EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 934-943
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Divisibility Property of Type 2 (p, q)-Analogue of
r-Whitney Numbers of the Second Kind

Roberto B. Corcino1,2,∗, Cristina B. Corcino1,2

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, the divisibility property of the type 2 (p, q)-analogue of the r-Whitney
numbers of the second kind is established. More precisely, a congruence relation modulo pq for
this (p, q)-analogue is derived.

2020 Mathematics Subject Classifications: 05A19, 05A30, 11B65

Key Words and Phrases: r-Whitney numbers, r-Dowling numbers, Stirling numbers, Bell
numbers, congruence relation, divisibility property, binomial transform, Hankel tranform

1. Introduction

The r-Whitney numbers of the second kind were introduced by Mezo [18] as coefficients
of the following generating function:

(mx+ r)n =

n∑
k=0

mkWm,r(n, k)x
k,

where xk = x(x− 1) . . . (x− k + 1). These numbers satisfy the following properties:

1. the exponential generating function

∞∑
n=0

Wm,r(n, k)
zn

n!
=

erz

k!

(
emz − 1

m

)k

,

2. the explicit formula

Wm,r(n, k) =
1

mkk!

k∑
i=0

(
k

i

)
(−1)k−i(mi+ r)n,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4715

Email addresses: rcorcino@yahoo.com (R. B. Corcino), corcinoc@cnu.edu.ph (C. C. Corcino)

https://www.ejpam.com 934 © 2023 EJPAM All rights reserved.



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 935

3. the triangular recurrence relation

Wm,r(n, k) = Wm,r(n− 1, k − 1) + (km+ r)Wm,r(n− 1, k).

These properties are exactly the same properties that the (r, β)-Stirling numbers in [7]
have possessed. This implies that the r-Whitney numbers of the second kind and the
(r, β)-Stirling numbers are equivalent. More properties of these numbers can be found in
[2, 4, 5, 7, 18].

One of the early studies on q-analogue of Stirling numbers of the second kind was
introduced by Carlitz in [1] in connection with a problem in abelian groups. This is
known as q-Stirling numbers of the second kind and is defined in terms of the following
recurrence relation

Sq[n, k] = Sq[n− 1, k − 1] + [k]qSq[n− 1, k], [k]q =
1− qk

1− q

such that, when q → 1, this gives the triangular recurrence relation for the classical Stirling
numbers of the second kind S(n, k)

S(n, k) = S(n− 1, k − 1) + kS(n− 1, k).

Another version of definition of this q-analogue was adapted in [17] as follows

Sq[n, k] = qk−1Sq[n− 1, k − 1] + [k]qSq[n− 1, k]. (1)

Through this definition, the Hankel transform of q-exponential polynomials and numbers
was successfully established, which may be considered as the Hankel transform of a certain
q-analogue of Bell polynomials and numbers.

There are many ways to define q-analogue of Stirling-type and Bell-type numbers (see
[6, 8–10, 12, 14]). However, in the desire to establish the Hankel transform of q-analogue
of generalized Bell numbers, Corcino et al. [11] were motivated to define a q-analogue of
r-Whitney numbers of the second kind parallel to that in (1) as follows:

Wm,r[n, k]q = qm(k−1)−rWm,r[n− 1, k − 1]q + [mk − r]qWm,r[n− 1, k]q. (2)

Two more forms of this q-analogue, denoted by W ∗
m,r[n, k]q and W̃m,r[n, k]q, were respec-

tively defined by

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

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

The corresponding q-analogues of generalized Bell numbers, also known as q-analogues of
r-Dowling numbers, were also defined in three forms as (see [3, 11, 13, 15])

Dm,r[n]q :=

n∑
k=0

Wm,r[n, k]q,



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 936

D∗
m,r[n]q :=

n∑
k=0

W ∗
m,r[n, k]q,

and

D̃m,r[n]q :=

n∑
k=0

W̃m,r[n, k]q.

where Dm,r[n]q, D
∗
m,r[n]q and D̃m,r[n]q denote the first, second and third form of the q-

analogues of r-Dowling numbers, respectively. The Hankel transforms ofDm,r[n]q, D
∗
m,r[n]q

and D̃m,r[n]q were successfully established in [3, 11, 15].

