Hankel Transform of (q,r)-Dowling Numbers

Abstract

In this paper, we establish certain combinatorial interpretation for $q$-analogue of $r$-Whitney numbers of the second kind defined by Corcino and Ca\~{n}ete in the context of $A$-tableaux. We derive convolution-type identities by making use of the combinatorics of $A$-tableaux. Finally, we define a $q$-analogue of $r$-Dowling numbers and obtain some necessary properties including its Hankel transform.

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

“…whose entries are the elements of the sequence A = (a n ) ∞ n=0 was defined in [16] as the Hankel matrix of order n of a sequence A, denoted by H n . This can also be written as H n = (a i+j ) 0≤i,j≤n .…”

“…This can also be written as H n = (a i+j ) 0≤i,j≤n . In the same paper [16], the Hankel determinant h n of order of n of A was defined as the determinant of the corresponding Hankel matrix of order n, (i.e. h n = det(H n )) and the Hankel transform of the sequence A, denoted by H(A), was defined as the sequence {h n } of Hankel determinants of A.…”

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

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

“…where T A r (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 in [16]:…”