EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 279-293
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Hankel Transform of (q, r)-Dowling Numbers

Roberto B. Corcino1, Mary Joy R. Latayada2,∗, Mary Ann Ritzell P. Vega3

1 Research Institute for Computational Mathematics and Physics, Cebu Normal University,
6000 Cebu City, Philippines
2 Department of Mathematics, Caraga State University, 8600 Butuan City, Philippines
3 Department of Mathematics and Statistics, College of Science and Mathematics,
Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines

Abstract. In this paper, the authors establish certain combinatorial interpretation for q-analogue
of r-Whitney numbers of the second kind defined by Corcino and Cañete in the context of A-
tableaux. They derive convolution-type identities by making use of the combinatorics of A-
tableaux. Finally, they define a q-analogue of r-Dowling numbers and obtain some necessary
properties including its Hankel transform.

Key Words and Phrases: Whitney numbers, Dowling numbers, generating function, q-analogue,
q-exponential function, A-tableau, convolution formula, Hankel transform, Hankel matrix, binomial
transform.

1. Introduction

The binomial transform B of a sequence A = {an} is the sequence {bn} defined by

bn =
n∑
k=0

(−1)k
(
n

k

)
ak.

That is, B(A) = bn. It is one of the common and useful transforms that frequently
appeared in the literature of integer sequences (see [16]). The inverse binomial transform
(or inverse transform) C of a sequence A is the sequence {cn} defined by

cn =

n∑
k=0

(
n

k

)
ak.

That is, C(A) = cn.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3406

Email addresses: rcorcino@yahoo.com (R. Corcino),
mrlatayada@gmail.com (MJ. Latayada), maryannritzel.vega@g.msuiit.edu.ph (MAR. Vega)

http://www.ejpam.com 279 c© 2019 EJPAM All rights reserved.



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 280

The Hankel matrix Hn of order n of a sequence A = {a0, a1, . . . , an} is given by
Hn = (ai+j)0≤i,j≤n. The Hankel determinant hn of order of n of A is the determinant of
the corresponding Hankel matrix of order n. That is, hn = det(Hn). The Hankel transform
of the sequence A, denoted by H(A), is the sequence {hn} of Hankel determinants of A.
For instance, the Hankel transform of the sequence of Catalan numbers C = { 1

n+1

(
2n
n

)
}∞n=1,

is given by
H(C) = {1, 1, 1, . . . , }

and the sequence of the sum of two consecutive Catalan numbers, an = cn + cn+1, with
cn the nth Catalan numbers, has the Hankel transform

H(an) = {F2n+1}∞n=0

where Fn is the nth Fibonacci numbers [12].
One remarkable property of Hankel transform is established by Layman [12], which

states that the Hankel transform of an integer sequence is invariant under binomial and
inverse transforms. That is, if A is an integer sequence, B is binomial transform of A and
C is the inverse transform of A, then

H(B(A)) = H(A) and H(C(A)) = H(A).

This property played an important role in proving that the Hankel transform of the se-
quence of Bell number {Bn)} [1] and that of r-Bell numbers {Bn,r} [14] are equal. Recently,
in the paper by R. Corcino and C. Corcino [7], this property has also been used in proving
that the Hankel transform of the sequence of generalized Bell numbers {Gn,r,β} is given
by

H(Gn,r,β) =
n∏
j=0

βjj!

where Gn,r,β is the sum of (r, β)-Stirling numbers

{
n
k

}
r,β

Gn,r,β =
n∑
k=0

{
n
k

}
r,β

(see [5, 8]), which are also known as (r, β)-Bell numbers. In the same paper, the authors
have made an attempt to establish the Hankel transform for the q-analogue of (r, β)-Bell
numbers. However, they are not successful with their attempt and have conjectured that
the Hankel transform for the q-analogue of (r, β)-Bell numbers when r = 0 is equal to

H
(
Gqn,β,0

)
=

n∏
k=0

qf(n,k)[β]kq [k]qβ ! (1)

for some number f(n, k), which is a function of n and k. With this, the present au-
thors have decided to use other method. Recently, R. Corcino et al.[9] have successfully



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 281

established the Hankel transform for the q-analogue of noncentral Bell numbers. This
motivates the present authors to use this method to establish the Hankel transform for
the q-analogue of (r, β)-Bell numbers Gn,r,β. It is important to note that the numbers
Gn,r,β are equivalent to the r-Dowling numbers Dm,r(n), which are defined as the sum of
r-Whitney numbers of the second kind, denoted by Wm,r(n, k). That is,

Dm,r(n) =

n∑
k=0

Wm,r(n, k).

