EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 1069-1081
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Some Theorems on Tauber’s Generalized Stirling, Lah,
and Bell Numbers

Roberto B. Corcino1,∗, Cristina B. Corcino2, Gladys Jane S. Rama2

1 Research Institute for Computational Mathematics and Physics, Cebu Normal University,
6000 Cebu City, Philippines
2 Department of Mathematics, Cebu Normal University, 6000 Cebu City, Philippines

Abstract. In this paper, the authors obtain some properties for Tauber’s generalized Stirling and
Lah numbers, including other forms of recurrence relations, orthogonality and inverse relations, ra-
tional generating function and explicit formula in symmetric function form. Moreover, the authors
derive a new explicit formula, which is analogous to the Qi formula.

2010 Mathematics Subject Classifications: 05A15, 11B65, 11B73
Key Words and Phrases: Stirling numbers, Whitney numbers, Dowling numbers, generating
function, recurrence relation

1. Introduction

The nth Bell numbers, denoted by Bn, are known by their combinatorial interpretation
as the number of ways to partition an n-set. This interpretation is based on the definition
of Bell numbers as the sum of the classical Stirling numbers of the second kind [2]. That
is,

Bn =
n∑
k=0

{
n

k

}
(1)

where

{
n

k

}
denotes the Stirling numbers of the second kind (the Karamata-Knuth no-

tation), which can be interpreted as the number of ways to partition an n-set into k
nonempty subsets.

The Bell numbers were expressed by Spivey [5] as

Bm+n =
n∑
k=0

m∑
j=0

jn−k
{
m

j

}(
n

k

)
Bk. (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3424

Email addresses: rcorcino@yahoo.com (R. Corcino),
cristinacorcino@yahoo.com (C. Corcino), gjsrama@yahoo.com (G. Rama)

http://www.ejpam.com 1069 c© 2019 EJPAM All rights reserved.



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1070

The above formula was proven combinatorially by considering another way of counting
the number of ways to partition the set with m+ n objects.

Recently, Feng Qi [4] established a new explicit formula for Bell numbers, which is

expressed in terms of Stirling numbers of the second kind

{
n

k

}
and Lah numbers L(n, k).

The formula is given by

Bn =

n∑
k=1

(−1)n−k

[
k∑
l=1

L(k, l)

]{
n

k

}
(3)

which may be referred to as the Qi formula for Bell numbers. The formula has been
derived in two ways:

• using Faa di Bruno’s formula and the formula for the n-th derivative of e
1
x containing

Lah numbers [3, 9];

• using the inverse relation for the classical Stirling numbers of the first and second
kind.

These methods have been applied to obtain new explicit formula for different generaliza-
tions of Stirling numbers.

Certain generalization of Stirling numbers was introduced by S. Tauber [6–8] as the
coefficients Cmk,n and Dm

k,n of the following expansions of the given two sequences of poly-
nomials Q1(x, n) and Q2(x, n) for n = 0, 1, 2, . . .,

Qk(x, n) =
n∑

m=0

Cmk,nx
m (4)

xn =

n∑
m=0

Dm
k,nQk(x,m) (5)

for k=1, 2. Also, these numbers are equal to zero if n < m, m < 0, n < 0. These
generalized Stirling numbers of the first and second kind on Q- polynomials satisfy the
following triangular recurrence relations in [8],

Cmn = M(n)Cmn−1 +N(n)Cm−1
n−1 (6)

Dm
n = −M(m+ 1)

N(m+ 1)
Dm
n−1 +

1

N(m+ 1)
Dm−1
n−1 , (7)

where M and N are two functions of the variable n, such that M(0) 6= 0 and for n a
positive integer or zero N(0) 6= 0. In the paper [6] of Tauber, a generalization of Lah
numbers Lmk,h,n was introduced as the coefficient of the following relation,

Qk(x, n) =

n∑
m=0

Lmk,h,nQh(x,m) (8)



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1071

for two sequences of polynomials Qk and Qh where k 6= h, and k, h ∈ {1, 2}. These
numbers satisfy the following triangular recurrence relation,

Lmk,h,n =

[
Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
Lmk,h,n−1 +

Nk(n)