To extend these research studies, a certain (p, q)-analogue of r-Whitney numbers of the
second kind, denoted by Wm,r[n, k]p,q, was defined in [16] as coefficients of the following
generating function:

[mt+ r]np,q =
n∑

k=0

Wm,r[n, k]p,q[mt|m]kp,q (3)

where

[t|m]np,q =
n−1∏
j=0

[t− jm]p,q. (4)

The orthogonality and inverse relations, an explicit formula, and a kind of exponential
generating function of Wm,r[n, k]p,q were already obtained. Unfortunately, its Hankel
transform was not successfully established using the method applied in [3, 11, 15]. This
motivated Corcino et al. [19] to define the type 2 (p, q)-analogue of r-Whitney numbers
of the second kind, denoted by Wm,r[n, k; t]p,q, as follows:

Wm,r[n+1, k; t]p,q = qm(k−1)+rWm,r[n, k− 1; t]p,q + [mk+ r]p,qp
mt−kmWm,r[n, k; t]p,q. (5)

The second form was then defined as follows:

W ∗
m,r[n, k; t]p,q := q−kr−m(k2)Wm,r[n, k; t]p,q. (6)

Several properties of these (p, q)-analogues were established in [19] including their Hankel
transforms, which are given by

det (Wm,r[s+ i+ j, s+ j; t]p,q)0≤i,j≤n =

n∏
k=0

qm(
s+k
2 )+(s+k)rpnmt[m(s+ k) + r]kp,q

det(W ∗
m,r[s+ i+ j, s+ j; t]p,q)0≤i,j≤n =

n∏
k=0

pnmt[m(s+ k) + r]kp,q.

On the other hand, the first, second and third forms of type 2 (p, q)-analogue of the
r-Dowling numbers, denoted by Dm,r[n]p,q, D

∗
m,r[n]p,q and D̃m,r[n]p,q were defined respec-

tively in [19] as follows:

Dm,r[n]p,q :=

n∑
k=0

Wm,r[n, k; t]p,q,



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 937

D∗
m,r[n]p,q :=

n∑
k=0

W ∗
m,r[n, k; t]p,q,

D̃m,r[n]p,q :=
n∑

k=0

W̃m,r[n, k; t]p,q,

where
W̃m,r[n, k; t]p,q = qkrW ∗

m,r[n, k; t]p,q (7)

denotes the third form of the (p, q)-analogue of the r-Whitney numbers of the second kind.
Among these three forms, only the second form was provided a Hankel transform, which
is given by

H(D∗
m,r[n]p,q) =

(
q

p

)n(n2+3n+8)
6

+r−1(n2)
([m] q

p
)(

n
2)

n−1∏
k=0

[k]( q
p

)m !.

The main objective of this study is to establish additional property of the type 2 (p, q)-
analogues of the r-Whitney numbers of the second kind. More precisely, the divisibility
property of these type 2 (p, q)-analogues will be discussed thoroughly.

2. Preliminary Results

This section provides a brief discussion on some relations that are necessary in deriving
the divisibility property of the type 2 (p, q)-analogue of the r-Whitney numbers of the
second kind W ∗

m,r[n, k; t]p,q.

Multiplying both sides of the recurrence relation in (5) by q−kr−m(k2) yields

q−kr−m(k2)Wm,r[n+ 1, k; t]p,q = q−kr−m(k2)qm(k−1)+rWm,r[n, k − 1; t]p,q

+ q−kr−m(k2)[mk + r]p,qp
mt−kmWm,r[n, k; t]p,q

q−kr−m(k2)Wm,r[n+ 1, k; t]p,q = q−(k−1)r−m(k−1
2 )Wm,r[n, k − 1; t]p,q

+ [mk + r]p,qp
mt−kmq−kr−m(k2)Wm,r[n, k; t]p,q.

Applying (6) consequently gives

W ∗
m,r[n+ 1, k; t]p,q = W ∗

m,r[n, k − 1; t]p,q + [mk + r]p,qp
mt−kmW ∗

m,r[n, k; t]p,q. (8)

This relation can be used to generate the following first few values of W ∗
m,r[n, k; t]p,q:

By repeated application of (8), we can easily derive the following vertical recurrence
relation.

