EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 746-759
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

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

Roberto 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 Hankel transform of the generalized q-exponential polynomial of the
first form (q, r)-Whitney numbers of the second kind is established using the method of Cigler.
Consequently, the Hankel transform of the first form (q, r)-Dowling numbers is obtained as special
case.

2020 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 r-Dowling numbers Dm,r(n) are defined in [6] as the sum of r-Whitney numbers
of the second Wm,r(n, k) [9, 11]. More precisely,

Dm,r(n) :=
n∑
k=0

Wm,r(n, k),

where n is a nonnegative integer and the parameters m and r may be real or complex
numbers. These numbers are certain generalization of ordinary Bell numbers Bn [3], r-
Bell numbers Br(n) [10], and noncentral Bell numbers Bn,a [7]. That is, when m = 1, the
r-Dowling numbers reduce to r-Bell numbers and noncentral Bell numbers. Furthermore,
when m = 1, r = 0, these yield the ordinary Bell numbers.

DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3953

Email address: rcorcino@yahoo.com (R. Corcino)

http://www.ejpam.com 746 © 2021 EJPAM All rights reserved.



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 747

J. Layman [8] defined the Hankel transform of an integers sequence (an) as a sequence
of the following determinants dn of Hankel matrix of order n

dn =

∣∣∣∣∣∣∣∣∣∣
a0 a1 a2 . . . an
a1 a2 a3 . . . an+1

a2 a3 a4 . . . an+2

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

∣∣∣∣∣∣∣∣∣∣
. (1)

Aigner [1] derived the Hankel transform of the ordinary Bell numbers to be

det(Bi+j)0≤i,j≤n =

n∏
k=0

k! (2)

which is exactly the Hankel transform obtained by Mezo [10] for r-Bell numbers using
Layman’s Theorem [8] on the invariance of Hankel transform.

Using the method of Aiger [1] and Layman’s Theorem [8], the sequence of (r, β)-Bell
numbers in [4, 12], denoted by {Gn,r,β}, has been shown to possess the following Hankel
transform (see [13])

H(Gn,r,β) =

n∏
j=0

βjj!.

It is worth mentioning that the (r, β)-Bell numbers are equivalent to the r-Dowling num-
bers Dm,r(n), which are defined in [6] 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 [9]. In [13], 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
[14]. 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 [16, 17] 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)

where n and k are nonnegative integers, the parameters m and r may be real of complex
numbers and

Wm,r[n, k]q =

{
1, n = k and n ≥ 0

0, n < k or n, k < 0.

From this definition, two more forms of the q-analogue were defined in [16, 17] as

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



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 748

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 these, three forms of q-analogues for r-Dowling numbers
(or (q, r)-Dowling numbers) were 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)

However, among the three forms of (q, r)-Dowling numbers, only the first form has not
been given a Hankel transform. The third form was thoroughly studied in [17] and its
Hankel transform was successfully derived, which 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 , (9)

using the Hankel transform of q-exponential polynomials in [5], the Layman’s Theorem in
[8] and the Spivey-Steil Theorem in [19]. 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 [2] can be used to derive the Hankel transform for the second
form of the (q, r)-Dowling numbers. The said Hankel transform was derived in [15], which
is given by

H(D∗m,r[n]q) = [m]
(n2)
q q(

n
3)+r(

n
2)
n−1∏
k=0

[k]qm !

Corcino et al. [18] have made a preliminary investigation for the first form (q, r)-
Dowling numbers Dm,r[n]q by establishing an explicit formula expressed in terms of the
first form (q, r)-Whitney numbers of the second kind and (q, r)-Whitney-Lah numbers.
In this present paper, the Hankel transform for the sequence (Dm,r[n]q)

∞
n=0 will be estab-

lished using Cigler’s method [2]. However, a more general form of Dm,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

Wm,r[n, k]qx
k (10)

such that, when x = 1, Φn[1, r,m]q = Dm,r[n]q.



