EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1676-1688
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Hankel Transform of the Second Form (q, r)-Dowling
Numbers

Roberto B. Corcino1,∗, Jay M. Ontolan1, Gladys Jane S. Rama1

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

Abstract. In this paper, using the rational generating for the second form of the q-analogue of
r-Whitney numbers of the second kind, certain divisibility property for this form is established.
Moreover, the Hankel transform for the second form of the q-analogue of r-Dowling numbers is
derived.

2010 Mathematics Subject Classifications: 05A15, 11B65, 11B73

Key Words and Phrases: r-Whitney numbers, r-Dowling numbers, generating function, q-
analogue, q-exponential function, A-tableau, convolution formula, Hankel transform, Hankel ma-
trix, k-binomial transform

1. Introduction

The matrix of the form 
a0 a1 a2 . . . an
a1 a2 a3 . . . an+1

a2 a3 a4 . . . an+2

. . . . . . . . . . . . . . . . . . . . . . . . . . .
an an+1 an+2 . . . a2n

 (1)

whose entries are the elements of the sequence A = (an)∞n=0 was defined in [16] as the
Hankel matrix of order n of a sequence A, denoted by Hn. This can also be written as
Hn = (ai+j)0≤i,j≤n. In the same paper [16], the Hankel determinant hn of order of n
of A was defined as the determinant of the corresponding Hankel matrix of order n, (i.e.
hn = det(Hn)) and the Hankel transform of the sequence A, denoted by H(A), was defined
as the sequence {hn} of Hankel determinants of A.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3583

Email addresses: rcorcino@yahoo.com (R. Corcino),
ontolanjay@gmail.com (J. Ontolan), gjsrama@yahoo.com (G. J. Rama)

http://www.ejpam.com 1676 c© 2019 EJPAM All rights reserved.



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1677

For example, the sequence of (r, β)-Bell numbers in [12, 15], denoted by {Gn,r,β}, has
possessed the following Hankel transform (see [14])

H(Gn,r,β) =

n∏
j=0

βjj!.

As mentioned in [16], one can easily verify that the (r, β)-Bell numbers are simply the
r-Dowling numbers Dm,r(n), which are defined in [5] as

Dm,r(n) =

n∑
k=0

Wm,r(n, k)

where Wm,r(n, k) denotes the r-Whitney numbers of the second kind introduced by Mezo
in [29]. In [14], the authors have also tried to derive the Hankel transform of the sequence
of q-analogue of (r, β)-Bell numbers. In this attempt, they used the q-analogue defined in
[17]. But they failed to derive it.

Just recently, another definition of q-analogue of r-Whitney numbers of the second
Wm,r[n, k]q was introduced in [13, 16] 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. (2)

From this definition, two more forms of the q-analogue were defined in [13, 16] as

W ∗m,r[n, k]q := q−kr−m(k2)Wm,r[n, k]q (3)

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

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 these, three forms of q-analogues for r-Dowling numbers
may be defined as follows:

Dm,r[n]q :=
n∑
k=0

Wm,r[n, k]q (5)

D∗m,r[n]q :=
n∑
k=0

W ∗m,r[n, k]q (6)

D̃m,r[n]q :=
n∑
k=0

W̃m,r[n, k]q. (7)

However, among these three forms, only the third form was considered in [16] and was
given the Hankel transform as follows

H(D̃m,r[n]q) = qm(n+1
3 )−rn(n+1)[0]qm ![1]qm ! . . . [n]qm ![m]

(n+1
2 )

q . (8)



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1678

This Hankel transform was derived using the Hankel transform of q-exponential polyno-
mials in [20], the Layman’s Theorem in [26] and the Spivey-Steil Theorem in [34]. This
method cannot be used to derive the Hankel transform of the first and second forms of
q-analogues for r-Dowling numbers. But the method used by Cigler in [8] is found to be
useful to derive the Hankel transforms for the second form of the q-analogue of r-Dowling
numbers.

