EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 1, 2021, 65-81
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Explicit Formulas for the First Form (q, r)-Dowling
Numbers and (q, r)-Whitney-Lah Numbers

Roberto B. Corcino1,2,∗, Jay M. Ontolan1,2, Maria Rowena S. Lobrigas2

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

Abstract. In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q, r)-Whitney-
Lah number, denoted by Lm,r[n, k]q is defined using the triangular recurrence relation. Several
fundamental properties for the q-analogue are established such as vertical and horizontal recurrence
relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-
Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-
analogue of Newton’s Interpolation Formula. Furthermore, an explicit formula for the first form
(q, r)-Dowling numbers is obtained which is expressed in terms of (q, r)-Whitney-Lah numbers and
(q, r)-Whitney numbers of the second kind.

2020 Mathematics Subject Classifications: 05A15, 11B65, 11B73

Key Words and Phrases: r-Whitney-Lah numbers, r-Whitney numbers, r-Dowling numbers,
Generating function, q-exponential function, q-difference operator, Newton’s Interpolation Formula

1. Introduction

The Lah numbers L(n, k) are the connection constants between the rising factorial
and falling factorial polynomial bases and count partitions of n distinct objects into k
blocks, where objects within a block are ordered (termed Laguerre configurations) [8].
The classical (signless) Lah numbers L(n, k) = n!

k! (
n−1
k−1) may be expressed in terms of the

Stirling numbers s(n, k) and S(n, k) of the first and second kind, respectively:

L(n, k) =
n∑
j=k

s(n, j)S(j, k). (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3900

Email addresses: rcorcino@yahoo.com (R. Corcino),
ontolanj@cnu.edu.ph (J. Ontolan), mariarowena.lobrigas@gmail.com (M. Lobrigas)

http://www.ejpam.com 65 c© 2021 EJPAM All rights reserved.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 66

Cheon and Jung [1] defined the r-Whitney-Lah numbers Lm,r(n, k) in terms of the r-
Whitney numbers of the first kind wm,r(n, k) and second kind Wm,r(n, k)

Lm,r(n, k) =

n∑
j=k

wm,r(n, j)Wm,r(j, k), (2)

which generalizes the identity for the Lah numbers in terms of the Stirling numbers of the
first kind s(n, k) and second kind S(n, k).

The importance of a q-analogue is related to a mathematical expression parameterized
by a quantity q that generalizes a known expression and reduces to the known expression
in the limit, as q approaches 1.

Corcino et al[5] established a q-analogue of Sn,k(a) denoted by σ[n, k]β,rq , given by

σ[n, k]β,rq = σ[n− 1, k − 1]β,rq + ([kβ]q + [r]q)σ[n− 1, k]β,rq

and obtained some properties including vertical and horizontal recurrence relation, hor-
izontal generating function, explicit formula, as well as exponential generating function,
rational generating function, and explicit formula in homogeneous symmetric form. It is
well-known that the Rucinski-Voigt numbers are also known as r-Whitney numbers of the
second kind, denoted by Wm,r(n, k) by Mező [10].

Corcino and Montero [5] defined a q-analogue of r-Whitney numbers of the second
kind which are exactly the same numbers with the Rucinski-Voigt numbers, in a form of
triangular recurrence relation.

On the other hand, Cheon and Jung [1] defined the r-Whitney-Lah numbers, denoted
by Lm,r(n, k), in terms of the r-Whitney numbers of the first kind wm,r(n, k) and the
second kind Wm,r(n, k)

Lm,r(n, k) =

n∑
j=k

wm,r(n, j)Wm,r(j, k),

which generalizes the identity for the Lah numbers in terms of the Stirling numbers of the
first kind and the second kind

(−1)nL(n, k) =
n∑
j=k

(−1)js(n, j)S(j, k).

Other properties for Lm,r(n, k) were established by means of its horizontal generating
function:

〈x+ 2r|m〉n =
n∑
k=0

Lm,r(n, k)(x|m)k,

where

〈x+ 2r|m〉n =

{
(x+ 2r) . . . (x+ 2r + (n− 1)m), n ≥ 1

0, n = 0,



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 67

and the triangular recurrence relation

Lm,r(n, k) = Lm,r(n− 1, k − 1) + (2r + (n+ k − 1)m)Lm,r(n− 1, k),

with Lm,r(n, n) = 1 for n ≥ 0 and Lm,r(n, n) = 0 for n < k, or n, k < 0. We can use the
triangular recurrence relation to generate the first values of Lm,r(n, k).