The term “r-Dowling numbers” was introduced by Cheon and Jung [3].

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

A q-analogue of both kinds of Stirling numbers was first defined by Carlitz in [2]. The
second kind of which, known as q-Stirling numbers of the second kind, is defined in terms
of the following recurrence relation

Sq[n, k] = Sq[n− 1, k − 1] + [k]qSq[n− 1, k] (2)

in connection with a problem in abelian groups, 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). (3)

A different way of defining q-analogue of Stirling numbers of the second kind has been
adapted in the paper by [10] which is given as follows

Sq[n, k] = qk−1Sq[n− 1, k − 1] + [k]qSq[n− 1, k]. (4)

This type of q-analogue gives the Hankel transform of q-exponential polynomials and
numbers which are certain q-analogue of Bell polynomials and numbers. Recently, a q-
analogue of r-Whitney numbers of the second kind was defined by Corcino and Cañete [6]
parallel to the definition for q-analogue of noncentral Stirling numbers of the second kind
as follows:

Definition 1. For non-negative integers n and k, and real number a, a q-analogue Wm,r[n, k]q
of Wm,r(n, k) is defined by

Wm,r[n, k]q = qm(k−1)+rWm,r[n− 1, k − 1]q + [mk + r]qWm,r[n− 1, k]q. (5)

where Wm,r[0, 0]q = 1, Wm,r[n, k]q = 0 for n < k or n, k < 0 and [t− k]q = 1
qk

([t]q − [k]q).

Remark 1. When m = 1 and r = 0, the relation (5) reduces to (4). This implies that

W1,0[n, k]q = Sq[n, k]. (6)



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 282

The q-analogue Wm,r[n, k]q satisfies the following properties:

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; (7)

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; (8)

Horizontal Generating Function

n∑
k=0

Wm,r[n, k]q[t− r|m]k,q = [t]nq . (9)

Explicit Formula

Wm,r[n, k]q =
1

[k]qm ![m]kq

k∑
j=0

(−1)k−jqm(k−j2 )
[
k

j

]
qm

[jm+ r]nq (10)

=
1

[k]qm ![m]kq

[
∆k
qm,m[x+ r]nq

]
x=0

(11)

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

. (12)

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)

.

Explicit Formula in Symmetric Function Form

Wm,r[n, k]q = qm(k2)+kr
∑

S1+S2+···Sk=n−k

k∏
j=0

[mj + r]
Sj
q

=
∑

0≤j1≤j2≤···jn−k≤k
qm(k2)+kr

n−k∏
i=1

[mj + r]q.



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 283

We now define another form of q-analogue of r-Whitney numbers of the second, denoted
by W ∗m,r[n, k]q, as follows

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

Hence,

W ∗m,r[n, k] =
∑

0≤j1≤j2≤...≤jn−k≤k

n−k∏
i=1

[mji + r]q. (13)

All other properties parallel to those of Wm,r[n, k]q can easily be established by imbed-

ding the factor q−kr−m(k2) in the derivations or multiply directly to the resulting identi-
ties/formula.

Definition 2. [13] An A-tableau is 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.

Let ω be a function from the set of nonnegative integers N to a ring K. Suppose Φ is
an A-tableau with l columns of lengths |c| ≤ h. We use TAr (h, l) to denote the set of such
A-tableaux. Then, we set

ωA(Φ) =
∏
c∈Φ

ω(|c|).

Note that Φ might contain a finite number of columns whose lengths are zero since 0 ∈
A = {0, 1, 2, . . . , k} and if ω(0) 6= 0.

From this point onward, whenever an A-tableau is mentioned, it is always associated
with the sequence A = {0, 1, 2, . . . , k}.

We are now ready to mention the following theorem.

Theorem 1. Let ω : N → K denote a function from N to a ring K (column weights
according to length) which is defined by ω(|c|) = [m|c|+ r]q where r is a complex number,
and |c| is the length of column l of an A-tableau in TAr (k, n− k). Then

W ∗m,r[n, k] =
∑

φ∈TAr (k,n−k)

∏
c∈φ

ω(|c|).