Theorem 2.1. For nonnegative integers n and k, and real number r, the (p, q)-analogue
of r-Whitney numbers of the second kind satisfies the following vertical recurrence relation

W ∗
m,r[n+ 1, k + 1; t]p,q =

n∑
j=k

[m(k + 1) + r]n−j
p,q p(n−j)[mt−(k+1)m]W ∗

m,r[j, k; t]p,q. (9)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 938

n/k 0 1 2 3

0 1

1 [r]p,qp
mt 1

2 [r]2p,qp
2mt [r]p,qp

mt + [m+ r]p,qp
m(t−1) 1

3 [r]3p,qp
3mt [r]2p,qp

2mt + [r]p,q[m+ r]p,qp
m(2t−1) [r]p,qp

mt + 2[m+ r]p,qp
m(t−1) 1

+[m+ r]2p,qp
2m(t−1)

Table 1: The First Values of W ∗
m,r[n, k; t]p,q

One can easily verify relation (9) using the values of W ∗
m,r[n, k; t]p,q in Table 1.

Now, let us derive the rational generating function for W ∗
m,r[n, k; t]p,q. Suppose that

Ψ∗
k(x) =

∞∑
n=k

W ∗
m,r[n, k; t]p,qx

n−k.

When k = 0, (8) reduces to

W ∗
m,r[n+ 1, 0; t]p,q = [r]p,qp

mtW ∗
m,r[n, 0; t]p,q.

By repeated application of (8), this inductively gives

W ∗
m,r[n+ 1, 0; t]p,q = [r]p,qp

mtW ∗
m,r[n, 0; t]p,q =

(
[r]p,qp

mt
)2

W ∗
m,r[n− 1, 0; t]p,q

...

=
(
[r]p,qp

mt
)n+1

W ∗
m,r[0, 0; t]p,q =

(
[r]p,qp

mt
)n+1

.

Hence,

Ψ∗
0(x) =

∞∑
n=0

W ∗
m,r[n, 0; t]p,qx

n =
1

(1− xpmt[r]p,q)
.

When k > 0 and applying the triangular recurrence relation in (5), we have

Ψ∗
k(x) =

∞∑
n=k

W ∗
m,r[n, k; t]p,qx

n−k

=
∞∑

n−1=k−1

W ∗
m,r[n− 1, k − 1; t]p,qx

(n−1)(k−1)

+ xpmt−km[mk + r]p,q

∞∑
n−1=k

W ∗
m,r[n− 1, k; t]p,qx

n−1−k

= Ψ∗
k−1(x) + xpm(t−k)[mk + r]p,qΨ

∗
k(x)

Solving for Ψ∗
k(t) yields

Ψ∗
k(x) =

1

1− xpm(t−k)[mk + r]p,q
Ψ∗

k−1(x).

Applying backward substitution gives the following rational generating function forWm,r[n, k; t]p,q.



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 939

Theorem 2.2. For nonnegative integers n and k, and real number r, the (p, q)-analogue
Wm,r[n, k; t]p,q satisfies the following rational generating function

Ψ∗
k(x) =

∞∑
n=k

W ∗
m,r[n, k; t]p,qx

n−k =
1∏k

j=0(1− xpm(t−j)[mj + r]p,q)
. (10)

Remark 2.3. This rational generating function plays an important role in proving the
main result of the paper.

3. Divisibility Property

In this section, the congruence relation modulo pq for the type 2 (p, q)-analogue of
the r-Whitney numbers of the second kind W ∗

m,r[n, k; t]p,q will be established using the
rational generating function in (10).

Using the values of W ∗
m,r[n, k; t]p,q in Table 1, we observe that, with

[t]p,q = pt−1 + pt−2q + pt−3q2 + . . .+ pqt−2 + qt−1,

the polynomial expressions of W ∗
m,r[n, k]q from row 0 to row 3, if they are divided by pq,

the remainders form the following triangle of expressions in p:

1
pmt+r−1 1

p2(mt+r−1) 2pmt+r−1 1
p3(mt+r−1) 3p2(mt+r−1) 3pmt+r−1 1.

This can further be written as (
0
0

)(
1
0

)
pmt+r−1

(
1
1

)(
2
0

)
p2(mt+r−1)

(
2
1

)
pmt+r−1

(
2
2

)(
3
0

)
p3(mt+r−1)

(
3
1

)
p2(mt+r−1)

(
3
2

)
pmt+r−1

(
3
3

)
,

To generalize this observation, the next theorem contains the divisibility property of
W ∗

m,r[n, k; t]p,q.

