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

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.

A q-analogue of generalized translated Whitney numbers: Cigler's approach

“…For future research work, it would be interesting to investigate a q-analogue of Qi-type formula [13] for translated r-Dowling numbers by introducing a q-analogue of generalized translated Whitney-Lah numbers. Moreover, it would also be interesting to obtain the Hankel transform of the q-analogue of translated r-Dowling numbers [11,12,[14][15][16].…”

On Divisibility Property of Type 2 (p, q)-Analogue of r-Whitney Numbers of the Second Kind

“…The corresponding q-analogues of generalized Bell numbers, also known as q-analogues of r-Dowling numbers, were also defined in three forms as (see [3,11,13,15])…”

“…where D m,r [n] q , D * m,r [n] q and D m,r [n] q denote the first, second and third form of the qanalogues of r-Dowling numbers, respectively. The Hankel transforms of D m,r [n] q , D * m,r [n] q and D m,r [n] q were successfully established in [3,11,15].…”

A ( p,q )-analogue of Qi-type formula for r -Dowling numbers

“…Comparing the coefficients of [mt − r|m] k p,q yields (3.4). A q-analogue of r-Dowling numbers has been introduced and investigated in [11,15] in three forms, namely,…”