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

Abstract

A q-analogue of r-Whitney numbers of the second kind, denoted by Wm,r[n, k]q, is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the q-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for Wm,r[n, k]q is obtained.

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

“…Just recently, another definition of q-analogue of r-Whitney numbers of the second W m,r [n, k] q was introduced in [13,16] by means of the following triangular recurrence relation…”

“…The second form of q-analogue of W m,r (n, k) is a kind of generalization of the qanalogue considered by Cigler [8]. This q-analogue possessed several properties (see [13]) including certain combinatorial interpretation in terms of A-tableau, which is defined in [27] to be a list φ of column c of a Ferrer's diagram of a partition λ(by decreasing order of length) such that the lengths |c| are part of the sequence A = (r i ) i≥0 , a strictly increasing sequence of nonnegative integers. By making use of the following explicit formula in symmetric function form [13]…”

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].…”

Hankel Transform of (q,r)-Dowling Numbers

“…This type of q-analogue gives the Hankel transform of q-exponential polynomials and numbers which are certain q-analogue of Bell polynomials and numbers. Recently, a qanalogue of r-Whitney numbers of the second kind was defined by Corcino and Cañete [6] parallel to the definition for q-analogue of noncentral Stirling numbers of the second kind as follows:…”