Cillar and Corcino [2] defined the q-Analogue of r-Whitney-Lah numbers which is
parallel to Cheon and Jung’s definition of r-Whitney-Lah numbers

Lβ,r[n, k]q =
n∑
j=k

φβ,r[n, j]qσ[j, k]β,rq .

where φβ,r[n, j]q and σ[j, k]β,rq is equivalent to wβ,r[n, j]q and Wβ,r[n, j]q, respectively.
They also obtained some combinatorial properties. Their results are as follows:

(i) Horizontal generating function:

〈t+ 2[r]q|[β]q〉n =
n∑
k=0

Lβ,r[n, k]q(t|[β]q)k

where

〈t+ 2[r]q|[β]q〉n =

{
Πn−1
i=0 (t+ 2[r]q + [iβ]q), n > 0,

1, n = 0.

(ii) Recurrence relation:

Lβ,r[n, k]q = Lβ,r[n− 1, k − 1]q + (2[r]q + [kβ]q + [(n− 1)β]q)Lβ,r[n− 1, k]q,

with Lβ,r[0, 0]q = 1 and Lβ,r[n, k]q = 0 for n < k or n, k < 0.

Recently, a q-analogue of r-Whitney numbers of the second kind Wm,r[n, k]q, also
known as (q, r)-Whitney number of the second kind, was introduced in [3, 6] by means of
the following triangular recurrence relation:

Wm,r[n, k]q = qm(k−1)−rWm,r[n− 1, k − 1]q + [mk − r]qWm,r[n− 1, k]q. (3)

From this definition, two more forms of the q-analogue were defined in [3, 6] as

W ∗m,r[n, k]q := q−kr−m(k2)Wm,r[n, k]q (4)

W̃m,r[n, k]q := qkrW ∗m,r[n, k]q = q−m(k2)Wm,r[n, k]q, (5)

where W ∗m,r[n, k]q and W̃m,r[n, k]q denote the second and third forms of the q-analogue,
respectively. Corresponding to the q-analogues in equations (3), (4) and (5), three forms



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 68

of q-analogues for r-Dowling numbers (also known as (q, r)-Dowling numbers) were also
defined as follows:

Dm,r[n]q :=

n∑
k=0

Wm,r[n, k]q (6)

D∗m,r[n]q :=

n∑
k=0

W ∗m,r[n, k]q (7)

D̃m,r[n]q :=

n∑
k=0

W̃m,r[n, k]q. (8)

The focus in this paper is on the first form of the q-analogue. The following results are
obtained for the first form (q, r)-Whitney numbers of the second kind:

(i) Vertical and Horizontal Recurrence Relations

Wm,r[n+ 1, k + 1]q = qmk+r
n∑
j=k

[m(k + 1) + r]n−jq Wm,r[j, k]q (9)

Wm,r[n, k]q =

n−k∑
j=0

(−1)jq−r−m(k+j) rk+j+1,q

rk+1,q
Wm,r[n+ 1, k + j + 1]q (10)

respectively, where

ri,q =

i−1∏
h=1

q−r−mh+m[mh+ r]q

with initial value Wm,r[0, 0]q = 1.

(ii) Horizontal Generating Function

n∑
k=0

Wm,r[n, k]q[t− r|m]k,q = [t]nq . (11)

(iii) Explicit Formula

Wm,r[n, k]q =
1

[k]qm ![m]kq

k∑
j=0

(−1)k−jqm(k−j
2 )
[
k

j

]
qm

[jm+ r]nq . (12)

(iv) Exponential Generating Function

∑
n≥0

Wm,r[n, k]q
[t]nq
[n]q!

=
1

[k]qm ![m]kq
[∆qm,mkeq([x+ jm+ r]q[t]q)]x=0. (13)



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 69

(v) Rational Generating Function

Ψk(t) =
∑
n≥k

Wm,r[n, k]q[t]
n
q =

qm(k2)+kr[t]kq∏k
j=0(1− [mj + r]q[t]q)

. (14)

The purpose of introducing these new q-analogues of r-Whitney and r-Dowling numbers
is to address the conjecture in the paper of Corcino-Corcino [4]. That is, to establish
the Hankel transform of the q-analogue of (r, β)-Stirling numbers, which are exactly the
r-Dowling numbers. In establishing the Hankel transforms of these three forms of (q, r)-
Dowling numbers, the Hankel transform of the third form was the first one being estab-
lished and followed by the second form. For the first form, it is still on the process of
constructing the method that can possibly be used to derive it.