Proof. Let Φ ∈ TAr (k, n − k). This means that Φ has exactly n − k columns say
c1, c2, · · · , cn−k whose lengths are j1, j2, · · · , jn−k, respectively. Now, for each column
ci ∈ Φ, i = 1, 2, 3, · · · , n− k, we have |ci| = ji and

ω(|ci|) = [m|ji|+ r]q.



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 284

Then ∏
c∈Φ

ω(|c|) =
n−k∏
i=1

ω(|ci|) =
n−k∏
i=1

[m|ji|+ r]q.

Since Φ ∈ TAr (k, n− k), then∑
Φ∈TAr (k,n−k)

∏
c∈Φ

ω(|c|) =
∑

0≤j1≤j2≤...≤jn−k≤k

∏
c∈Φ

ω(|c|)

=
∑

0≤j1≤j2≤...≤jn−k≤k

n−k∏
i=1

[m|ji|+ r]q

= W ∗m,r[n, k].

�

Suppose that for some numbers r1 and r2, we have r = r1 + r2. Then, equation (13)
yields

W ∗m,r[n, k]q =
∑

0≤j1≤j2≤...≤jn−k≤k

n−k∏
i=1

[(mji + r1) + r2]q.

That is, for any φ ∈ TAr (k, n− k),

ωA(φ) =
∏
c∈φ

[(mji + r1) + r2]q,

where |c| ∈ {0, 1, 2, . . . , k}. Note that the weight of each column of φ can be considered as
a finite sum with additive constant r2, that is, for each c ∈ φ, we can write

ω(|c|) =
1

qr2
(ω∗(|c|) + [r2]q), (14)

where ω∗(|c|) = [m|c|+ r1]q. The following theorem determines how an additive constant
affects the recurrence formula for Wm,r[n, k]q. From Theorem 1,

W ∗m,r[n, k]q =
∑

φ∈TAr (k,n−k)

ωA(φ) =
∑

φ∈TAr (k,n−k)

∏
c∈φ

ω(|c|)

where

ωA(φ) =
∏
c∈φ

[m|c|+ r]q, where |c| ∈ {0, 1, . . . , k}

=
n−k∏
i=1

[mji + r]q, where ji ∈ {0, 1, . . . , k}.



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 285

If r = r1 + r2 for some r1 and r2, then by (14),

ωA(φ) =
n−k∏
i=1

1

qr2
(ω∗(ji) + [r2]q) , where ω∗(ji) = [mji + r1]q

= q−(n−k)r2 (ω∗(j1) + [r2]q) (ω∗(j2) + [r2]q) · · · (ω∗(jn−k) + [r2]q))

= q−(n−k)r2

n−k∑
l=0

([r2]q)
n−k−l

∑
j1≤j1≤j2≤...≤jl≤jn−k

l∏
i=1

ω∗(ji).

Suppose Bφ is the set of all A-tableaux corresponding to φ such that for each ψ ∈ Bφ,
either

ψ has no column whose weight is [r2]q, or

ψ has one column whose weight is [r2]q, or

ψ has two columns whose weights are [r2]q, or

...

ψ has (n− k) columns whose weights are [r2]q.

Then, we may write

ωA(φ) =
∑
ψ∈Bφ

ωA(ψ).

Now, if l columns in ψ have weights other than [r2]q, then

ωA(ψ) =
∏
c∈ψ

ω∗(|c|) = q−(n−k)r2([r2]q)
n−k−r

r∏
i=1

ω∗(qi)

where q1, q2, . . . , qr ∈ {j1, j2, . . . , jn−k}. Note that for each l, there corresponds(
n− k
l

)
tableaux with l columns having weights ω∗(ji) = [mji+r1]q. It can be easily verified that,

|TAr (k, n− k)| =
(

(n− k) + k

n− k

)
=

(
n

n− k

)
=

(
n

k

)
.

