An Asymptotic Formula for -Bell Numbers with Real Arguments

The -Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for -Bell numbers with real arguments is obtained. 1. Introduction The -Stirling numbers of the second kind, denoted by , are defined by Broder in [1], combinatorially, to be the number of partitions of the set into nonempty subsets, such that the numbers are in distinct subsets. Several properties of these numbers are established in [1–3]. Further generalization was established in [4] which is called -Stirling numbers. These numbers are equivalent to the -Whitney numbers of the second kind [5] and the Rucinski-Voigt numbers [6]. The sum of -Stirling numbers of the second kind for integral arguments was first considered by Corcino in [7] and was called the -Bell numbers. Corcino obtained an asymptotic approximation of -Bell numbers using the method of Moser and Wyman. Here, we use to denote the -Bell numbers; that is, In a followup study of Mez？ [8], the -Bell numbers were given more properties. One of these is the following exponential generating function: A more general form of Bell numbers, denoted by , was defined in [9] as where the parameters , , and are complex numbers with , , and . In this paper, we define the -Bell numbers with complex argument using the concept of Hankel contour and establish some properties parallel to those obtained by Mez？ in [8]. Moreover, an asymptotic formula of these numbers for real arguments will be derived using the method of Moser and Wyman [10]. 2. -Stirling Numbers of the Second Kind Graham et al. [11] proposed another way of generalizing the Stirling numbers by extending the range of values of the parameters and to complex numbers. This problem was first considered by Flajolet and Prodinger [12] by defining the classical Stirling numbers with complex arguments using the concept of Hankel contour. Recently, the -Stirling numbers with complex arguments, denoted by , were defined in [9] by means of the following integral representation over a Hankel contour : where and are complex numbers with , , and . We know that, for integral case, the -Stirling numbers of the second kind may be obtained by taking . Hence, using (4), we can define the second-kind -Stirling numbers with complex arguments as follows. Definition 1. The -Stirling numbers of the second kind of complex arguments and are defined by where is complex number with and , and the logarithm involved in the functions and is taken to be the principal branch. The Hankel