Nh(m)
Lm−1
k,h,n−1, (9)

where Mk(n) and Nk(n) are the subscripted version of the functions M(n) and N(n)
that correspond to Qk(x, n). Furthermore, the generalized Lah numbers Lmk,h,n have been
expressed in terms of generalized Stirling numbers of the first kind and the second kind
as follows

Lmk,h,n =

n∑
s=m

Csk,nD
m
h,s. (10)

The generalized Bell numbers, denoted by Bh,n, may be defined as the sum of the
generalized Stirling numbers of the second kind Dm

h,n,

Bh,n =
n∑

m=0

Dm
h,n. (11)

The Lah numbers can be defined as the coefficients of the following relations:

(−x)n =

n∑
k=0

Ln,k(x)k (12)

(x)n =

n∑
k=0

Ln,k(−x)k (13)

where (x)n is the falling factorial, defined by

(x)n =

{
x(x− 1) . . . (x− n+ 1), n ≥ 1

1, n = 0
.

Also,
(−x)n = (−1)n(x)(x+ 1) . . . (x+ n− 1) = (−1)nx(n)

where x(n) is the rising factorial, defined by

(x)(n) =

{
x(x+ 1) . . . (x+ n− 1), n ≥ 1

1, n = 0
.

Given polynomial Qk(x, n) = ((−1)k−1x)n, Tauber’s generalized Lah numbers Lmk,h,n in
(8) reduce to the ordinary Lah numbers Ln,m in (12) and (13) for h = 1 and k = 2. That
is,

Lm2,1,n = Ln,m = Lm1,2,n



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1072

Moreover, (4) and (5) give{
Cm1,n, D

m
1,n

}
=

{
(−1)n−m

[
n

m

]
,

{
n

m

}}
{
Cm2,n, D

m
2,n

}
=

{
(−1)n

[
n

m

]
, (−1)n

{
n

m

}}
,

where

[
n

k

]
is the Karamata-Knuth notation for the unsigned Stirling numbers of the first

kind.

The Whitney numbers of the first kind wα(n,m) and second kind Wα(n,m) of Dowling
lattices are defined in the paper of Benoumhani [1] as the coefficient of the following
equations,

αn
(
x− 1

α

)
n

=
n∑

m=0

wα(n,m)xm (14)

xn =
n∑

m=0

αmWα(n,m)

(
x− 1

α

)
m

(15)

which can be written as

(x− 1|α)n =

n∑
m=0

wα(n,m)xm (16)

xn =
n∑

m=0

Wα(n,m)(x− 1|α)m (17)

where α is a positive integer and (x|α)m =
∏m−1
i=0 (x− iα). It can easily be shown that

n∑
k=j

Wα(n, k)wα(k, j) =
n∑
k=j

wα(n, k)Wα(k, j) = δn,j (18)

fn =
n∑
k=0

wα(n, k)gk ⇐⇒ gn =

n∑
k=0

Wα(n, k)fk. (19)

When Qk(x, n) = ((−1)k−1x− 1|α)n, equations (4) and (5) yield equations (16) and (17),
respectively. This implies that{

Cm1,n, D
m
1,n

}
= {wα(n,m),Wα(n,m)} and{

Cm2,n, D
m
2,n

}
= {(−1)mwα(n,m), (−1)nWα(n,m)} .

From the definition of Whitney numbers, we may also define Whitney-Lah numbers
denoted by LWn,m(α) as the coefficients of the following relations,

(−x− 1|α)n =

n∑
m=0

LWn,m(α)(x− 1|α)m (20)



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1073

(x− 1|α)n =

n∑
m=0

LWn,m(α)(−x− 1|α)m. (21)

By assigning Qk(x, n) = ((−1)k−1x− 1|α)n , (8) gives

Lm2,1,n = LWn,m(α) = Lm1,2,n.

Hence, with k = 2 and h = 1, (10) yields

LWn,j(α) =
n∑
k=j

(−1)kwα(n, k)Wα(k, j). (22)

The Dowling numbers, denoted by Dn(α), can be defined as the sum of Whitney numbers
of the second kind Wα(n,m), i.e.