In this paper, the Hankel transform for the sequence
(
D∗m,r[n]q

)∞
n=0

will be estab-
lished using Cigler’s method [8]. However, a more general form of D∗m,r[n]q, denoted by
ϕn[x, r,m]q, is considered, which is defined in polynomial form as follows:

ϕn[x, r,m]q =
n∑
k=0

W ∗m,r[n, k][x]nq , (9)

such that, when x = 1, ϕn[1, r,m]q = D∗m,r[n]q.

2. A q-Analogue of Wm,r(n, k): Second Form

The second form of q-analogue of Wm,r(n, k) is a kind of generalization of the q-
analogue considered by Cigler [8]. This q-analogue possessed several properties (see [13])
including certain combinatorial interpretation in terms of A-tableau, which is defined in
[27] to be a list φ of column c of a Ferrer’s diagram of a partition λ(by decreasing order of
length) such that the lengths |c| are part of the sequence A = (ri)i≥0, a strictly increasing
sequence of nonnegative integers. By making use of the following explicit formula in
symmetric function form [13]

Wm,r[n, k]q = qm(k2)+kr
∑

S1+S2+···Sk=n−k

k∏
j=1

[mj + r]
Sj
q

=
∑

0≤j1≤j2≤···jn−k≤k
qm(k2)+kr

n−k∏
i=1

[mji + r]q, (10)

we have

W ∗m,r[n, k]q =
∑

0≤j1≤j2≤···≤jn−k≤k

n−k∏
i=1

[mji + r]q. (11)

In [16], W ∗m,r[n, k] was expressed as

W ∗m,r[n, k] =
∑

φ∈TA
r (k,n−k)

∏
c∈φ

ω(|c|)

where TAr (h, l) denotes the set of A-tableau with l columns of lengths |c| ≤ h and ω(|c|) =
[m|c|+r]q. Using the combinatorics of A-tableau, the following identities were established



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1679

in [16]:

W ∗m,r[n, k]q =

n∑
j=k

(−1)n−j
(
n

j

)
q−nr2 [r2]

n−j
q W ∗m,r1 [j, k]q (12)

W ∗m,r[n+ 1,m+ j + 1]q =
n∑
k=0

W ∗m,r[k,m]qW
∗
m,r−m−1[n− k, j]q (13)

W ∗m,r[s+ p, t]q =

min{t,s}∑
k=max{0,t−p}

W ∗m,r[s, k]qW
∗
m,r+mk[p, t− k]q. (14)

Moreover, the convolution-type identity (14) has been used in [13] to derive the following
Hankel determinant

det
(
W ∗m,r[s+ i+ j, s+ j]q

)
0≤i,j≤n =

n∏
k=0

[m(s+ k) + r]kq .

Another interesting property of W ∗m,r[n, k]q is the divisibility property. One can easily
observe that, using the triangular recurrence relation of Wm,r[n, k]q in (2), we can generate
the following table of values

n/k 0 1 2 3

0 1

1 [r]q qr

2 [r]2q qr ([r]q + [m+ r]q) qm+2r

2 [r]2q qr ([r]q + [m+ r]q) qm+2r

3 [r]3q qr[r]2q + qr[r]q[m+ r]q qm+2r ([r]q + [m+ r]q) q3m+3r

+qr[m+ r]2q qm+2r (+[2m+ r]q)

Then, we can generate the first values of W ∗m,r[n, k]q as follows

n/k 0 1 2 3

0 1

1 [r]q 1

2 [r]2q [r]q + [m+ r]q 1

3 [r]3q [r]2q + [r]q[m+ r]q + [m+ r]2q [r]q + [m+ r]q + [2m+ r]q 1

Note that [n]q = 1 + q+ q2 + . . .+ qn−1. Based on the preceding table, the constant values
of W ∗m,r[n, k]q from row 0 to row 3 form the following triangle of numbers

1
1 1

1 2 1
1 3 3 1.



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1680

This can be written as (
0
0

)(
1
0

) (
1
1

)(
2
0

) (
2
1

) (
2
2

)(
3
0

) (
3
1

) (
3
2

) (
3
3

)
,

which is a portion of Pascal’s triangle. The following theorem generalizes the above ob-
servation.

