denoted by Sa(x, y), are defined by

Sa(x, y) =
1

2πi

x!

y!

∫
H

e−au(eu − 1)ydu

ux+1
,

where a is a complex number with Re(a) < 0, x! = Γ(x+1) and the logarithm involved in
the functions (eu − 1)y and ux+1 is taken to be the principal branch.

Following this definition, we explore a fundamental relationship that is essential to
the theory of Stirling-type numbers. The first theorem we present establishes a relation
commonly employed to define these numbers.

Theorem 3.1. For complex numbers x and a with Re(a) < 0, the non-central Stirling
numbers for complex arguments satisfy the following relation:

(t− a)x =

∞∑
k=0

Sa(x, k)(t)k.

Proof. First, observe that

(t)k = [t(t− 1)(t− 2) · · · (t− k + 1)]
k!

k!

(t− k)!

(t− k)!
= k!

(
t

k

)
.



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 6 of 10

Now, applying Definition 3.1, we get

∞∑
k=0

Sa(x, k)(t)k =
∞∑
k=0

{
1

2πi

x!

k!

∫
H

e−au(eu − 1)kdu

ux+1

}
(t)k

=
x!

2πi

∫
H
e−au

{ ∞∑
k=0

(eu − 1)k

k!
(t)k

}
du

ux+1
.

Then by the generalized binomial theorem due to Newton,
∞∑
k=0

(eu − 1)k

k!
(t)k =

∞∑
k=0

(
t

k

)
(eu − 1)k

= [(eu − 1) + 1]t = eut.

Therefore, by Lemma 4,
∞∑
k=0

Sa(x, k)(t)k =
x!

2πi

∫
H
e−aueut

du

ux+1
= (t− a)x.

Theorem 3.2. For complex numbers x and a with Re(a) < 0, Re(x) > 0, the non-central
Stirling numbers Sa(x, y) satisfy the following recurrence relation:

Sa(x, y) = Sa(x− 1, y − 1) + (y − a)Sa(x− 1, y).

Proof. By Definition 3.1,

Sa(x, y) =
1

2πi

x!

y!

∫
H

e−au(eu − 1)ydu

ux+1
.

Now let s = e−au(eu − 1)y and dt =
du

ux+1
. Then,

ds =

(
eue−auy(eu − 1)y−1 − ae−au(eu − 1)y

)
du and t =

1

−xux
.

Thus, integration by parts yields

Sa(x, y) =
1

2πi

x!

y!

{[e−au(eu − 1)y

−xux

]
H
−
∫
H

1

−xux

(
eue−auy(eu−1)y−1−ae−au(eu−1)y

)
du

}
.

However, as u → −∞,
1

−xux
→ 0. Hence,

Sa(x, y) =
1

2πi

x!

y!

[
−
∫
H

1

−xux
e−au(eu − 1)y−1

(
euy − a(eu − 1)

)
du

]

=
1

2πi

(x− 1)!

y!

∫
H

(
ye−au(eu − 1)y−1

ux
+

(y − a)e−au(eu − 1)y

ux

)
du

=
1

2πi

(x− 1)!

(y − 1)!

∫
H

e−au(eu − 1)y−1du

u(x−1)+1
+ (y − a)

1

2πi

(x− 1)!

y!

∫
H

e−au(eu − 1)ydu

u(x−1)+1
.



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 7 of 10

Thus, by Definition 3.1,

Sa(x, y) = Sa(x− 1, y − 1) + (y − a)Sa(x− 1, y).

The next result is an explicit formula for Sa(x, k).

Theorem 3.3. For complex numbers x and a with Re(a) < 0, and a nonnegative integer
k, we have

Sa(x, k) =
1

k!

k∑
j=0

(−1)k−j

(
k

j

)
(j − a)x.

Proof. By Definition 3.1, we have

Sa(x, k) =
1

2πi

x!

k!

∫
H

e−au(eu − 1)kdu

ux+1
.

Applying the binomial theorem,

(eu − 1)k =
k∑

j=0

(
k

j

)
(−1)k−j(eu)j

=
k∑

j=0

(−1)k−j(e)uj
(
k

j

)
.

Thus,

Sa(x, k) =
1

2πi

x!

k!

∫
H
e−au

( k∑
j=0

(−1)k−jeuj
(
k

j

))
du

ux+1

=
1

k!

k∑
j=0

[
(−1)k−j

(
k

j

)
· x!

2πi

∫
H
e−aueuj

du

ux+1

]
.

Moreover, by Lemma 4,

Sa(x, k) =
1

k!

k∑
j=0

(−1)k−j

(
k

j

)
(j − a)x.

To illustrate this, let us consider the case where a = −1. Using the explicit formula
above, we compute the values of S−1(i, 0), S−1(i, 1), and S−1(1 + i, 1) as follows:

S−1

(
i, 0
)
=

1

0!

0∑
j=0

(−1)0−j

(
0

j

)
(j + 1)i

= 1(−1)0
(
0

0

)
(0 + 1)i = 1i = 1;



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 8 of 10

S−1

(
i, 1
)
=

1

1!

1∑
j=0

(−1)1−j

(
1

j

)
(j + 1)i

= 1

[
(−1)1

(
1

0

)
(0 + 1)i + (−1)0

(
1

1

)
(1 + 1)i

]
= −1 + cos (ln 2) + i sin (ln 2);

S−1

(
1 + i, 1

)
=

1

1!