Thus, ∀φ ∈ TAr (k, n− k), Bφ contains a total of(
n

k

)(
n− k
l

)



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 286

tableaux with l columns of weights ω∗(ji). However, only
(
l+k
l

)
tableaux with l columns

in Bφ are distinct. Hence, every distinct tableaux ψ with l columns of weights other than
[r2]q appears (

n
k

)(
n−k
l

)(
l+k
l

) =

(
n

l + k

)
times in the collection. Thus,

∑
φ∈TAr (k,n−k)

ωA(φ) =
n−k∑
l=0

(
n

l + k

)
q−(n−k)r2([r2]q)

n−k−l
∑
ϕ∈B̄l

∏
c∈ϕ

ω∗(|c|)

where B̄l denotes the set of all tableaux ϕ having l columns of weights ω∗(ji) = [mji+r1]q.
Reindexing the double sum, we get

∑
φ∈TAr (k,n−k)

ωA(φ) =

n∑
j=k

(
n

j

)
q−nr2([r2]q)

n−j
∑

ϕ∈B̄j−k

∏
c∈ϕ

ω∗(|c|)

where B̄j−k is the set of all tableaux ϕ with j − k columns of weights ω∗(ji) = [mji + r1]q
for each i = 1, 2, . . . , j − k. Clearly B̄j−k = TAr1(k, j − k). Hence,

∑
φ∈TAr (k,n−k)

ωA(φ) =
n∑
j=k

(
n

j

)
q−nr2([r2]q)

n−j
∑

ϕ∈TAr1 (k,j−k)

ωA(ϕ).

Applying Theorem 1, we obtain the following theorem.

Theorem 2. The q-analogue W ∗m,r[n, k]q satisfies the following identity

W ∗m,r[n, k]q =

n∑
j=k

(−1)n−j
(
n

j

)
q−nr2 [r2]n−jq W ∗m,r1 [j, k]q

where r = r1 + r2 for some numbers r1 and r2.

Suppose

φ1 is a tableau with k − s columns whose lengths are in the set

{0, 1, . . . , s}, and

φ2 be a tableau with n− k − j columns whose lengths are in the

set {s+ 1, s+ 2, . . . , s+ j + 1}

Then
φ1 ∈ TA1(s, k − s) and φ2 ∈ TA2(j, n− k − j)



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 287

where A1 = {0, 1, . . . , s} and A2 = {s+ 1, s+ 2, . . . , s+ j + 1}. Notice that by joining the
columns of φ1 and φ2, we obtain an A-tableau φ with n− s− j columns whose lengths are
in the set A = A1 ∪A2 = {0, 1, . . . , s+ j+ 1}. That is, φ ∈ TA(s+ j+ 1, n− s− j). Then,

∑
φ∈TA(s+j+1,n−s−j)

ωA(φ) =

n−j∑
k=s

 ∑
φ1∈TA1 (s, k−s)

ωA1(φ1)


 ∑
φ2∈TA2 (j, n−k−j)

ωA2(φ2)

 .

Note that ∑
φ2∈TA2 (j, n−k−j)

ωA2(φ2) =
∑

φ2∈TA2 (j, n−k−j)

∏
c∈φ2

[m|c|+ r]q

=
∑

s+1≤g1≤...≤g n−k−j≤ s+j+1

n−k−j∏
i=1

[mgi + r]q

=
∑

0≤g1≤...≤g n−k−j≤j

n−k−j∏
i=1

[mgi +m(s+ 1) + r]q.

Thus,

∑
0≤g1≤...≤gn−s−j≤s+j+1

n−s−j∏
i=1

[mgi + r]q

=

n−j∑
k=s

 ∑
0≤g1≤...≤gk−s≤s

k−s∏
i=1

[mgi + r]q


 ∑

0≤g1≤...≤gn−k−j≤j

n−k−j∏
i=1

[mgi +m(s+ 1) + r]q

.
By (13), we obtain the following theorem.

Theorem 3. The q-analogue W ∗m,r[n, k] satisfies the following convolution-type identity

W ∗m,r[n+ 1, s+ j + 1]q =
n∑
k=0

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

The next theorem provides another form of convolution-type identity.

Theorem 4. The q-analogue W ∗m,r[n, k]q satisfies the following second form of convolution
formula

W ∗m,r[s+ j, n]q =

n−j∑
k=s

W ∗m,r[s, k]qW
∗
m,r+mk[j, n− k]q.



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 288

Proof. Let

φ1 be a tableau with s− k columns whose lengths are in

A1 = {0, 1, . . . , k}, and

φ2 be a tableau with j − n+ k columns whose lengths are in

A2 = {k, k + 1, . . . , n}.