Dn(α) =
n∑

m=0

Wα(n,m). (23)

Using the inverse relation in (19), we have

Wα(n, k) =
n∑
j=0

(−1)nWα(n, j)LWj,k(α)

Thus, the Dowling numbers equal

Dn(α) =
n∑
j=0

(−1)n−j

[
j∑

k=0

(−1)jLWj,k(α)

]
Wα(n, j). (24)

In this paper, we establish more properties of Tauber’s generalized Stirling numbers
as well as some properties of the generalized Bell numbers.

2. Some Recurrence Relations

Tauber’s generalized Stirling numbers of the first and second kind and Tauber’s gener-
alized Lah numbers are known to have triangular recurrence relations. In this section, we
establish other forms of recurrence relations for Tauber’s generalized Stirling numbers of
the first and second kind and Tauber’s generalized Lah numbers. The following theorems
contain the vertical recurrence relation.

Theorem 2.1. Tauber’s generalized Stirling numbers of the first kind Cnm satisfy the fol-
lowing vertical recurrence relation,

Cmn =
n−m+1∑
i=1

i−1∏
j=0

M(n+ 1− j)

N(n− i+ 1)Cm−1
n−i (25)

where
M(n+ 1) = 1,M(n+ 2) = 1.



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1074

Proof. From the triangular recurrence relation (6), that is

Cmn = M(n)Cmn−1 +N(n)Cm−1
n−1 = N(n)Cm−1

n−1 +M(n)Cmn−1.

Since
Cmn−1 = M(n− 1)Cmn−2 +N(n− 1)Cm−1

n−2

then
Cmn = N(n)Cm−1

n−1 +M(n)N(n− 1)Cm−1
n−2 +M(n)M(n− 1)Cmn−2.

Continuing in this manner,

Cmn =N(n)Cm−1
n−1 +M(n)N(n− 1)Cm−1

n−2

+ . . .+M(n) . . .M(m+ 1)N(m)Cm−1
m−1 +M(n) . . .M(m+ 1)M(m)Cmm−1.

By definition, the last term is equal to zero. Thus,

Cmn =
n−m+1∑
i=1

[M(n+ 1) . . .M(n− i+ 2)N(n− i+ 1)]Cm−1
n−i

where
M(n+ 1) = 1,M(n+ 2) = 1.

Theorem 2.2. Tauber’s generalized Stirling numbers of the second kind Dm
n satisfy the

following vertical recurrence relation,

Dm
n =

n−m+1∑
i=1

(−1)i−1 (M(m+ 1))i−1

(N(m+ 1))i
Dm−1
n−i . (26)

Proof. From the triangular recurrence relation (7),

Dm
n = −M(m+ 1)

N(m+ 1)
Dm
n−1 +

1

N(m+ 1)
Dm−1
n−1

Since

Dm
n−1 = −M(m+ 1)

N(m+ 1)
Dm
n−2 +

1

N(m+ 1)
Dm−1
n−2 ,

then

Dm
n =

1

N(m+ 1)
Dm−1
n−1 −

M(m+ 1)

(N(m+ 1))2
Dm−1
n−2 +

(M(m+ 1))2

(N(m+ 1))2
Dm
n−2.

Continuing in this manner,

Dm
n =

1

N(m+ 1)
Dm−1
n−1 −

M(m+ 1)

(N(m+ 1))2
Dm−1
n−2

+ . . .+ (−1)n−m
(M(m+ 1))n−m

(N(m+ 1))n−m+1
Dm−1
m−1



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1075

+ (−1)n−m+1 (M(m+ 1))n−m+1

(N(m+ 1))n−m+1
Dm
m−1.

By definition, the last term is equal to zero. Thus,

Dm
n =

n−m+1∑
i=1

(−1)i−1 (M(m+ 1))i−1

(N(m+ 1))i
Dm−1
n−i .