In this paper, (q, r)-Whitney-Lah numbers will be introduced and properties of these
number will be establshed using the approach employed in [6]. The method used in the
paper of Cillar and Corcino [2] will also be used to obtain the properties of the (q, r)-
Whitney-Lah numbers. Moreover, this paper is concluded by deriving an explicit formula
for the first form (q, r)-Dowling numbers expressed in terms of (q, r)-Whitney-Lah numbers
and (q, r)-Whitney numbers of the second kind.

2. (q, r)-Whitney-Lah Numbers and Their Recurrence Relations

The definition of the q-analogue of r-Whitney-Lah numbers, also known as the (q, r)-
Whitney-Lah numbers, is given as follows.

Definition 2.1. The (q, r)-Whitney-Lah numbers Lm,r[n, k]q are defined by

Lm,r[n, k]q = q2r+m(k−1)+m(n−1)Lm,r[n− 1, k− 1]q + [2r+ km+ (n− 1)m]qLm,r[n− 1, k]q,
(15)

with

Lm,r[n, k]q =

{
0 if n < k or n, k < 0,
1 if n = k and n ≥ 0.

and

[t− k]q =
1

qk
([t]q − [k]q).

Remark 2.2. When q → 1, we obtain the triangular recurrence relation of r-Whitney-Lah
numbers, Lm,r(n, k), defined by Cheon and Jung [1]:

Lm,r(n, k) = Lm,r(n− 1, k − 1) + (2r + km+ (n− 1)m)Lm,r(n− 1, k)

with Lm,r(n, k) = 1 for n ≥ 0 and n = k,
and Lm,r(n, k) = 0 for n < k or n, k < 0.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 70

Remark 2.3. It can easily be verified that

Lm,r[n, 0] = [2r + (n− 1)m]nq .

By making use of (15), we can easily obtain the following two other forms of recurrence
relations and generating function.

Theorem 2.4. A q-analogue of r-Whitney-Lah numbers, Lm,r[n, k]q, satisfies the follow-
ing vertical recurrence relations,

Lm,r[n+ 1, k + 1]q =

n∑
j=k

q[2r+mk+m(n−j)]

(
k∏
i=0

[2r + (k + 1)m+ (n− i)m]q

)
Lm,r[j, k]q

with initial value Lm,r[0, 0]q = 1.

Proof. We replace n by n+ 1 and k by k + 1, then using (15), we have

Lm,r[n+ 1, k + 1]q = q2r+mk+mnLm,r[n, k]q + [2r + (k + 1)m+ (n)m]qLm,r[n, k + 1]q.

Using iteration method on (15) we have,

Lm,r[n+ 1, k + 1]q = q2r+mk+mnLm,r[n, k]q + [2r + (k + 1)m+ nm]q

{q2r+mk+m(n−1)Lm,r[n− 1, k]q + [2r + (k + 1)m+ (n− 1)m]qLm,r[n− 1, k + 1]q}
= q2r+mk+mnLm,r[n, k]q + q2r+mk+m(n−1)[2r + (k + 1)m+ nm]qLm,r[n− 1, k]q

+ q2r+mk+m(n−2)[2r + (k + 1)m+ nm]q[2r + (k + 1)m+ (n− 1)m]qLm,r[n− 2, k]q

+ . . .+ q2r+mk+m(k+1){[2r + (k + 1)m+ nm]q[2r + (k + 1)m+ (n− 1)m]q

. . . [2r + (k + 1)m+ (n− k)m]q}Lm,r[k + 1, k + 1]q.

Using the fact that Lm,r[k + 1, k + 1]q = Lm,r[k, k]q we have,

Lm,r[n+ 1, k + 1]q =

n∑
j=k

q[2r+mk+m(n−j)]

(
k∏
i=0

[2r + (k + 1)m+ (n− i)m]q

)
Lm,r[j, k]q.