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 749

2. Generalized q-Exponential Polynomials

We may call Φn [x, r,m]q to be the generalized q-exponential polynomial of q-analogue
of r-Whitney numbers of the second kind. Note that we can rewrite (10) as

Φn−1 [x, r,m]q =
n−1∑
k=0

Wm,r[n− 1, k]qx
k

Φn−1 [qx, r,m]q =
n−1∑
k=0

Wm,r[n− 1, k]qq
rk+m(k2)+kxk. (11)

The following theorem contains a recursive relation for Φn [x, r,m]q.

Theorem 2.1. The generalized q-exponential polynomials Φn [x, r,m]q of q-analogue of
r-Whitney numbers of the second kind satisfy the following relation

Φn[x, r,m]q =
[
qrx+ (qm − 1)qrx2Dqm + [r]q + qr[m]qxDqm

]
Φn−1[x, r,m]q. (12)

Proof. Using (3), equation (10) can be written as

Φn [x, r,m]q =
n∑
k=0

Wm,r[n, k]qx
k

=
n∑
k=0

qmk−m+rWm,r[n− 1, k − 1]qx
k +

n∑
k=0

[mk + r]qWm,r[n− 1, k]qx
k

=
n−1∑
k=0

qm(k+1)−m+rWm,r[n− 1, k]qx
k+1 + ([r]q + qr[m]qxDqm) Φn−1[x, r,m]q

= x

n−1∑
k=0

qmk+rWm,r[n− 1, k]qx
k + ([r]q + qr[m]qxDqm) Φn−1[x, r,m]q

= xqr
n−1∑
k=0

qmkWm,r[n− 1, k]qx
k + ([r]q + qr[m]qxDqm) Φn−1[x, r,m]q,

where Dq denotes the q-derivative operator defined by

Dqf(x) =
f(x)− f(qx)

(1− q)x
. (13)

Hence, using (11), we have

Φn[x, r,m]q = xqrΦn−1[q
mx, r,m]q + ([r]q + qr[m]qxDqm) Φn−1[x, r,m]q. (14)

Note that (13) can be expressed as

f(qx) = (q − 1)xDqf(x) + f(x)



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 750

f(qmx) = (qm − 1)xDqmf(x) + f(x)

f(qmx) = ((qm − 1)xDqm + 1)f(x).

This implies that

Φn−1[q
mx, r,m]q = (1 + (qm − 1)xDqm) Φn−1[x, r,m]q.

Thus, equation (14) can further be written as

Φn[x, r,m]q = xqr (1 + (qm − 1)xDqm) Φn−1[x, r,m]q

+ ([r]q + qr[m]qxDqm) Φn−1[x, r,m]q

=
[
qrx+ (qm − 1)qrx2Dqm

+[r]q + qr[m]qxDqm ] Φn−1[x, r,m]q,

which is exactly the desired relation.

Remark 2.2. Let D̂qx =
[
qrx+ (qm − 1)qrx2Dqm + [r]q + qr[m]qxDqm

]
. Then, (12) can

be written as

Φn[x, r,m]q = D̂qxΦn−1[x, r,m]q. (15)

By repeated application of (15),

Φn[x, r,m]q = D̂qxΦn−1[x, r,m]q

= D̂qx

(
D̂qxΦn−2[x, r,m]q

)
= D̂2

qxΦn−2[x, r,m]q

...

= D̂n
qxΦ0[x, r,m]q

= D̂n
qx.

3. Hankel Transform of Dm,r[n]q

Let 〈〈x〉〉r,m,k =
k−1∏
j=0

(x−[r+jm]q)
qr+jm = q−rk−m(k2)〈x〉r,m,k.

The horizontal generating function of W ∗m,r[n, k]q is given by:

n∑
k=0

Wm,r[n, k]q〈x〉r,m,k = xn.

where x = [t]k. Define a linear functional Gr,q by

Gr,q (〈〈x〉〉r,m,n) = an



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 751

and a linear operator Vr,q by
Vr,q (〈〈x〉〉r,m,n) = xn.

Then

Vr,q (xn) =
n∑
k=0

W ∗m,r[n, k]qVr,q (〈x〉r,m,k)

=
n∑
k=0

W ∗m,r[n, k]qq
rk+m(k2)Vr,q (〈〈x〉〉r,m,k)

=
n∑
k=0

W ∗m,r[n, k]qq
rk+m(k2)xk

= Φn[x, r,m]q

Consider the polynomial

gn,q(x, a, r,m) =

n∑
k=0

(−a)kq(
k
2)
[
n

k

]
q

〈〈x〉〉r,m,n−k.