Theorem 2.3. Tauber’s generalized Lah numbers Lmk,h,n satisfy the following vertical re-
currence relation,

Lmk,h,n =
n−m+1∑
i=1


i−1∏
j=0

[
Mk(n+ 1− j)− Nk(n+ 1− j)Mh(m+ 1)

Nh(m+ 1)

] Nk(n− i+ 1)

Nh(m)
Lm−1
k,h,n−i

where [
Mk(n+ 1)− Nk(n+ 1)Mh(m+ 1)

Nh(m+ 1)

]
= 1.

Proof. From the triangular recurrence relation (9), that is

Lmk,h,n =

[
Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
Lmk,h,n−1 +

Nk(n)

Nh(m)
Lm−1
k,h,n−1

=
Nk(n)

Nh(m)
Lm−1
k,h,n−1 +

[
Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
Lmk,h,n−1.

Since

Lmk,h,n−1 =

[
Mk(n− 1)− Nk(n− 1)Mh(m+ 1)

Nh(m+ 1)

]
Lmk,h,n−2 +

Nk(n− 1)

Nh(m)
Lm−1
k,h,n−2,

then

Lmk,h,n =
Nk(n)

Nh(m)
Lm−1
k,h,n−1 +

Nk(n− 1)

Nh(m)

[
Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
Lm−1
k,h,n−2

+

[
Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

] [
Mk(n− 1)− Nk(n− 1)Mh(m+ 1)

Nh(m+ 1)

]
×
[
Lmk,h,n−2

]
.

Continuing in this manner,

Lmk,h,n =
Nk(n)

Nh(m)
Lm−1
k,h,n−1 +

[
Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
Nk(n− 1)

Nh(m)
Lm−1
k,h,n−2 + . . .+[

Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
. . .

[
Mk(m+ 1)− Nk(m+ 1)Mh(m+ 1)

Nh(m+ 1)

]



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1076

Nk(m)

Nh(m)
Lm−1
k,h,m−1+[

Mk(n)− Nk(n)Mh(m+ 1)

Nh(m+ 1)

]
. . .

[
Mk(m+ 1)− Nk(m+ 1)Mh(m+ 1)

Nh(m+ 1)

]
[
Mk(m)− Nk(m)Mh(m+ 1)

Nh(m+ 1)

]
Lmk,h,m−1.

By definition, the last term is equal to zero. Thus,

Lmk,h,n =
n−m+1∑
i=1

[
Mk(n+ 1)− Nk(n+ 1)Mh(m+ 1)

Nh(m+ 1)

]
. . .

[
Mk(n− i+ 2)− Nk(n− i+ 2)Mh(m+ 1)

Nh(m+ 1)

]
Nk(n− i+ 1)

Nh(m)
Lm−1
k,h,n−i

where [
Mk(n+ 1)− Nk(n+ 1)Mh(m+ 1)

Nh(m+ 1)

]
= 1.

3. Some Explicit Formulas

Consider the generating function ψm(t) for Tauber’s generalized Stirling numbers of
the second kind given by

ψm(t) =
∑
n≥m

Dm
k,nt

n (27)

where ψ0(t) = 1. Using the recurrence relation in (7), we have

ψm(t) =
∑
n≥m

{
−M(m+ 1)

N(m+ 1)
Dm
k,n−1 +

1

N(m+ 1)
Dm−1
k,n−1

}
tn

= −M(m+ 1)

N(m+ 1)
t
∑
n≥m

Dm
k,n−1t

n−1 +
t

N(m+ 1)

∑
n≥m

Dm−1
k,n−1t

n−1

= −M(m+ 1)

N(m+ 1)
tψm(t) +

t

N(m+ 1)
ψm−1(t).

Hence,

ψm(t) =
t

N(m+ 1 +M(m+ 1)t
ψm−1(t).

By backward substitution, we have

ψm(t) =
tm∏m−1

i=0 [N(m+ 1− i) +M(m+ 1− i)t]
.

This result is stated formally in the following theorem.



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1077

Theorem 3.1. The rational generating function for Tauber’s generalized Stirling numbers
of the second kind is given by∑

n≥m
Dm
k,nt

n =
tm∏m−1

i=0 [N(m+ 1− i) +M(m+ 1− i)t]
.

As a consequence of this rational generating function, we have the following explicit
formula in symmetric function form.