Theorem 2.5. A q-analogue Lm,r[n, k]q satisfies the horizontal recurrence relation

Lm,r[n, k]q =

n−k∑
j=0

(−1)j q−2r−m(k+j)−nm rk+j+1,q

rk+1,q
Lm,r[n+ 1, k + j + 1]q

where

ri,q =

i−1∏
h=1

q−2r−mh−nm+m[mh+ 2r + nm]q

with initial value Lm,r[0, 0]q = 1.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 71

Proof. Using iteration method on (15) we have,

RHS =
n−k∑
j=0

(−1)jq−2r−m(k+j)−nm rk+j+1,q

rk+1,q

{q2r+m(k+j+1−1)+m(n+1−1)Lm,r[n, k + j]q + [2r + (k + j + 1)m+ nm]qLm,r[n, k + j + 1]q}

=
n−k∑
j=0

(−1)j
rk+j+1,q

rk+1,q
Lm,r[n, k + j]q

+
n−k+1∑
j=1

(−1)j−1q−2r−m(k+j−1)−nm rk+j,q
rk+1,q

[2r + (k + j)m+ nm]qLm,r[n, k + j]q

=
n−k∑
j=0

(−1)j
rk+j+1,q

rk+1,q
Lm,r[n, k + j]q +

n−k∑
j=1

(−1)j−1q−2r−m(k+j−1)−nm

∏k+j−1
h=1 q−2r−mh−nm+m[mh+ 2r + nm]q∏k
h=1 q

−2r−mh−nm+m[mh+ 2r + nm]q
[2r + (k + j)m+ nm]qLm,r[n, k + j]q.

Note that

k+j−1∏
h=1

q−2r−mh−nm+m[mh+ 2r + nm]q = (q−2r−m(1)−nm+m[m(1) + 2r + nm]q)

(q−2r−m(2)−nm+m[m(2) + 2r + nm]q)(q
−2r−m(3)−nm+m[m(3) + 2r + nm]q)

. . . (q−2r−m(k+j−1)−nm+m[m(k + j − 1) + 2r + nm]q).

Now, simplifying the RHS we have,

RHS =
n−k∑
j=0

(−1)j
rk+j+1,q

rk+1,q
Lm,r[n, k + j]q +

1∏k
h=1 q

−2r−mh−nm+m[mh+ 2r + nm]q

=
n−k∑
j=0

(−1)j
rk+j+1,q

rk+1,q
Lm,r[n, k + j]q +

1∏k
h=1 q

−2r−mh−nm+m[mh+ 2r + nm]q

n−k∑
j=1

(−1)j−1{(q−2r−m(1)−nm+m[m(1) + 2r + nm]q)(q
−2r−m(2)−nm+m[m(2) + 2r + nm]q)

(q−2r−m(3)−nm+m[m(3) + 2r + nm]q) . . . (q
−2r−m(k+j−1)−nm+m[m(k + j − 1) + 2r + nm]q)}

q−2r−m(k+j)−nm+m[2r + (k + j)m+ nm]qLm,r[n, k + j]q

=
n−k∑
j=0

(−1)j
rk+j+1,q

rk+1,q
Lm,r[n, k + j]q +

n−k∑
j=1

(−1)j−1
rk+j+1,q

rk+1,q
Lm,r[n, k + j]q

=
rk+1,q

rk+1,q
Lm,r[n, k]q = Lm,r[n, k]q.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 72

3. Explicit Formula and Generating Functions

The ordinary generating function for the sequence (ar) is defined to be the power series

A(x) =
∞∑
r=0

arx
r.

It is a way of uniting the power series in compact form which describes the sequence as
coefficients of the variable xr determined by its power. Generating function is useful in
drawing combinatorial interpretation of a given sequence of numbers as well as in finding
asymptotic expansion of a given number. Motivated by the power series expansion of the
exponential function

e(x) =

∞∑
r=0

xr

r!
,

a generating function that expresses the sequence of numbers (ar) as coefficients of xr

r! ,
say,

E(x) =
∞∑
r=0

ar
xr

r!
,

is called an exponential generating function for (ar). If a number a(n, k) is a function of
two parameters n and k, then the generating function for a(n, k) can be expressed in two
forms: the horizontal and vertical generating functions.

The following generating function is the horizontal generating function for Lm,r[n, k]q
that expresses the values in the nth row of the array of numbers Lm,r[n, k]q (i.e., fixing
the value of n), as coefficients of the variable [t|m]k,q. This is necessary to obtain the
exponential generating function and explicit formula of Lm,r[n, k]q.

Theorem 3.1. A horizontal generating function of Lm,r[n, k]q is given by

n∑
k=0

Lm,r[n, k]q [t|m]k,q = [t+ 2r|m]n,q (16)

where

[t+ 2r|m]n,q =

{ ∏n−1
i=0 [t+ 2r + im]q, if n ≥ 1,
1, if n = 0.

Proof. To prove this, we will use induction. We now verify that (16) holds when n=0.

Lm,r[0, 0]q [t|m]0,q = 1 · 1 = 1 = [t+ 2r|m]0,q.