1∑
j=0

(−1)1−j

(
1

j

)
(j + 1)1+i

= 1

[
(−1)1

(
1

0

)
(0 + 1)1+i + (−1)0

(
1

1

)
(1 + 1)1+i

]
= −1 + 2 cos (ln 2) + 2i sin (ln 2).

The same value is obtained for S−1(1 + i, 1) using the recurrence relation in Theorem
3.2 as shown below:

S−1

(
1 + i, 1

)
= S−1

(
(1 + i)− 1 , (1− 1)

)
+
(
1− (−1)

)
S−1

(
(1 + i)− 1 , 1

)
= S−1

(
i , 0

)
+ (2) S−1

(
i , 1

)
= 1 + 2

[
− 1 + cos (ln 2) + i sin (ln 2)

]
= −1 + 2 cos (ln 2) + 2i sin (ln 2).

Theorem 3.3 can be generalized further by substituting k with a complex number y,
as presented in the subsequent theorem.

Theorem 3.4. For complex numbers x, y and a with Re(a) < 0, we have

Sa(x, y) =
1

y!

∞∑
j=0

(−1)y−j

(
y

j

)
(j − a)x.

Proof. Since y is a complex number, we use the generalized binomial theorem due to
Newton to obtain

(eu − 1)y =
∞∑
j=0

(
y

j

)
(−1)y−j(eu)j .



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 9 of 10

In effect, using Definition 3.1, we have

Sa(x, y) =
1

2πi

x!

y!

∫
H
e−au

( ∞∑
j=0

(−1)y−jeuj
(
y

j

))
du

ux+1

=
1

2πi

x!

y!

∞∑
k=0

∫
H
e−au(−1)y−jeuj

(
y

j

)
du

ux+1

=
1

y!

∞∑
k=0

[
(−1)y−jeuj

(
y

j

)
· x!

2πi

∫
H
e(j−a)u du

ux+1

]
.

Consequently, by Lemma 4,

Sa(x, y) =
1

y!

∞∑
j=0

(−1)y−j

(
y

j

)
(j − a)x.

4. Conclusion

In this work, we extended the non-central Stirling numbers of the second kind to com-
plex arguments using an integral representation with a Hankel contour. This extension
broadens the use of these numbers and offers new insights into their properties in the com-
plex domain. We also explored how important properties, such as recurrence relations, are
preserved and adapted, showing the usefulness of this approach for studying combinatorial
structures.

Looking ahead, it would be beneficial to extend non-central Bell numbers (as defined in
[10]) to complex arguments, building on the work in [11]. These Bell numbers are impor-
tant for counting partitions and other combinatorial structures, and extending them could
uncover new identities, recurrence relations, and lead to deeper insights in areas like num-
ber theory, asymptotics, and complex analysis. Additionally, expanding the parameters of
(r1, · · · , rp)-Stirling numbers of the second kind from [12] and the generalized Apostol-type
Frobenius-Euler polynomials in [13] to complex arguments is also an interesting direction
for future research.

Acknowledgements

The authors would like to express their gratitude to Dr. Mary Joy D. Regidor-
Latayada for her valuable critiques and recommendations, which greatly contributed to
the refinement and direction of this research.

References

[1] M. Koutras. Non-central stirling numbers and some applications. Discrete Mathe-
matics, 42:73–89, 1982.



E.N. Arlan, M.B. Montero / Eur. J. Pure Appl. Math, 18 (2) (2025), 6069 10 of 10

[2] R.L. Graham, D.E. Knuth, and O. Patashnik. Discrete Mathematics. Addison-Wesley,
1989.

[3] P. Flajolet and H. Prodinger. On stirling numbers for complex arguments and hankel
contours. SIAM Journal on Discrete Mathematics, 12(2):155–159, 1999.

[4] C-C. Chen and K-M. Koh. Principles and Techniques in Combinatorics. World
Scientific, 1992.

[5] L. Comtet. Advanced Combinatorics. Reidel, Dordrecht, The Netherlands, 1974.
[6] R.V. Churchill and J.W. Brown. Complex Variables and Applications. McGraw-Hill,

Inc., 5th edition, 1990.
[7] F.W.J. Olver. Introduction to Asymptotics and Special Functions. Academic Press,

1974.
[8] L.L. Pennisi. Elements of Complex Variables. Cacho Hermanos, Inc., 2nd edition,

1976.
[9] F.W.J. Olver, D. Lozier, R. Boisvert, et al. NIST Handbook of Mathematical Func-

tions. Cambridge Press, 2010.
[10] Roberto B. Corcino, Harren Jaylo-Campos, and Amila P. Macodi-Ringia. On noncen-

tral bell numbers and their hankel transforms. The Journal of Algebra and Number
Theory, 2(2):1–10, 2014.

[11] Roberto B. Corcino, Cristina B. Corcino, and Maribeth B. Montero. On generalized
bell numbers for complex argument. Journal of Applied Mathematics and Computa-
tion, 270:60–68, 2015.

[12] Miloud Mihoubi and Mohammed Said Maamra. The (r1, ..., rp)-stirling numbers of
the second kind. Integers, 12(5):1047–1059, 2012.

[13] Letelier Castilla and William Ramı́rez. A new class of generalized apostol–type frobe-
nius–euler polynomials. AIMS Mathematics, 10(2):3623–3641, 2025.