#### Abstract

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 Mező [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 Mező 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 contour starts from below the negative axis, surrounds the origin counterclockwise, and returns to in the half plane , such that it has a distance from the nonpositive real axis.

*Remark 2. *Since and , the integral in (5) converges for all values of and . Also, is a meromorphic function of (for any fixed ) with poles at the nonpositive integers, and it is entire as a function of (for any fixed not a negative integer).

*Remark 3. *By the change of variable on the integral in (4), say , we can express in terms of as follows:
Because of relation (6), every property of -Stirling numbers with complex arguments will have a corresponding property for -Stirling numbers with complex arguments and vice versa. For instance, the -Stirling numbers in [9] satisfy the following relation:
Replacing with , we get
Using (6), we obtain
Thus, we have

On the other hand, the -Stirling numbers in [9] satisfy the relation Again, replacing with , we obtain By (6), we get Thus, we have

Let’s state these relations formally in the following theorem.

Theorem 4. *The -Stirling numbers of the second kind with complex arguments satisfy the following relations:
*

*Remark 5. *We can give an alternative proof of this theorem. That is, by applying integration by parts on (5) and using the fact that , we obtain
This implies (15) immediately. On the other hand, using Newton’s Binomial Theorem, (5) can be expressed as
We know that Hankel’s contour integral is a unit for gamma function over the set of complex numbers; that is,
This implies that
This completes the proof of (16).

The next theorem contains a property for which is analogous to the identity that usually defines Stirling-type numbers.

Theorem 6. *The -Stirling numbers of the second kind with complex arguments satisfy the following relation:
**
where . *

*Proof. *Using (5) with , a nonnegative integer, we obtain
Thus, by (20), we prove the theorem.

*Remark 7. *It is worth mentioning that this type of property was not established for -Stirling numbers with complex arguments. However, replacing with and with in (21), we obtain
Using relation (6), we get
This is the corresponding property for -Stirling numbers with complex arguments.

*Remark 8. *In a separate paper of the present authors, an asymptotic formula for -Stirling numbers of the second kind with real arguments was established using the method of Chelluri et al. [13].

#### 3. -Bell Numbers

Using Cauchy’s integral formula, (2) can be transformed into the following integral representation of : where the integral contour is a small contour encircling the origin. Since is nonnegative in (25), the contour can be deformed into a Hankel contour that starts from below the negative axis, surrounds the origin counterclockwise, and returns to in the half plane . We assume that it is at distance ≤1 from the real axis. Now, let us consider the following definition for the generalization of -Bell numbers where is a complex number.

*Definition 9. *The -Bell numbers of complex argument are defined by
where is complex numbers with and .

Since and , clearly, the integral in (26) converges for all values of . Moreover, is a meromorphic function of with poles at the nonpositive integers.

It can easily be shown that By (16) and (20), we obtain Hence, we have the following theorem.

Theorem 10. *The -Bell numbers are equal to
**
where and are complex numbers.*

By Theorem 10, we have verified that identity in (1) also holds for with complex argument . This means that the numbers in Definition 9 belong to the family of Bell numbers.

The next theorem asserts that also possess a kind of Dobiński’s formula which can easily be shown using Definiton 9 and the expansion of .

Theorem 11. *The -Bell numbers are equal to
**
where and are complex numbers. *

Using (26) and (30), we obtain the following corollary which is a kind of extension of the integral formula obtained by Mező in [8].

Corollary 12. *The following integral identity holds:
*

The -Bell polynomials of Mező [8] satisfy the recurrence relation where and are nonnegative integers. This will reduce to when . Analogous to this relation, we have the following relation which can easily be shown using (28), (15), and (16).

Theorem 13. *The -Bell numbers satisfy the following relation:
**
where and are complex numbers.*

*Proof. *Summing up both sides of (15) gives
By applying (16), we get
Hence, we have
Substituting this to (35) completes the proof of the theorem.

Note that, when the parameters , and are nonnegative integers, Definition 1 is just equivalent to the integral representation of the -Stirling numbers of the second kind in [1]. Hence, the value of is equal to 0.

#### 4. Asymptotic Formula

An asymptotic formula for -Bell numbers was first established by Corcino in [7]. But the formula holds only when is a nonnegative integer. Here, we aim to obtain an asymptotic formula for when is a real number.

Using Definition 9, To obtain an asymptotic formula, we deform the path into the following contour: , where (i) is the line , and , is a small positive number; (ii) is the line segment , going from to the circle ; (iii) and are the reflections in the real axis of and , respectively;(iv) is the portion of the circle , meeting and . The new contour is in the counterclockwise sense. This idea of deforming the contour is also done in [13]. The integrals along , , , and are seen to be It will also be shown that these integrals go to 0 as provided that . To see this, we consider . For the other contours, the estimate can be seen similarly. We look at Note that Turning to , Thus, Consider Choose , so that . Then This implies that

Consequently, we obtain where

Since as , we have the integral along goes to 0 as . So what remains is the integral along . That is, where is a semicircle , . Hence, by Laplace method or following the analysis in [7], In [7], the integration is along a circle about zero with radius . This number is shown to be the unique solution to as a function of (see Lemma 3 [7]). We see that the asymptotic formula for the -Bell numbers obtained in [7] holds for real argument . Thus, we have the following asymptotic formula.

Theorem 14. *The -Bell numbers with real arguments and have the following asymptotic formula:
**
where
**
and is the unique positive solution to
**
as a function of . *

An asymptotic formula for -Bell numbers has already been established in [14]. However, the formula holds only when is a nonnegative integer. Here, using the same method as employed above, we can show that this asymptotic formula will also work for the case in which the parameters , , and are real numbers.

Now, for real parameters , , and , we have where the path can also be deformed into the following contour: , such that (i) is the line , , and , is a small positive number; (ii) is the line segment , going from to the circle ; (iii) and are the reflections in the real axis of and , respectively; (iv) is the portion of the circle , meeting and .

Also, the new contour is in the counterclockwise sense. It can easily be shown that, along the path , Hence, as goes to , This can be done similarly along , and . Thus, we have where is a semicircle , . Then

This implies that the asymptotic formula for the -Bell numbers obtained in [14] holds for real argument. Thus, from [14], we have the following asymptotic formula for -Bell numbers with real arguments , , and : where and is the unique positive solution to as a function of .

#### 5. Summary and Recommendation

In this paper, we have defined -Stirling numbers of the second kind and -Bell numbers with complex arguments using the concept of Hankel contour and established some properties parallel to those of the classical Stirling and Bell numbers. Moreover, we have derived an asymptotic formula for -Bell numbers as well as for -Bell numbers for real arguments using the method of Chelluri and that of Moser and Wyman.

We observe that, by employing those methods, one can possibly establish an asymptotic formula for -Stirling numbers of the second kind with real arguments and, consequently, using relation (6), an asymptotic formula for -Stirling numbers with real arguments.

#### Acknowledgment

The authors would like to acknowledge the support from the Office of the Vice Chancellor for Research andDevelopment of the University of the Philippines Diliman for this research project. The authors also wish tothank the referees for reading and evaluating the paper thoroughly.