Suppose that it is true for some n > 0. Then,

n∑
k=0

Lm,r[n, k]q [t|m]k,q = [t+ 2r|m]n,q.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 73

Next, we show that

n+1∑
k=0

Lm,r[n+ 1, k]q [t|m]k,q = [t+ 2r|m]n+1,q.

Using (15) and the fact that. [t|m]k+1,q = [t|m]k,q[t− km]q, we get

n+1∑
k=0

Lm,r[n+1, k]q [t|m]k,q =
n+1∑
k=0

{q2r+m(k−1)+mnLm,r[n, k−1]q+[2r+km+nm]qLm,r[n, k]q}[t|m]k,q

=

n+1∑
k=0

q2r+m(k−1)+mnLm,r[n, k − 1]q[t|m]k,q +

n+1∑
k=0

[2r + km+ nm]qLm,r[n, k]q[t|m]k,q

=

n∑
k=0

q2r+mk+mnLm,r[n, k]q[t|m]k+1,q +

n∑
k=0

[2r + km+ nm]qLm,r[n, k]q[t|m]k,q

=

n∑
k=0

q2r+mk+mnLm,r[n, k]q[t|m]k,q[t− km]q +

n∑
k=0

[2r + km+ nm]qLm,r[n, k]q[t|m]k,q.

Note that [t − k]q = 1
qk

([t]q − [k]q). Since [t − km]q = [t + 2r + nm − 2r − nm − km]q =

[(t+ 2r + nm)− (2r + km+ nm)]q, then

[t− km]q =
1

q2r+mk+mn
([t+ 2r + nm]q − [2r + km+ nm]q).

Thus,

n+1∑
k=0

Lm,r[n+1, k]q [t|m]k,q =

n∑
k=0

q2r+mk+mnLm,r[n, k]q[t|m]k,q

{q−(2r+mk+mn)([t+ 2r+nm]q− [2r+km+nm]q)}+
n∑
k=0

[2r+km+nm]qLm,r[n, k]q[t|m]k,q

=
n∑
k=0

Lm,r[n, k]q[t|m]k,q([t+ 2r + nm]q − [2r + km+ nm]q)

+

n∑
k=0

[2r + km+ nm]qLm,r[n, k]q[t|m]k,q

=

n∑
k=0

Lm,r[n, k]q[t|m]k,q{[t+ 2r + nm]q − [2r + km+ nm]q + [2r + km+ nm]q}

= [t+ 2r|m]nq [t+ 2r + nm]q = [t+ 2r|m]n+1,q.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 74

This proves the theorem.

Next, we establish the explicit formula for Lm,r[n, k]q using the horizontal generating
function of Lm,r[n, k]q. Also, the exponential generating function for Lm,r[n, k]q is obtained
using the explicit formula.

The explicit formula for the q-difference operator is as follows

4h
q,kf(x) =

k∑
j=0

(−1)k−jq(
k−j
2 )[kj ]f(x+ jh). (17)

The new q-analogue of Newton’s Interpolation Formula in [9] is given by

fq(x) = a0 + a1[x− xo]q + . . .+ am[x− xo]q[x− x1]q . . . [x− xm−1]q,

which is equivalent to

fq(x) = fq(x0) +
4qh,hfq(x0)[x− x0]q

[1]qh ![h]q
+
4qh2,hfq(x0)[x− x0]q[x− x1]q

[2]qh ![h]2q

+ . . .+
4qhm,hfq(x0)[x− x0]q[x− x1]q . . . [x− xm−1]q

[m]qh ![h]mq

where xk = x0 + kh, k = 1, 2, . . . such that if x0 = 0 and h = m, we have

fq(x) = fq(x0) +
4qm,mfq(0)[x]q

[1]qm ![m]q
+
4qm2,mfq(0)[x]q[x−m]q

[2]qm ![m]2q

+ . . .+
4qmm,mfq(0)[x]q[x−m]q . . . [x−m(m− 1)]q

[m]qm ![m]mq

By (16) with t = x, we get

n∑
k=0

Lm,r[n, k]q[x|m]k,q = [x+ 2r|m]n,q

which can be rewritten as

n∑
k=0

Lm,r[n, k]q[x]q[x−m]q[x− 2m]q . . . [x− (k − 1)m]q = [x+ 2r|m]n,q.

Suppose fq(x) = [x+ 2r|m]n,q and

Lm,r[n, k]q =
4k
qm,mfq(0)

[k]qm ![m]kq
=

1

[k]qm ![m]kq
· 4k

qm,mfq(0)



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 75

Note that applying the above Newton’s Interpolation Formula and the identity in (17),
we have

4f
qm,mfq(x) =

k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qmfq(x+ jm)

=
k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qm [x+ jm+ 2r|m]n,q.