Theorem 2.1. The q-analogue W ∗m,r[n, k]q satisfies the following congruence relations

W ∗m,r[n, k]q ≡
(
n

k

)
(mod q). (15)

Proof. We recall the rational generating function [13] for W ∗m,r[n, k]q is given by

Ψ∗k(t) =
∑
n≥0

W ∗m,r[n, k]q[t]
n
q =

[t]kq∏k
j=0(1− [mj + r]q[t]q)

.

Since

1

1− [mj + r]q[t]q
=
∑
n≥0

[mj + r]nq [t]nq

=
∑
n≥0

(1 + q + q2 + ...+ qmj+r−1)n[t]nq

=
∑
n≥0

(1 + qy)n[t]nq ,

where y in q. Then
1

1− [mj + r]q[t]q
=
∑
n≥0

(1 + qzn)[t]nq

for some polynomial zn in q. Hence,

1

1− [mj + r]q[t]q
=
∑
n≥0

[t]nq + q
∑
n≥0

zn[t]nq

≡
∑
n≥0

[t]nq (mod q) ≡
(

1

1− [t]q

)
(mod q)

Then

Ψ∗k(t) =
∑
n≥0

W ∗m,r[n, k]q[t]
n
q =

[t]kq∏k
j=0(1− [mj + r]q[t]q)



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1681

≡ [t]kq

(
1

(1− [t]q)k+1

)
(mod q).

Using the Newton’s Binomial Theorem, we have∑
n≥0

W ∗m,r[n, k]q[t]
n
q ≡ [t]kq

∑
n≥0

(
n+ (k + 1)− 1

n

)
[t]nq (mod q)

≡
∑
n≥0

(
n+ k

n

)
[t]n+kq (mod q)

≡
∑
n≥k

(
n− k + k

n− k

)
[t]n+k−kq (mod q)

≡
∑
n≥k

(
n

k

)
[t]nq (mod q).

Comparing the coefficients of [t]nq completes the proof of the theorem.

3. Hankel Transform of D∗m,r[n]q

We recall that the horizontal generating function for Wm,r[n, k]q is given by

n∑
k=0

Wm,r[n, k]q[x− r|m]k,q = [x]nq . (16)

Using the fact that

[x− r|m]k,q = q−kr−m(k2)〈x〉r,m,k,

where 〈x〉r,m,k =
∏n−1
j=0 ([x]q − [r + jm]q), we can write (16) as follows

n∑
k=0

q−kr−m(k2)Wm,r[n, k]q〈x〉r,m,k = [x]nq

n∑
k=0

W ∗m,r[n, k]q〈x〉r,m,k = [x]nq .

Using the method of Cigler [8], let d[n, k] = det (ai+j+k)
n−1
i,j=0 denote the kth Hankel

determinant. That is, the 0th Hankel determinant is given by

d[n, 0] = det


a0 a1 a2 . . . an−1
a1 a2 a3 . . . an
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
an−1 an an+1 . . . a2n−2





R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1682

and the 1st Hankel determinant is given by

d[n, 1] = det


a1 a2 a3 . . . an
a2 a3 a4 . . . an+1

. . . . . . . . . . . . . . . . . . . . . . . . . . . .
an an+1 an+2 . . . a2n−1

 .
Now, define a linear functional F on the polynomial by

F (xn) = an

By Gram-Schmidt orthogonalization process, there exists a sequence of orthogonal poly-
nomials

pn(x) = c0,n + c1,nx+ . . .+ cn−1,nx
n−1 + xn (cn,n = 1)

with respect to F such that

pn(x) =
1

d[n, 0]
det


a0 a1 a2 . . . an−1 1
a1 a2 a3 . . . an x
a2 a3 a4 . . . an+1 x2

. . . . . . . . . . . . . . . . . . . . . . . . . . .
an an+1 an+2 . . . a2n. xn

 (17)

where pn(x) := 1. This means that

F (pnpk) = dn[n = k] with dn 6= 0.

Then

d[n, 0] =
n−1∏
i=0

di.

Clearly, from (17), we have

pn(0) = c0,n =
1

d[n, 0]
(−1)nd[n, 1].

Hence, we have
d[n, 1] = d[n, 0](−1)npn(0). (18)

First, let us consider the Hankel transform of ϕn[x, r,m]q corresponding to the 0th Hankel
determinant.