Then

Vr,q (gn,q(x, a, r,m)) =
n∑
k=0

(−a)kq(
k
2)
[
n

k

]
q

Vr,q (〈〈x〉〉r,m,n−k)

=

n∑
k=0

(−a)kq(
k
2)
[
n

k

]
q

xn−k

= pn,q(x, a).

This implies that V −1r,q pn,q(x, a) = gn,q(x, a, r,m). Now,

Vr,qxgn,q(x, a, r,m) = Vr,qxV
−1
r,q pn,q(x, a).

Applying the operator to pn,q(x, a), we get

Vr,qxgn,q(x, a, r,m) = Vr,qxV
−1
r,q pn,q(x, a)

= qrxpn,q(x, a) + (qm − 1)qrx2Dqmpn,q(x, a) + [r]qpn,q(x, a)

+ qr[m]qxDqmpn,q(x, a).

Note that

xpn,q(x, a) =

n∑
k=0

(−a)kq(
k
2)
[
n

k

]
xn+1−k

=

n∑
k=0

(−a)kq(
k
2)
([
n+ 1

k

]
− qn+1−k

[
n

k − 1

])
xn+1−k



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 752

=

n∑
k=0

(−a)kq(
k
2)
[
n+ 1

k

]
xn+1−k −

n∑
k=0

(−a)kq(
k
2)qn+1−k

[
n

k − 1

]
xn+1−k

+ (−a)n+1q(
n+1
2 ) − (−a)n+1q(

n+1
2 ).

xpn,q(x, a) =
n+1∑
k=0

(−a)kq(
k
2)
[
n+ 1

k

]
xn+1−k −

n−1∑
k=−1

(−a)k+1q(
k+1
2 )qn−k

[
n

k

]
xn−k

− (−a)n+1q(
n+1
2 )

= pn+1,q(x, a)−
n−1∑
k=0

(−a)k+1q(
k+1
2 )+n−k

[
n

k

]
xn−k

− (−a)n+1q(
n+1
2 )

= pn+1,q(x, a)−
n∑
k=0

(−a)k+1q(
k+1
2 )+n−k

[
n

k

]
xn−k

= pn+1,q(x, a) + aqn
n∑
k=0

(−a)kq(
k
2)
[
n

k

]
xn−k

= pn+1,q(x, a) + aqnpn,q(x, a).

So, qrxpn,q(x, a) = qrpn+1,q(x, a) + aqn+rpn,q(x, a). With Dqpn,q(x, a) = [n]qpn−1,q(x, a),
we have

Dqmpn,q(x, a) = [n]qmpn−1,q(x, a).

Hence,

(qm − 1)qrx2Dqmpn,q(x, a) = (qm − 1)qrx2[n]qmpn−1,q(x, a) = (qmn − 1)qrx2pn−1,q(x, a)

and
qr[m]qxDqmpn,q(x, a) = qr[m]qx[n]qmpn−1,q(x, a)

Thus,

Vr,qxgn,q(x, a, r,m) = Vr,qxV
−1
r,q pn,q(x, a)

= qrpn+1,q(x, a) + aqn+rpn,q(x, a) + [r]qpn,q(x, a)

+ (qmn − 1)qrx2pn−1,q(x, a) + qr[m]qx[n]qmpn−1,q(x, a)

= qr(pn+1,q(x, a) + qnapn,q(x, a)) + qrpn+1,q(x, a) + aqn+rpn,q(x, a) + [r]qpn,q(x, a)

+ (qmn − 1)qr[pn+1,q(x, a) + (qna+ qn−1a)pn,q(x, a) + q2n−2a2pn−1,a(x, a)]

+ qr[m]q[n]qm(pn,q(x, a) + qn−1apn−1,q(x, a)

= qr(2 + (qmn − 1))pn+1,q(x, a)

+ (qn+ra+ aqn+r + [r]q + (qmn − 1)qr(qna+ qn−1a) + qr[m]q[n]qm)pn,q(x, a)



