A -Analogue of Rucinski-Voigt Numbers

A -analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some properties including the orthogonality and inverse relations with the -analogue of the limit of the differences of the generalized factorial are obtained. 1. Introduction Rucinski and Voigt [1] defined the numbers satisfying the relation where is the sequence and and proved that these numbers are asymptotically normal. We call these numbers Rucinski-Voigt numbers. Note that the classical Stirling numbers of the second kind in [2–4] and the -Stirling numbers of the second kind of？？Broder [5] can be expressed in terms of as follows: where and are the sequences and , respectively. With these observations, may be considered as certain generalization of the second kind Stirling-type numbers. Several properties of Rucinski-Voigt numbers can easily be established parallel to those in the classical Stirling numbers of the second kind. To mention a few, we have the triangular recurrence relation the exponential and rational generating function and explicit formulas The explicit formula in can be used to interpret as the number of ways to distribute distinct balls into the cells ( one ball at a time ), the first of which has distinct compartments and the last cell with distinct compartments, such that(i)the capacity of each compartment is unlimited;(ii)the first cells are nonempty. The other explicit formula can also be used to interpret as the number of ways of assigning people to groups of tables where all groups are occupied such that the first group contains distinct tables and the rest of the group each contains distinct tables. The Rucinski-Voigt numbers are nothing else but the -Whitney numbers of the second kind, denoted by , in Mez？ [6]. That is, . It is worth-mentioning that the -Whitney numbers of the second kind are generalization of Whitney numbers of the second kind in Benoumhani's papers [7–9]. On the other hand, the limit of the differences of the generalized factorial [10] was also known as a generalization of the Stirling numbers of the first kind. That is, all the first kind Stirling-type numbers may also be expressed in terms of by a special choice of the values of and . It was shown in [10] that where is the sequence . Recently, -analogue and -analogue of , denoted by and , respectively, were established by Corcino and Hererra in [10] and obtained several properties including the horizontal generating function for where The numbers are equivalent to the -Whitney numbers of the first kind, denoted by , in [6]. More precisely, . These numbers are