Evaluate at x = 0,

4f
qm,mfq(0) =

k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qm [2r + jm|m]n,q.

Thus,

Lm,r[n, k]q =
1

[k]qm ![m]kq

k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qm [2r + jm|m]n,q.

So, the explicit formula for Lm,r[n, k]q is as follows:

Theorem 3.2. The explicit formula for Lm,r[n, k]q is given by

Lm,r[n, k]q =
1

[k]qm ![m]kq

k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qm [2r + jm|m]n,q (18)

Remark 3.3. When q → 1, the above theorem reduces to,

Lm,r(n, k) =
1

k!mk

k∑
j=0

(−1)k−j(kj )(2r + jm|m)n,

which is the explicit formula of Lm,r(n, k), where

(t|m)n = t(t+m) . . . (t+ (n− 1)m).

Remark 3.4. The Eq. (18) can also be written as

Lm,r[n, k]q =
1

[k]qm ![m]kq
[4k

qm,m[x+ 2r|m]n,q]x=0

Theorem 3.5. For nonnegative integers n and k, and real number a, the q-analogue
Lm,r[n, k]q has an exponential generating function.∑

n≥o
Lm,r[n, k]q

[t]nq
[n]q!

=
1

[k]qm ![m]kq
[4k

qm,m(F [x+ 2r + jm,m, t])x=0, (19)

where

F [x,m, t] =
∑
n≥0

[x|m]n,q
[t]nq
[n]q!

.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 76

Proof. Using the explicit formula in (18), we obtained

∑
n≥0

Lm,r[n, k]
[t]nq
[n]q!

=
∑
n≥0

1

[k]qm ![m]kq

k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qm [2r + jm|m]n,q

[t]nq
[n]q!

=
k∑
j=0

1

[k]qm ![m]kq
(−1)k−jqm(k−j

2 )[kj ]qm
∑
n≥0

[2r + jm|m]n,q
[t]nq
[n]q!

.

Then, we have

=
1

[k]qm ![m]kq

k∑
j=0

(−1)k−jqm(k−j
2 )[kj ]qmF [2r + jm,m, t]

=
1

[k]qm ![m]kq
[4k

qm,m(F [x+ 2r + jm,m, t])x=0.

Remark 3.6. When q → 1, (19) becomes

∑
n≥0

Lm,r(n, k)
tn

n!
=

1

k!mk

k∑
j=0

(−1)k−j [kj ]F (2r + jm,m, t)

where

F (2r + jm,m, t) =
∑
n≥0

(2r + jm|m)n,q
[t]nq
[n]q!

.

4. An Explicit Formula for (q, r)-Dowling Numbers

One of the common properties of Lah-type numbers is their relation with both kinds of
Stirling-type or Whitney-type numbers. Analogous to this, it is also interesting to express
(q, r)-Whitney-Lah numbers in terms of (q, r)-Whitney numbers. To do this, we need to
define the (q, r)-Whitney numbers of the first kind.

Definition 4.1. The (q, r)-Whitney numbers of the first kind wm,r[n, k]q are defined as
coefficients of the following generating function

[t− r|m]n,q =

n∑
k=0

wm,r[n, k]q[t]
k
q . (20)

Note that

n+1∑
k=0

wm,r[n+ 1, k]q[t]
k
q = [t− r|m]n+1,q = [t− r|m]n,q[t− r − nm]q



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 77

=
1

qr+nm
([t]q − [r + nm]q)

n∑
k=0

wm,r[n, k]q[t]
k
q

=
1

qr+nm

(
n∑
k=0

wm,r[n, k]q[t]
k+1
q −

n∑
k=0

[r + nm]qwm,r[n, k]q[t]
k
q

)

=
1

qr+nm

(
n+1∑
k=0

wm,r[n, k − 1]q[t]
k
q −

n+1∑
k=0

[r + nm]qwm,r[n, k]q[t]
k
q

)

=
n+1∑
k=0

1

qr+nm
(wm,r[n, k − 1]q − [r + nm]qwm,r[n, k]q) [t]kq .

Comparing the coefficients of [t]kq completes the proof the following theorem.

