A q-Analogue of Rucinski-Voigt Numbers

Abstract

A q-analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some properties including the orthogonality and inverse relations with the q-analogue of the limit of the differences of the generalized factorial are obtained.

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

“…In [14], 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 [18]. But they failed to derive it.…”

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

“…The next theorem contains an explicit formula for W m,r [j, n] p,q which is analogous to [13,Equation P4]. This can easily be derived using the inverse relation for the (p, q)-binomial coefficients in [9].…”

“…For instance, the q-analogues of the classical Stirling numbers are defined using different motivations by Carlitz [2], Gould [18], Cigler [4], and Ehrenborg [17]. For the q-analogues of r-Whitneytype numbers, some notable works are that of Corcino et al [16], Corcino and Montero [13], Bent-Usman et al [1], and Mangontarum and Katriel [22]. On the other hand, the (p, q)-analogues of the generalized Stirling numbers by Hsu and Shiue [19] were done separately by Remmel and Wachs [25] and Corcino and Montero [12].…”

A q-analogue of r-Whitney numbers of the second kind and its Hankel transform

“…These numbers are also known as (r, β)-Bell numbers. In the same paper [9], the authors have tried to establish the Hankel transform for the q-analogue of Rucinski-Voigt numbers [12], which are also equivalent to (r, β)-Bell numbers. However, the authors were not successful in their attempt.…”