Theorem 3.1. The Hankel transform of ϕn[x, r,m]q corresponding to the 0th Hankel de-
terminant is given by

H(ϕn[x, r,m]q) = ([m]q[x]q)
(n2) qr(

n
2)+(n3)

n−1∏
k=0

[k]qm !



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1683

Proof. We prove this theorem using the method of Cigler [8]. First, consider a linear
operator Ur,q on the polynomials defined by

Ur,q〈x〉r,m,n = [x]nq where Ur,q[x]qU
−1
r,q = [x]q(1 + [x]−rq D[x]rq)

Then, we have

Ur,q[x]qU
−1
r,q [x]nq = Ur[x]q〈x〉r,m,n

= Ur,q(〈x〉r,m,n+1 + [r + n]q〈x〉r,m,n)

= [x]n+1
q + [r + n]q[x]nq

= [x]q(1 + [x]−rq D[x]rq)[x]nq .

Let Fr,q be the linear function defined by

Fr,q(〈x〉r,m,n) = [a]nq .

The orthogonal polynomial with respect to Fr,q is given by

hn,q(x, a, r,m) =
n∑
k=0

(−[a]q)
kq(

k
2)
[
n

k

]
q

〈x〉r,m,n−k,

which is a kind of q-Poisson-Charlier polynomials satisfying the following recurrence rela-
tion

hn+1,q(x, a, r,m) = ([x]q − [mn+ r]q − qn[a]q)hn,q(x, a, r,m)

− qr+mn−1[a]q[n]qhn−1,q(x, a, r,m).

Now, consider the following polynomial in [x]q

pn,q(x, a) =

n−1∏
k=0

(
[x]q − qk[a]q

)
=

n∑
k=0

(−[a]q)
kq(

k
2)
[
n

k

]
q

[x]n−kq .

By applying the linear operator Ur,q : 〈x〉r,m,k 7→ [x]kq to hn,q(x, a, r,m),

Urhn,q(x, a, r,m) =
n∑
k=0

(−[a]q)
kq(

k
2)
[
n

k

]
q

[x]n−kq = pn,q(x, a).

This implies that
U−1r,q (pn,q(x, a)) = hn,q(x, a, r,m).

Then

Ur[x]qhn,q(x, a, r,m) = Ur[x]qU
−1
r,q (pn,q(x, a))

= [x]q(1 + [x]−rq D[x]rq)pn,q(x, a)



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1684

= [x]qpn,q(x, a) + [r +mn]qpn,q(x, a)

Note that

pn+1,q(x, q) =
n∏
k=0

(
[x]q − qk[a]q

)
= ([x]q − qn[a]q) pn,q(x, q).

Hence, [x]qpn,q(x, a) = pn+1,q(x, a) + [a]qq
npn,q(x, a). Using the fact that

[r +mn]q = [r]q + qr[mn]q,

we have

Ur[x]hn,q(x, a, r,m) = pn+1,q(x, a) + [a]qq
npn,q(x, a)

+ ([r]q + qr[mn]q)pn,q(x, a)

= pn+1,q(x, a) + [a]qq
npn,q(x, a) + [r]qpn,q(x, a) + qr[mn]qpn,q(x, a)

= pn+1,q(x, a) + [a]qq
npn,q(x, a) + [r]qpn,q(x, a) + qr[mn]q[x]qpn−1,q(x, a).

Also, [x]qpn−1,q(x, a) = pn,q(x, a) + [a]qq
n−1pn−1,q(x, a). Then

Ur[x]hn,q(x, a, r,m) = pn+1,q(x, a) + [a]qq
npn,q(x, a)

+ [r]qpn,q(x, a) + qr[mn]q(pn,q(x, a) + [a]qq
n−1pn−1,q(x, a))

= pn+1,q(x, a) + [a]qq
npn,q(x, a) + [r]qpn,q(x, a) + qr[mn]qpn(x, a)

+ [a]q[mn]qq
r+n−1pn−1,q(x, a)

Applying U−1r,q yields

[x]qhn,q(x, a, r,m) = hn+1,q(x, a, r,m) + ([a]qq
n + [r]q + qr[mn]q)hn,q(x, a, r,m)

+ [a]q[mn]qq
r+n−1hn−1,q(x, a, r,m).

