axioms
Article
Hankel Transform of the Type 2 ( p, q)-Analogue of
r -Dowling Numbers
Roberto Corcino
1,
*
,†
, Mary Ann Ritzell Vega
2,†
and Amerah Dibagulun
3,†
Citation: Corcino, R.; Vega, M.A.R.;
Dibagulun, A. Hankel Transform of
the Type 2 ( p, q)-Analogue of
r-Dowling Numbers. Axioms 2021, 10,
343. https://doi.org/10.3390/
axioms10040343
Academic Editor: Stevan Pilipovi´ c
Received: 19 October 2021
Accepted: 9 December 2021
Published: 16 December 2021
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1
Research Institute for Computational Mathematics and Physics, Cebu Normal University,
Cebu City 6000, Philippines
2
Department of Mathematics and Statistics, Mindanao State University-Iligan Institute of Technology,
Iligan City 9200, Philippines; maryannritzell.vega@g.msuiit.edu.ph
3
Department of Mathematics, Mindanao State University-Main Campus, Marawi City 9700, Philippines;
a.dibagulun@yahoo.com
* Correspondence: rcorcino@yahoo.com
† These authors contributed equally to this work.
Abstract: In this paper, type 2 ( p, q)-analogues of the r-Whitney numbers of the second kind is
defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover,
some convolution-type identities, which are useful in deriving the Hankel transform of the type
2 ( p, q)-analogue of the r-Whitney numbers of the second kind are obtained. Finally, the Hankel
transform of the type 2 ( p, q)-analogue of the r-Dowling numbers are established.
Keywords: r-Whitney numbers; r-Dowling numbers; A-tableaux; convolution identities; binomial
transform; Hankel transform
1. Introduction
Several mathematicians were attracted to work on Hankel matrices because of their
connections and applications to some areas in mathematics, physics, and computer science.
Several theories and applications of these matrices were established including the Hankel
determinant and Hankel transform. The notion of Hankel transform was first introduced
in Sloane’s sequence A055878 [1] and was later on studied by Layman [2].
The Hankel matrix H
n
of order n of a sequence A = {a
0
, a
1
,..., a
n
} is defined by
H
n
=( a
i+j
)
0≤i, j≤n
.
On the other hand, the Hankel determinant h
n
of order n of A is defined to be the
determinant of the corresponding Hankel matrix of order n. That is, h
n
= det( H
n
). The
Hankel transform of the sequence A, denoted by H( A), is the sequence {h
n
} 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, a
n
= c
n
+ c
n+1
with c
n
,
the nth Catalan numbers, has the Hankel transform
H( a
n
)= {F
2n+1
}
∞
n=0
,
where F
n
is the nth Fibonacci numbers [2].
One remarkable property of the Hankel transform was established by Layman [2],
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 the binomial transform of
A and C is the inverse transform of A, then,
Axioms 2021, 10, 343. https://doi.org/10.3390/axioms10040343 https://www.mdpi.com/journal/axioms