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 753

+ ((qmn − 1)qrq2n−2a2 + qr[m]q[n]qmq
n−1a)pn−1,q(x, a)

Applying the operator V −1r,q : pn,q(x, a) 7→ gn,q(x, a, r,m), then

xgn,q(x, a, r,m) = qr(2 + (qmn − 1))gn+1,q(x, a, r,m)

+ (2qn+ra+ [r]q + (qmn − 1)qr(qna+ qn−1a) + qr[m]q[n]qm)gn,q(x, a, r,m)

+ ((qmn − 1)qrq2n−2a2 + qr[m]q[n]qmq
n−1a)gn−1,q(x, a, r,m)

We set
hn,q(x, a, r,m) = qm(n2)+rngn,q(x, a, r,m).

That is,

gn,q(x, a, r,m) = q−m(n2)−rnhn,q(x, a, r,m).

Then,

xq−m(n2)−rnhn,q(x, a, r,m) = qr(2 + (qmn − 1))q−m(n+1
2 )−r(n+1)hn+1,q(x, a, r,m)

+ (2qn+ra+ [r]q + (qmn − 1)qr(qna+ qn−1a) + qr[m]q[n]qm)q−m(n2)−rnhn,q(x, a, r,m)

+ ((qmn − 1)qrq2n−2a2 + qr[m]q[n]qmq
n−1a)q−m(n−1

2 )−r(n−1)hn−1,q(x, a, r,m)

xhn,q(x, a, r,m) = qr(2 + (qmn − 1))q−mn−rhn+1,q(x, a, r,m)

+ (2qn+ra+ [r]q + (qmn − 1)qr(qna+ qn−1a) + qr[m]q[n]qm)hn,q(x, a, r,m)

+ ((qmn − 1)qrq2n−2a2 + qr[m]q[n]qmq
n−1a)qm(n−1)+rhn−1,q(x, a, r,m)

xhn,q(x, a, r,m) = (2 + (qmn − 1))q−mnhn+1,q(x, a, r,m)

+ (2qn+ra+ [r]q + (qmn − 1)qr(qna+ qn−1a) + qr[m]q[n]qm)hn,q(x, a, r,m)

+ ((qmn − 1)qrq2n−2a2 + qr[m]q[n]qmq
n−1a)qm(n−1)+rhn−1,q(x, a, r,m) (16)

It is clear that

Gr,q(hn,q(x, a, r,m)) = Gr,q

(
qm(n2)+rngn,q(x, a, r,m)

)
= qm(n2)+rn

n∑
k=0

(−a)kq(
k
2)
[
n

k

]
q

Gr,q (〈〈x〉〉r,m,n−k)

= qm(n2)+rn
n∑
k=0

(−a)kq(
k
2)
[
n

k

]
q

an−k

= qm(n2)+rnpn,q(a, a)

= 0,

Gr,q (〈〈x〉〉r,m,0) = a0 = 1



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 754

which implies
Gr,q(1) = 1.

and

g0,q(x, a, r,m) =

0∑
k=0

(−a)kq(
k
2)
[
0

k

]
q

〈〈x〉〉r,m,0−k

= (−a)0q(
0
2)
[
0

0

]
q

〈〈x〉〉r,m,0

= 1.

It follows that
h0,q(x, a, r,m) = qm(02)+0g0,q(x, a, r,m) = 1

and
Gr,q (h0,q(x, a, r,m)) = Gr,q(1) = 1.

Clearly, Gr,q ([x]qhn,q(x, a, r,m)) = 0 and from (16),

xhn,q(x, a, r,m) = g(n)hn+1,q(x, a, r,m) + f(n)hn,q(x, a, r,m) + c(n)hn−1,q(x, a, r,m)

where

g(n) = (2 + (qmn − 1))q−mn

f(n) = (2qn+ra+ [r]q + (qmn − 1)qr(qna+ qn−1a) + qr[m]q[n]qm)

c(n) = ((qmn − 1)qrq2n−2a2 + qr[m]q[n]qmq
n−1a)qm(n−1)+r.