Theorem 4.2. The (q, r)-Whitney numbers of the first kind wm,r[n, k]q satisfy the follow-
ing triangular recurrence relation:

wm,r[n+ 1, k]q[t]
k
q =

1

qr+nm
(wm,r[n, k − 1]q − [r + nm]qwm,r[n, k]q) (21)

with initial condition wm,r[0, 0]q = 1, wm,r[n, 0]q = (−1)nq−r−(n−1)m[r + (n − 1)m]q and
wm,r[n, k]q = 0 if n < k.

n/k 0 1 2

0 1 0 0

1 −q−r[r]q q−r 0

2 q−(r+m)[r +m]q −q−(2r+m)([r]q + [r +m]q) q−(2r+m)

The next theorem contains the orthogonality relation of (q, r)-Whitney numbers of the
first and second kinds.

Theorem 4.3. The following orthogonality relation holds

n∑
k=j

wm,r[n, k]qWm,r[k, j]q =

n∑
k=j

Wm,r[n, k]qwm,r[k, j]q = δnj =

{
0, if j 6= n

1, if j = n
(22)

Proof. Applying equation (11) to (20) yields

[t− r|m]n,q =

n∑
k=0

wm,r[n, k]q[t]
k
q =

n∑
k=0

wm,r[n, k]q

k∑
j=0

Wm,r[k, j]q[t− r|m]j,q

=

n∑
j=0

n∑
k=j

wm,r[n, k]qWm,r[k, j]q[t− r|m]j,q =

n∑
j=0

 n∑
k=j

wm,r[n, k]qWm,r[k, j]q

 [t− r|m]j,q.



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 78

By comparing coefficients, we get

n∑
k=j

wm,r[n, k]qWm,r[k, j]q =

{
0, if j 6= n

1, if j = n.
(23)

Similarly, we have

[t]nq =
n∑
k=0

Wm,r[n, k]q[t− r|m]k,q =
n∑
k=0

Wm,r[n, k]q

k∑
j=0

wm,r[k, j]qj[t]
n
q

=

n∑
j=0

n∑
k=j

Wm,r[n, k]qwm,r[k, j]q[t]
j
q =

n∑
j=0

 n∑
k=j

Wm,r[n, k]qwm,r[k, j]q

 [t]jq.

Hence,
n∑
k=j

Wm,r[n, k]qwm,r[k, j]q =

{
0, if j 6= n

1, if j = n.
(24)

Using the orthogonality relations in (23) and (24), we can easily prove the following
inverse relation.

Theorem 4.4. The following inverse relations hold:

fn =
n∑
k=0

wm,r[n, k]qgk ⇔ gn =
n∑
k=0

Wm,r[n, k]qfk (25)

fk =
∞∑
n=0

wm,r[n, k]qgn ⇔ gk =
∞∑
n=0

Wm,r[n, k]qfk. (26)

Remark 4.5. Using the inverse relation in (26) and the exponential generating function
in (13), we can easily obtain the following identity∑

n≥0

[k]q!wm,r[n, k]q[∆qm,mneq([x+ jm+ r]q[t]q)]x=0

[t]kq [n]qm ![m]nq
= 1. (27)

Also, using the inverse relation in (26) and the rational generating function in (14), we
obtain ∑

n≥k

wm,r[n, k]qq
m(n2)+nr[t]n−kq∏n

j=0(1− [mj + r]q[t]q)
= 1. (28)

Thus, combining (27) and (28) yields

∑
n≥0

wm,r[n, k]q

{
[k]q![∆qm,mneq([x+ jm+ r]q[t]q)]x=0

[n]qm ![m]nq
−

qm(n2)+nr[t]nq∏n
j=0(1− [mj + r]q[t]q)

}
= 0.

(29)



R. Corcino, J. Ontolan, M. Lobrigas / Eur. J. Pure Appl. Math, 14 (1) (2021), 65-81 79

Now, we can establish the relation between (q, r)-Whitney-Lah numbers and both
kinds of (q, r)-Whitney numbers.

Theorem 4.6. The (q, r)-Whitney-Lah numbers satisfy the following relation

Lm,r[n, j]q =
n∑
k=j

wm,−r[n, k]qWm,r[k, j]q. (30)

Proof. Using the horizontal generating functions for (q, r)-Whitney numbers and (q, r)-
Whitney numbers, we have

n∑
k=0

Lm,r[n, k]q[t− r|m]k,q = [t+ r|m]n,q =
n∑
k=0

wm,−r[n, k]q[t]
k
q

=
n∑
k=0

wm,−r[n, k]q

k∑
j=0

Wm,r[k, j]q[t− r|m]j,q

=

n∑
j=0

n∑
k=j

wm,−r[n, k]qWm,r[k, j]q[t− r|m]j,q

n∑
j=0

Lm,r[n, j]q[t|m]j,q =
n∑
j=0

 n∑
k=j

wm,−r[n, k]qWm,r[k, j]q

 [t− r|m]j,q.