Then φ1 ∈ TA1(k, s− k) and φ2 ∈ TA2(n− k, j−n+ k). Using the same argument above,
we can easily obtain the convolution formula. �

3. (q, r)−Dowling Number and Its Hankel Transform

In this section, we define a q-analogue of the r-Dowling numbers and obtain some
combinatorial properties that will be used to establish its Hankel transform.

A q-analogue of the r-Dowling numbers, denoted by D̃m,r[n]q, is defined by

D̃m,r[n]q =

n∑
k=0

W̃m,r[n, k]q

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

For brevity, we use the term (q, r)-Dowling numbers for D̃m,r[n]q.

Remark 2. When m = 1 and r = 0, (6) yields

W̃1,0[n, k]q = q−(k2)W1,0[n, k] = q−(k2)Sq[n, k] = S̃q[n, k]. (15)

It follows that the (q, r)-Dowling numbers reduces to

D̃1,0[n]q = ẽq,n[1] (16)

where ẽq,n[z] is the q-exponential polynomial in [11] defined by

ẽq,n[z] =
n∑
k=0

S̃q[n, k]zk. (17)

Remark 3. We recall that the Hankel transform of the q-exponential polynomial ẽq,n[z]
is given by

H (ẽq,n(z)) = q(
n+1
3 )[0]![1]! . . . [n]!(z)(

n+1
2 ).

It can easily be verified that the Hankel transform of

ēq,n[z] =

n∑
k=0

S̃q[n, k]zn−k (18)

is equal to that of ẽq,n[z].



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 289

Remark 4. Since

Wm,0[n, k]q = [m]n−kq

 1

[k]qm !

k∑
j=0

(−1)k−jqm(k−j2 )
[
k

j

]
qm

[j]nqm


= [m]n−kq Sqm [n, k],

we have

W̃m,0[n, k]q = q−m(k2)Wm,0[n, k] = [m]n−kq (qm)−(k2) Sqm [n, k] = [m]n−kq S̃qm [n, k].

This implies that

D̃m,0[n]q =

n∑
k=0

W̃m,0[n, k]q =

n∑
k=0

S̃qm [n, k][m]n−kq . (19)

Thus, using Remark 3, the Hankel transform of D̃m,0[n]q is given by

H
(
D̃m,0[n]q)

)
= H (ēqm,n([m]q)) = qm(n+1

3 )[0]qm ![1]qm ! . . . [n]qm ![m]
(n+1

2 )
q (20)

Clearly, when q → 1, D̃m,r[n]q → D̃m,r(n), the r-Dowling numbers. By making use of
Theorem 2, with r1 = r − 1 and r2 = 1 and multiplying both sides by q−kr,we have

W̃m,r[n, k]q =
n∑
j=k

(−1)n−j
(
n

j

)
q−nW̃m,r−1[j, k]q. (21)

Summing up both sides of (21), we have

D̃m,r[n]q =

n∑
k=0

n∑
j=k

(−1)n−j
(
n

j

)
q−nW̃m,r−1[j, k]q

=

n∑
j=0

j∑
k=0

(−1)n−j
(
n

j

)
q−nW̃m,r−1[j, k]q

=
n∑
j=0

(−1)n−j
(
n

j

)
q−n

j∑
k=0

W̃m,r−1[j, k]q

=

n∑
j=0

(−1)n−j
(
n

j

)
q−nD̃m,r−1[j]q.

The following theorem states formally the above recurrence relation for D̃m,r[n]q.

Theorem 5. The (q, r)-Dowling numbers satisfy the following relation

qnD̃m,r[n]q =

n∑
j=0

(−1)n−j
(
n

j

)
D̃m,r−1[j]q. (22)



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 290

The following corollary is a direct consequence of Theorem 5 which can be proved
using the inversion formula by Riordan [4, 15].

Corollary 1. The (q, r)-Dowling numbers satisfy the following relations

D̃m,r−1[n]q =
n∑
j=0

(
n

j

)
qjD̃m,r[j]q. (23)

To establish the Hankel transform of D̃m,r[n]q, we need the concept of rising k-binomial
transform by Spivey and Steil [17] as well as its property in relation to Hankel transform.