Then,

x2hn,q(x, a, r,m) = xxhn,q(x, a, r,m)

= x [g(n)hn+1,q(x, a, r,m) + f(n)hn,q(x, a, r,m) + c(n)hn−1,q(x, a, r,m)]

= g(n)xhn+1,q(x, a, r,m) + f(n)xhn,q(x, a, r,m)

+ c(n)xhn−1,q(x, a, r,m)

= g(n)g(n+ 1)hn+2,q(x, a, r,m) + g(n)f(n+ 1)hn+1,q(x, a, r,m)

+ g(n)c(n+ 1)hn,q(x, a, r,m) + g(n)f(n)hn+1,q(x, a, r,m)

+ f2(n)hn,q(x, a, r,m) + f(n)c(n)hn−1,q(x, a, r,m)

+ c(n)g(n− 1)hn,q(x, a, r,m) + c(n)f(n− 1)hn−1,q(x, a, r,m)

+ c(n)c(n− 1)hn−2,q(x, a, r,m)

= g(n)g(n+ 1)hn+2,q(x, a, r,m)

+ [g(n)f(n+ 1) + g(n)f(n)]hn+1,q(x, a, r,m)

+
[
g(n)c(n+ 1) + f2(n) + c(n)g(n− 1)

]
hn,q(x, a, r,m)

+ [f(n)c(n) + c(n)f(n− 1)]hn−1,q(x, a, r,m)



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 755

+ c(n)c(n− 1)hn−2,q(x, a, r,m).

Applying the linear functional Gr,q to [x]2qhn,q(x, a, r,m) gives,

Gr,q
(
x2hn,q(x, a, r,m)

)
= 0

...

Gr,q

(
xkhn,q(x, a, r,m)

)
= 0

for k < n. For k = n,

xnhn,q(x, a, r,m) =g(n)xn−1hn+1,q(x, a, r,m) + f(n)xn−1hn,q(x, a, r,m)

+ c(n)xn−1hn−1,q(x, a, r,m).

Then,

Gr,q (xnhn,q(x, a, r,m)) =g(n)Gr,q
(
xn−1hn+1,q(x, a, r,m)

)
+ f(n)Gr,q

(
xn−1hn,q(x, a, r,m)

)
+ c(n)Gr,q

(
xn−1hn−1,q(x, a, r,m)

)
=c(n)Gr,q

(
xn−1hn−1,q(x, a, r,m)

)
=c(n)c(n− 1)Gr,q

(
xn−2hn−2,q(x, a, r,m)

)
=c(n)c(n− 1)c(n− 2)Gr,q

(
xn−3hn−3,q(x, a, r,m)

)
...

=c(n)c(n− 1)c(n− 2) . . . c(1)Gr,q
(
x0h0,q(x, a, r,m)

)
=

[
n∏
i=1

c(i)

]
(1)

=

n∏
i=1

c(i)

Since xnhn,q(x, a, r,m) is a sequence of orthogonal polynomials with respect to linear
functional Gr,q,

dn,q = Gr,q (xnhn,q(x, a, r,m)) =
n∏
i=1

c(i)

where
c(i) = ((qmi − 1)qrq2i−2a2 + qr[m]q[i]qmq

i−1a)qm(i−1)+r

Then

dn,q(n, 0) = Gr,q
[
[x]nqhn,q(x, a, r,m)

]
=

n−1∏
i=0

di,q



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 756

=
n−1∏
i=0


i∏

j=1

[
((qmj − 1)qrq2j−2a2 + qr[m]q[j]qmq

j−1a)qm(j−1)+r
]

=

n−1∏
i=0


i∏

j=1

[
aqm(j−1)+2r((qmj − 1)q2j−2a+ [m]q[j]qmq

j−1)
]

=
n−1∏
i=0

q2riqm(1+2+3+···+(i−1))ai
i∏

j=1

[
((qmj − 1)q2j−2a+ [m]q[j]qmq

j−1)
]

=
n−1∏
i=0

q2riqm(i
2)ai

i∏
j=1

[
((qmj − 1)q2j−2a+ [m]q[j]qmq

j−1)
]

= q2r(0+1+2+3+···+(n−1))+m[(22)+(32)+···+(n−1
2 )]a0+1+2+···+(n−1)

n−1∏
i=0

i∏
j=1

[
((qmj − 1)q2j−2a+ [m]q[j]qmq

j−1)
]