Comparing the coefficients of [t|m]j,q completes the proof.

The following theorem contains the main result of this paper.

Theorem 4.7. The explicit formula for the first form (q, r)-Dowling numbers Dm,r[n]q is
given by

Dm,r[n]q =
n∑
k=0

Wm,−r[n, k]q

 k∑
j=0

Lm,r[k, j]q

 ,

Proof. Using the inverse relation (25) with

fn = Lm,r[n, j]q, gk = Wm,r[k, j]q,

relation (30) can be transformed as

Wm,r[n, j]q =

n∑
k=j

Wm,−r[n, k]qLm,r[k, j]q. (31)

Then summing up both sides over j yields

Dm,r[n]q =

n∑
j=0

Wm,r[n, j]q =

n∑
j=0

n∑
k=j

Wm,−r[n, k]qLm,r[k, j]q



REFERENCES 80

=
n∑
k=0

Wm,−r[n, k]q

 k∑
j=0

Lm,r[k, j]q

 ,

which is exactly the desired explicit formula for (q, r)-Dowling numbers.

The above explicit formula may also be called a Qi-type formula for (q, r)-Dowling
numbers, which is analogous to explicit formula obtained F. Qi [11].

5. Conclusion

The explicit formula for the first form (q, r)-Dowling numbers has already been derived
by establishing an appropriate (q, r)-Whitney-Lah numbers and (q, r)-Whitney numbers
of the first kind. To establish explicit formulas for the two other forms of (q, r)-Dowling
numbers, it entails defining appropriate versions of (q, r)-Whitney-Lah numbers and (q, r)-
Whitney numbers of the first kind. It is left to the readers to derive these explicit formulas.

This paper is also related to [7, 12, 13]. With these, more properties and relations can
further be established.

Acknowledgements

This research has been funded by Cebu Normal University (CNU) and the Commission
on Higher Education - Grants-in-Aid for Research (CHED-GIA).

References

[1] G.S. Cheon and J. H. Jung. r-whitney number of dowling lattices. Discrete Math.,
312:2337–2348, 2012.

[2] JA. Cillar and R. Corcino. A q-analogue of qi formula for r-dowling numbers. Com-
mun. Korean Math. Soc., 35(1):21–31, 2020.

[3] R. Corcino, MJ. Latayada, and MAR. Vega. Hankel transform of (q, r)-dowling num-
bers. Eur. J. Pure Appl. Math, 12(2):279–293, 2019.

[4] R.B. Corcino and C.B. Corcino. The hankel transform of generalized bell numbers
and its q-analogue. Util. Math., 89:297–309, 2012.

[5] R.B. Corcino and C.B. Montero. A q-analogue of rucinski-voigt numbers. ISRN
Discrete Math., 2012:18 Pages, 2012.

[6] R.B. Corcino, J.M. Ontolan, J. Cañete, and M.R. Latayad. A q-analogue of r-whitney
numbers of the second kind and its hankel transform. J. Math. Computer Sci., 21:258–
272, 2020.



REFERENCES 81

[7] U. Duran, M. Acikgoz, and S. Araci. On (q, r, w)-stirling numbers of the second kind.
J. Inequal. Spec. Funct., 9(1):9–16, 2018.

[8] Lindsay J. et. al. A new combinatorial interpretation of a q-analogue of the lah
numbers. J. Combi., 2(2):245–264, 2011.

[9] M. Kim and J. Son. A note on q-difference operators. Commun. Korean Math. Soc.,
17(3):423–430, 2002.

[10] I. Mező. A new formula for the bernoulli polynomials. Result. Math, 58:330–333,
2010.

[11] F. Qi. An explicit formula for the bell numbers in terms of lah and stirling numbers.
Mediterr. J. Math., 13(5):2795–2800, 2016.

[12] Srivastava, HM., M. Riyasat, S. Khan, S. Araci, and M. Acikgoz. A new approach
to legendre-truncated-exponential-based sheffer sequences via riordan arrays. Appl.
Math. Comput., 369:124683, 2020.

[13] H.M. Srivastava, S. Khan, S. Araci, M. Acikgoz, and M. Riyasat. A general class
of the three-variable unified apostol-type q-polynomials and multiple power q-sums.
Bull. Iran. Math. Soc., 46:519–542, 2020.