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.
In this paper, (p, q)-analogues of r-Whitney numbers of the first and second kinds are defined using horizontal generating functions. Several fundamental properties such as orthogonality and inverse relations, an explicit formula, and a kind of exponential generating function are obtained. Moreover, a (p, q)-analogue of r-Whitney-Lah numbers is also defined in terms of a horizontal generating function, where necessary properties are obtained. These properties help develop a (p, q)-analogue of the r-Dowling numbers, particularly, a (p, q)-analogue of a Qi-type formula.
In this paper, we define Hurwitz–Lerch multi-poly-Cauchy numbers using the multiple polylogarithm factorial function. Furthermore, we establish properties of these types of numbers and obtain two different forms of the explicit formula using Stirling numbers of the first kind.
Most identities of Genocchi numbers and polynomials are related to the well-knownBenoulli and Euler polynomials. In this paper, multi poly-Genocchi polynomials withparameters a, b and c are dened by means of multiple parameters polylogarithm. Several properties of these polynomials are established including some recurrence relations and explicit formulas.
Using a certain combinatorial interpretation in terms of set partition, a q-analogue of generalized translated Whitney numbers of the second kind is defined in this paper. Some properties such as the recurrence relation, explicit formula, and certain symmetric formula are obtained. Moreover, a q-analogue of generalized translated Whitney numbers of the first kind is introduced to obtain a q-analogue of the orthogonality and inverse relations of the two kinds of generalized translated Whitney numbers.
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