Theorem 3.2. Tauber’s generalized Stirling numbers of the second kind are given by

Dm
k,n =

1∏m−1
i=0 N(m+ 1− i)

∑
s1+s2+...+sm=n−m

m∏
i=0

[
−M(m+ 1− i)
N(m+ 1− i)

]si
.

Proof. Using the rational generating function in Theorem 3.1, we get∑
n≥m

Dm
k,nt

n =
tm∏m−1

i=0 [N(m+ 1− i) +M(m+ 1− i)t]

=
tm∏m−1

i=0 N(m+ 1− i)

m−1∏
i=0

1[
1−

(
−M(m+1−i)
N(m+1−i)

)
t
]

=
tm∏m−1

i=0 N(m+ 1− i)

m−1∏
i=0

∑
n≥0

(
−M(m+ 1− i)
N(m+ 1− i)

)n
tn

=
tm∏m−1

i=0 N(m+ 1− i)

∑
n≥m

∑
s1+s2+...+sm=n−m

m∏
i=0

[
−M(m+ 1− i)
N(m+ 1− i)

]si
tsi

=
∑
n≥m

{
tm∏m−1

i=0 N(m+ 1− i)

∑
s1+s2+...+sm=n−m

m∏
i=0

[
−M(m+ 1− i)
N(m+ 1− i)

]si}
tn

Comparing coefficients of tn completes the proof of the theorem.

One of the objectives of this study is to establish a new explicit formula for Tauber’s
generalized Bell numbers, which is analogous to the Qi formula. To derive the desired
formula, we need to establish the following relations.

Theorem 3.3. Tauber’s generalized Stirling numbers of the first and second kind satisfy
the following orthogonality relation for a given sequence of polynomials Qk(x, n),

n∑
m=s

C m
k,nD

s
k,m =

n∑
m=s

D m
k,nC

s
k,m = δsn =

{
0, if n 6= s

1, if n = s

where k = 1, 2.



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1078

Proof. Note that (5) can be written as

xm =

m∑
s=0

D s
k,mQk(x, s). (28)

Substitute (28) into (4), then

Qk(x, n) =

n∑
m=0

C m
k,nx

m =

n∑
m=0

C m
k,n

m∑
s=0

D s
k,mQk(x, s)

=

n∑
m=0

m∑
s=0

C m
k,nD

s
k,mQk(x, s)

=
n∑
s=0

{
n∑

m=s

C m
k,nD

s
k,m

}
Qk(x, s).

Thus,
n∑

m=s

C m
k,nD

s
k,m = δsn =

{
0, if n 6= s

1, if n = s
.

Similarly, (4) can be written as

Qk(x,m) =
m∑
s=0

C s
k,mx

s. (29)

Substitute (29) into (5), then

xn =

n∑
m=0

D m
k,nQk(x,m) =

n∑
m=0

D m
k,n

m∑
s=0

C s
k,mx

s

=
n∑

m=0

m∑
s=0

D m
k,nC

s
k,mx

s

=

n∑
s=0

{
n∑

m=s

D m
k,nC

s
k,m

}
xs.

Thus,
n∑

m=s

D m
k,nC

s
k,m = δsn =

{
0, if n 6= s

1, if n = s
.

Consequently,
n∑

m=s

C m
k,nD

s
k,m =

n∑
m=s

D m
k,nC

s
k,m = δsn. (30)



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1079

Theorem 3.4. The Tauber’s generalized Stirling numbers of the first and second kind are
inverses,

fn =
n∑

m=0

C m
k,ngm ⇐⇒ gn =

n∑
m=0

D m
k,nfm (31)

for k = 1, 2.