Clearly,

Fr,q(hn,q(x, a, r,m)) =
n∑
k=0

(−[a]q)
k q(

k
2)
[
n

k

]
q

[a]nq = pn,q(a, a) = 0,

which implies

dn,q = Fr,q([x]nq hn,q(x, a, r,m))

= qr+n−1[mn]q[a]q Fr,q([x]n−1q hn−1,q(x, a, r,m))

=

n∏
k=1

qr+k−1[mk]q[a]q =

n∏
k=1

qr+k−1[k]qm [m]q[a]q

= (qr[a]q[m]q)
n q(

n
2)[n]qm !

Hence, we have

d[n, 0]q =
n−1∏
k=0

dk,q



R. Corcino, J. Ontolan, G. J. Rama / Eur. J. Pure Appl. Math, 12 (4) (2019), 1676-1688 1685

=
n−1∏
k=0

(qr[m]q[x]q)
k q(

k
2)[k]qm !

= (qr[m]q[x]q)
0+1+2+...+n−1 q(

0
2)+(12)+(22)+...+(n−1

2 )
n−1∏
k=0

[k]qm !

= (qr[m]q[x]q)
(n2) q(

n
3)
n−1∏
k=0

[k]qm !.

This is exactly the desired Hankel transform.

As an immediate consequence of Theorem 3.1, we have the following corollary.

Corollary 3.2. The Hankel transform of D∗m,r[n]q is given by

H(D∗m,r[n]q) = [m]
(n2)
q q(

n
3)+r(

n
2)
n−1∏
k=0

[k]qm !

Proof. This can easily be derived from Theorem 3.1 by letting x = 1.

Remark 3.3. When m = 1, the Hankel tranform in Corollary 3.2 yields

H(D∗1,r[n]q) = q(
n
3)+r(

n
2)
n−1∏
k=0

[k]q!,

which is exactly the Hankel transform of the second form of q-noncentral Bell numbers
B̂q
n,a when r = −a in [11] defined by

B̂q
n,a =

n∑
k=0

S∗a[n, k].

Remark 3.4. When q → 1, Corollary 3.2 gives

H(D∗m,r(n)) = m(n2)
n−1∏
k=0

k!,

which is exactly the Hankel transform of (r, β)-Bell numbers Gn,β,r with β = m in [14].

Theorem 3.5. The Hankel transform of ϕn[x, r,m]q corresponding to the 1st Hankel
determinant d[n, 1]q is given by

H (ϕn[x, r,m]q) = d[n, 1]q

= ([m]q[x]q)
(n2) qr(

n
2)+(n3)

n−1∏
k=0

[k]qm !

n∑
k=0

(−1)n[x]kqq
(k2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q.



REFERENCES 1686

Proof. Taking [pn(x)]q = hn,q(x, a, r,m), we can compute the desired Hankel transform
using (18) with

[pn(0)]q = hn,q(0, a, r,m) =
n∑
k=0

(−[a]q)
k q(

k
2)
[
n

k

]
q

[0− r|m]k,q

=
n∑
k=0

(−1)k[a]kqq
(k2)
[
n

k

]
q

(−1)k
k−1∏
j=0

[r + jm]q

=
n∑
k=0

[a]kqq
(k2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q .

Hence, we have

H (ϕn[x, r,m]q) = d[n, 1]q = d[n, 0]q(−1)n[pn(0)]q

= ([m]q[x]q)
(n2) qr(

n
2)+(n3)

n−1∏
k=0

[k]qm !

n∑
k=0

(−1)n[x]kqq
(k2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q.

4. Recommendation

We observe that the Hankel transform of the second and third forms of the q-analogue
of r-Dowling numbers are obtained using different methods. It would be interesting to
find a method that can be used to establish the Hankel transform of the first form of the
q-analogue of r-Dowling numbers. It may be possible that this method is closely related
to the one being applied in this paper.

Data Availability. No data were used to support this study.

Acknowledgements

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

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