Definition 3. (Spivey-Steil [17]) The rising k-binomial transform R of a sequence A =
{an} is the sequence R(A; k) = {rn}, where rn is given by

rn =

n∑
j=0

(
n

j

)
kjaj , k 6= 0. (24)

We use R(A, k) to denote the set of rising k-binomial transform of A. That is,
R(A, k) = {rn}. Then we have the following theorem by Spivey and Steil.

Theorem 6. (Spivey-Steil [17]) Given a sequence A = {a0, a1, . . . , }. Let H(A) = {hn}.
Then

H(R(A, k)) = {a0, 0, 0, . . . , }.

If k 6= 0,
H(R(A, k)) = {kn(n+1)hn}.

Now, we are ready to state the main result of the paper.

Theorem 7. The Hankel transform of the sequence of (q, r)-Dowling numbers {D̃m,r[n]q}
is given by

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

(n+1
2 )

q . (25)

Proof. Using equation (18) in Remark 4, we have

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

(n+1
2 )

q . (26)

From Corollary 1, we say that D̃m,r−1[n]q is the binomial transform of qnD̃m,r[n]q. This
means that

B(qnD̃m,r[n]q) = D̃m,r−1[n]q.

Hence, by Layman’s Theorem [12],

H(B(qnD̃m,r[n]q)) = H(qnD̃m,r[n]q).



R. Corcino, MJ. Latayada, MAR. Vega / Eur. J. Pure Appl. Math, 12 (2) (2019), 279-293 291

That is,
H(D̃m,r−1[n]q) = H(qnD̃m,r[n]q).

Now, Corollary 1 can also be stated as D̃m,r−1[n]q is the rising q-binomial transform of

D̃m,r[n]q. Using Spivey-Steil Theorem, with A = {D̃m,r[n]q}, hn = H(D̃m,r[n]q) and

rn = D̃m,r−1[n]q, we have

H(D̃m,r−1[n]q) = qn(n+1)H(D̃m,r[n]q).

We observe that, when r = 1 and using (26), we have

H(D̃m,1[n]q) = q−n(n+1)H(D̃m,0[n]q)

= q−n(n+1)qm(n+1
3 )[0]qm ![1]qm ! . . . [n]qm ![m]

(n+1
2 )

q

= qm(n+1
3 )−n(n+1)[0]qm ![1]qm ! . . . [n]qm ![m]

(n+1
2 )

q

Also, when r = 2,

H(D̃m,2[n]q) = qm(n+1
3 )−2n(n+1)[0]qm ![1]qm ! . . . [n]qm ![m]

(n+1
2 )

q .

Continuing this argument, we obtain

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

(n+1
2 )

q

�

Remark 5. When m = 1, the Hankel transform in (25) reduces to

H(D̃1,r[n]q) = q(
n+1
3 )−rn(n+1)[0]![1]! . . . [n]!,

which is exactly the Hankel transform for the q-noncentral Bell numbers in [9].

Remark 6. When q → 1, the Hankel transform in (25) yields

H(D̃m,r[n]q) = [0]![1]! . . . [n]!m(n+1
2 ),

which is exactly the Hankel transform for the q-analogue of (r, β)-Bell numbers in [9].

Remark 7. The Hankel transform in (25) can also be written as

H(D̃m,r[n]q) = qm(n+1
3 )

n∏
k=0

q−2rk[m]kq [k]qm !

such that, when r = 0, we have

H(D̃m,0[n]q) = qm(n+1
3 )

n∏
k=0

[m]kq [k]qm !,

which is exactly the conjectured Hankel transform in (1) with m = β and

n∏
k=0

f(n, k) = qm(n+1
3 ).



REFERENCES 292

Acknowledgements

The first author would like to thank Cebu Normal University (CNU) and the Com-
mission on Higher Education-Grants-in-Aid for Research (CHED-GIA) for the financial
support extended to this project. The second author would also like to thank the Com-
mission on Higher Education-Faculty Development Program (CHED-FDP) for providing
her financial support in pursuing Ph.D. Math Program at the Mathematics Department
of MSU - Iligan Institute of Technology. Lastly, the authors would like to thank the two
anonymous referees for spending their free time in reviewing our paper. Their comments
and suggestions have helped a lot in improving the paper.

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