Proof.

n∑
m=0

D m
k,nfm =

n∑
m=0

D m
k,n

m∑
s=0

C s
k,mgs

=

n∑
m=0

m∑
s=0

D m
k,nC

s
k,mgs

=
n∑
s=0

{
n∑

m=s

D m
k,nC

s
k,m

}
gs

=

n∑
s=0

δsngs = δnngn = gn

Conversely,

n∑
m=0

C m
k,ngm =

n∑
m=0

C m
k,n

m∑
s=0

D s
k,mfs

=
n∑

m=0

m∑
s=0

C m
k,nD

s
k,mfs

=

n∑
s=0

{
n∑

m=s

C m
k,nD

s
k,m

}
fs

=
n∑
s=0

δsnfs = δnnfn = fn

Theorem 3.5. The Bell numbers Bh,n can be computed in terms of Tauber’s generalized
Lah and Stirling numbers of the second kind. That is,

Bh,n =

n∑
s=0

{
n∑

m=0

Lmk,h,s

}
Ds
k,n. (32)

Proof. From the inverse relation (31) of Tauber’s generalized Stirling numbers of the
first and second kind,

fn =
n∑
s=0

Csk,ngs ⇐⇒ gn =
n∑
s=0

Ds
k,nfs. (33)



R. Corcino, C. Corcino, G. Rama / Eur. J. Pure Appl. Math, 12 (3) (2019), 1069-1081 1080

Consider (10) and let
gs = Dm

h,s

and
fn = Lmk,h,n,

by (33),

Dm
h,n =

n∑
s=0

Ds
k,nL

m
k,h,s. (34)

Substituting (34) to (11) yields

Bh,n =
n∑

m=0

n∑
s=0

Ds
k,nL

m
k,h,s

which implies

Bh,n =
n∑
s=0

{
n∑

m=0

Lmk,h,s

}
Ds
k,n. (35)

Note that (32) is analogous to the Qi formula. For h = 2, and k = 1, the first five
values of B2,n are

B2,0 = 1

B2,1 = 1

B2,2 = 0

B2,3 = −1

B2,4 = 1.

By switching the values of k and h (i.e., k = 2 and h = 1), the explicit formula for
generalized Bell numbers

Bh,n =

n∑
s=0

{
n∑

m=0

Lmk,h,s

}
Ds
k,n

generates the ordinary Bell numbers for n = 1, 2, . . .,

B1,0 = 1

B1,1 = 1

B1,2 = 2

B1,3 = 5

B1,4 = 15.

Remark 3.6. Tauber’s generalized Bell numbers Bh,n are equal to eiDLe, where D and
L are the generalized Stirling and Lah matrices, ei is the i-th unit vector, and e is the
vector with all entries equal to 1.



REFERENCES 1081

Remark 3.6 is equivalent to the following matrix relation[
Di
h,j

]
n×n =

[
Di
k,j

]
n×n

[
Ljk,h,i

]
n×n

. (36)

Now, the orthogonality relation in Theorem 3.3 implies the following matrix relation[
C j
k,i

]
n×n

[
D i
k,j

]
n×n = In, (37)

where In is the identity matrix of order n. That is,[
D i
k,j

]−1

n×n =
[
C j
k,i

]
n×n

,

which implies [
C j
k,i

]
n×n

[
D i
h,j

]
n×n =

[
Ljk,h,i

]
n×n

. (38)

The matrix equation (38) is equivalent to equation (10).

References

[1] Benoumhani, M., On Whitney numbers of Dowling lattices, Discrete Math, 133
(1996), 199–218.

[2] Comtet, L., Advanced Combinatorics, D.Reidel Publishing Company Inc.. Dordrecht,
Holland, 1974.

[3] Daboul, S., Mangaldan, J., Spivey, M., & Taylor, P., The Lah Numbers and the nth

Derivative of e
1
x , Math. Mag. 86(1) (2013), 39–47.

[4] Qi, F., An Explicit Formula for the Bell Numbers in terms of Lah and Stirling Num-
bers, Mediterr. J. Math, 13 (2016), 2795–2800.

[5] Spivey, M., A Generalized Recurrence for Bell Numbers, Journal of Integer Sequences,
11 (2008).

[6] Tauber, S., On Generalised Lah-Numbers, Portland State College, 1964.

[7] Tauber, S., Lah Numbers for R-Polynomials, Portland State College, 1968.

[8] Tauber, S., On Quasi-Orthogonal Numbers, Amer. Math. Monthly, 69 (1962), 365–
372.

[9] Zhang, X-J., Qi, F., Li, & W-H., Properties of Three Functions Relating to
the Exponentiasl Function and the Existence of Partitions of Unity, Int. J.
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