= q2r(
n
2)+(m+1)(n3)a(n2)

n−1∏
i=0

i∏
j=1

[
((qmj − 1)q2j−2a+ [m]q[j]qmq

j−1)
]

= q2r(
n
2)+m(n3)a(n2)

n−1∏
i=0

q(
i
2)

i∏
j=1

[
[mj]q

(
1− qj

(
1− q
q

)
a

)]

= q2r(
n
2)+(m+1)(n3)a(n2)

n−1∏
i=0

i∏
j=1

[
[mj]q

(
1− qj−1(1− q)a

)]
.

This result is stated formally in the following theorem.

Theorem 3.1. The Hankel transform of Φn[x, r,m]q corresponding to the 0th Hankel
determinant is given by

H (Φn[x, r,m]q) = q2r(
n
2)+(m+1)(n3)a(n2)

n−1∏
i=0

i∏
j=1

[
[mj]q

(
1− qj−1(1− q)a

)]
. (17)

Note that when m = 1, (17) yields

H (Φn[x, r, 1]q) = q2r(
n
2)+2(n3)a(n2)

n−1∏
i=0

[i]q!((1− q)a; q)i

where

(x; q)i =

i−1∏
j=0

(
1− qjx

)
.

This is exactly the result obtained by Cigler [2].



R. Corcino / Eur. J. Pure Appl. Math, 14 (3) (2021), 746-759 757

As a direct consequence of Theorem 3.1, we have the following corollary, which contains
the main result of this paper.

Corollary 3.2. The Hankel transform of the sequence (Dm,r[n]q)
∞
n=0 is given by

H (Dm,r[n]q) = q2r(
n
2)+(m+1)(n3)

n−1∏
i=0

((1− q)a; q)i

i∏
j=1

[mj]q.

Theorem 3.3. The Hankel transform of Φn[x, r,m]q corresponding to the 1st Hankel
determinant is given by

dn,q(n, 1) = q2r(
n
2)+(m+1)(n3)a(n2)

n−1∏
i=0

((1− q)a; q)i

i∏
j=1

[mj]q

n∑
k=0

(−1)n[x]kqq
(k2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q
qr+jm

.

Proof. From Gram-Schmidt orthogonalization process, we obtain

dn,q(n, 1) = dn,q(n, 0)(−1)npn,q(0)

where pn,q(0) is a sequence of orthogonal polynomials i.e.,

gn,q(x, a, r,m) =
n∑
k=0

(−a)k q(
k
2)
[
n

k

]
q

〈〈x〉〉r,m,k = pn,q(x)

which implies

pn,q(0) =
n∑
k=0

(−a)k q(
k
2)
[
n

k

]
q

〈〈0〉〉r,m,k.

Since,

〈〈0〉〉r,m,k =
k−1∏
j=0

([0]q − [r + jm]q)

qr+jm

=
k−1∏
j=0

−[r + jm]q
qr+jm

=

(
−[r]q
qr

)(
−[r + j]q
qr+m

)(
−[r + (k − 1)m]q

qr+(k−1)m

)
= (−1)k

k−1∏
j=0

[r + jm]q
qr+jm

.



REFERENCES 758

Then,

pn,q(0) =
n∑
k=0

(−a)k q(
k
2)
[
n

k

]
q

(−1)k
k−1∏
j=0

[r + jm]q
qr+jm

=
n∑
k=0

(−1)kakq(
k
2)
[
n

k

]
q

(−1)k
k−1∏
j=0

[r + jm]q
qr+jm

=
n∑
k=0

akq(
k
2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q
qr+jm

which implies

(−1)npn,q(0) =

n∑
k=0

(−1)n[x]kqq
(k2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q
qr+jm

.

Hence,

dn,q(n, 1) = dn,q(n, 0)(−1)npn,q(0)

= q2r(
n
2)+(m+1)(n3)a(n2)

n−1∏
i=0

((1− q)a; q)i

i∏
j=1

[mj]q

n∑
k=0

(−1)n[x]kqq
(k2)
[
n

k

]
q

k−1∏
j=0

[r + jm]q
qr+jm

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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