Theorem 3.1. For nonnegative integers n and k, the type 2 (p, q)-analogue of the r-
Whitney numbers of the second kind Wm,r[n, k; t]p,q satisfies the following congruence re-
lation

W ∗
m,r[n, k; t]p,q ≡

(
n

k

)
p(n−k)(mt+r−1) mod pq. (11)

Proof. The polynomial [t]p,q can be written as

[t]p,q = pt−1 + qt−1 + pqy,



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 940

where y is a polynomial in p and q. Then, we have

1∏k
j=0(1− xpm(t−j)[mj + r]p,q)

=

∞∑
n=0

(
xpm(t−j)[mj + r]p,q

)n
=

∞∑
n=0

pnm(t−j)(pmj+r−1 + qmj+r−1 + pqy)nxn

=
∞∑
n=0

pn(mt+r−j)xn + pq
∞∑
n=0

ẑnx
n,

where ẑn is a polynomial in p and q. It follows that

1∏k
j=0(1− xpm(t−j)[mj + r]p,q)

=
∞∑
n=0

pn(mt+r−j)xn mod pq

=
1

1− pmt+r−1x
mod pq.

Thus, using (10), we have

∞∑
n=k

W ∗
m,r[n, k; t]p,qx

n−k ≡ 1

(1− pmt+r−1x)k+1
mod pq

≡
∞∑
n=0

(
n+ (k + 1)− 1

n

)
pn(mt+r−1)xn mod pq

≡
∞∑
n=k

(
n

k

)
p(n−k)(mt+r−1)xn−k mod pq.

Comparing the coefficients of xn−k completes the proof of the theorem.

Remark 3.2. Using (6) and Theorem 3.1, the first form of the type 2 (p, q)-analogues
of the r-Whitney numbers of the second kind satisfies the following congruence relation
modulo pq:

Wm,r[n, k; t]p,q ≡
(
n

k

)
p(n−k)(mt+r−1)qkr+m(k2) mod pq (12)

≡

{
qnr+m(n2) mod pq, for n = k

0 mod pq, otherwise.

Moreover, using (7) and Theorem 3.1, the third form of the type 2 (p, q)-analogues of the
r-Whitney numbers of the second kind satisfies the following congruence relation modulo
pq:

W̃m,r[n, k; t]p,q ≡
(
n

k

)
p(n−k)(mt+r−1)qkr mod pq (13)



R. B. Corcino, C. B. Corcino / Eur. J. Pure Appl. Math, 16 (2) (2023), 934-943 941

≡

{
qnr mod pq, for n = k

0 mod pq, otherwise.

Remark 3.3. When p = 1, the congruence relation in (11) reduces to

W ∗
m,r[n, k]q = W ∗

m,r[n, k; t]1,q ≡
(
n

k

)
mod q,

which is exactly the congruence relation in [15, Theorem 2.1] for the second form of (q, r)-
Whitney numbers of the second kind. Moreover, the congruence relations in (12) and (13)
reduce to

Wm,r[n, k]q = Wm,r[n, k; t]1,q ≡
(
n

k

)
qkr+m(k2) ≡ 0 mod q

W̃m,r[n, k]q = W̃m,r[n, k; t]1,q ≡
(
n

k

)
qkr ≡ 0 mod q,

which are the congruence relations for the first and third forms of (q, r)-Whitney numbers
of the second kind. We recall that, for a prime p, the p-adic valuation νp(n) of n is defined
to be the largest exponent k such that pk|n. Moreover, the p-adic valuation of the rational
number n

m is defined by

νp

( n

m

)
= νp(n)− νp(m).

Furthermore, the p-adic absolute value |n|p of n is defined by

|n|p =
1

pνp(n)
.

Clearly, when q is prime,

νq (Wm,r[n, k]q) = kr +m

(
k

2

)
νq

(
W̃m,r[n, k]q

)
= kr.

Consequently,

νq

(
Wm,r[n, k]q

W̃m,r[n, k]q

)
= νq (Wm,r[n, k]q)− νq

(
W̃m,r[n, k]q

)
= m

(
k

2

)
.

Also, one can easily see that

νq

(
W ∗

m,r[n, k]q −
(
n

k

))
= qνp(W

∗
m,r[n,k]q−(

n
k))
∣∣∣∣W ∗

m,r[n, k]q −
(
n

k

)∣∣∣∣
q

= 1.



REFERENCES 942

Acknowledgements

The authors are grateful to Cebu Normal University (CNU) for partially funding
this research project through its Research Institute for Computational Mathematics and
Physics (RICMP). They are also grateful to the two referees for their valuable time in
reviewing the paper.

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