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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 918513, 7 pages
http://dx.doi.org/10.1155/2013/918513
Research Article

Asymptotic Estimates for Second Kind Generalized Stirling Numbers

1Institute of Mathematics, University of the Philippines, Diliman, 1101 Quezon City, Philippines
2Department of Mathematics, Mindanao State University Main Campus, 9700 Marawi City, Philippines

Received 5 June 2013; Accepted 1 August 2013

Academic Editor: Zhijun Liu

Copyright © 2013 Cristina B. Corcino and Roberto B. Corcino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained and ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie's unified generalization.

1. Introduction

Asymptotic formulas of the classical Stirling numbers have been done by many authors like Temme [1], Moser, and Wyman [2, 3] due to the importance of the formulas in computing values of the numbers under consideration when parameters become large. The Stirling numbers and their generalization, on the other hand, are important due to their applications in statistics, life science, and physics.

The -Stirling numbers [4], the -Whitney numbers of the second kind [5], and the numbers considered by Ruciński and Voigt [6] are exactly the same numbers which can be classified as generalization of the classical Stirling numbers of the second kind. This generalization is in line with the generalization of Hsu and Shuie [7]. For brevity, we can use to denote these numbers. These numbers satisfy the following exponential generating function: where and are positive integers. When , (1) reduces to a generating function of -Stirling numbers [8] of the second kind and further reduces to a generating function of the classical Stirling numbers of the second when . Combinatorial interpretation and probability distribution involving are discussed in [9].

The behavior of the numbers was shown to be asymptotically normal in [4, 6]. That means that the distribution of these numbers when the parameters and are large will follow a bell-shaped distribution. The unimodality of this distribution was also discussed in [4]. Moreover, the bound for the index in which the maximum value of these numbers occurs has been established in [10]. With these properties of the numbers , it is necessary to consider the asymptotic formula for the numbers to be able to compute large values of the numbers.

Applying Cauchy Integral Formula to (1), the following integral representation is obtained: where is a circle about the origin.

The primary purpose of the present paper is to investigate if the analysis in [3] can be extended to the generalized Stirling numbers of the second kind. The authors are motivated by the work of Chelluri et al. [11] which proved the asymptotic equivalence of the formulas obtained by Temme [1] and those obtained by Moser and Wyman in [2, 3]. Moreover, it was also shown in [11] that the formulas obtained apply not only for integral values of the parameters and , but for real values as well. To be able to do a similar investigation with that in [11] for the generalized Stirling numbers of the first and second kinds, it is necessary to come up with asymptotic formulas for each kind generalized Stirling numbers which may have used a method similar to that in [2, 3]. The necessary asymptotic formulas for the generalized Stirling numbers of the first kind can be found in [12, 13]. In this paper, an asymptotic formula for the generalized Stirling numbers of the second kind with integral values of and are obtained using a similar analysis as that in [3]. The formula is proved to be valid when and , where . Moreover, other asymptotic formulas are obtained for , where and are real numbers under the conditions and .

2. Preliminary Results

The integral representation in (2) can be written in the form where , .

Using the representation , for the circle , where and is a positive real number, (3) becomes

Multiplying and dividing the right-hand side of (4) by the correction constant it reduces to where and is the function,

Observe that can be written in the form where is the same function which appeared in [14], except for the value of .

The Maclaurin expansion of is where where is the operator .

Lemma 1. There is a unique such that

Proof. Equation (14) can be written in the form
Let and . Note that is a continuous function on and while . On the other hand, is a line with intercept at and intercept at and is decreasing when and are positive real numbers. Thus, and , as functions of , surely intersect at some point. The value of at the intersection point is the desired solution. Moreover, it can be seen graphically that .

Lemma 2. defined in (12) obeys the inequalities

Proof. This lemma follows from Lemma 2.2 in [3].

Lemma 3. There exists a constant independent of and such that

Proof. defined in (13) can be written as where is the same as that defined in in [3]. Dividing both sides of the preceding equation by and taking the absolute value, we have
With a fixed parameter and as and using Lemma 3.2 in [3], the desired result is obtained.
Define and consider the integral defined by

Lemma 4. A constant exists such that

Proof. We claim that
Using the definition of given by (9), we have
Because , it remains to show that .
Note that . The claim now follows from the fact that and .
From (9),
Thus,
The result now follows from the previous claim and the fact that is the same function that appeared in [3] except for the value of .

Observe that when as , Lemma 4 will imply that as . Indeed, as under some conditions, as shown in the following lemma.

Lemma 5. as provided , .

Proof. Write (14) in the form
Now the application of the mean value theorem to the function over the interval will yield where is a number within the interval . Then,
Consequently,
The last inequality will yield provided . The previous inequality can be written in the form when . The second inequality previous follows if . Indeed, when and because , where and are the functions in the proof of Lemma 1. Hence, which is the -coordinate of the point of intersection of the two functions must be less than . The last inequality shows that as under the given restriction of .
Returning to the condition that , note that . Thus, under the restriction that . So that whenever .

For example, when and , in Lemma 5 must satisfy . Thus, .

3. Asymptotic Formula with Integral Parameters

In the discussion that follows, denotes the unique positive solution to (14). It will be seen later that our final asymptotic formula is expressed in terms of powers of . Anticipating this result and in view of Lemma 4, we write

Substituting (10) for and noting that , (33) becomes

The change of variable will yield ,

Now, (34) becomes where , ,

The equation in (37) can be written in the from where

Note here that is within the radius of convergence of the series in (38). On the other hand, the Maclaurin series of is where is a polynomial in containing only even powers of , and is a polynomial in containing only odd powers of . Now, (36) can be written in the form where and .

We write (41) as where

In view of Lemma 2, ; hence,

Since is a polynomial in , we can replace with in (41). Following the discussion in [3], it can be shown that ; hence, we have the asymptotic formula

Note that

This is why no odd subscript of appears in (45).

An approximation is obtained by taking the first two terms of the sum in (45). Thus,

It can be computed that where while where

Thus, can be written as follows:

Let

Then,

Finally, we obtain the approximation formula

In view of Lemmas 4 and 5 and (33), the previous asymptotic approximation is valid for . The formula obtained in the previous discussion is formally stated in the following theorem.

Theorem 6. The formula behaves as an asymptotic approximation for the generalized Stirling numbers of the second kind with positive integral parameters for , and such that as .

Table 1 displays the exact values and approximate values for , , . The exact values are obtained using recurrence formula while the approximate values are obtained using Theorem 6. In view of Lemma 5, the formula is valid for .

tab1
Table 1

Values on the table affirm that the asymptotic formula in Theorem 6 gives a good approximation for when .

4. Asymptotic Formula with Real Parameters

Recently, the -Stirling numbers for complex arguments were defined in [15] parallel to the definition of Flajolet and Prodinger as where and is a Hankel contour that starts from below the negative axis surrounds the origin counterclockwise and returns to in the half plane . Note that by change of variable, say , we can express as follows: where

The numbers are certain generalization of -Stirling numbers of the second kind [8] in which the parameters involved are complex numbers. These numbers satisfy the following properties.

Theorem 7. For nonnegative real number , one has

Theorem 8. For nonnegative real numbers and and , one has
Furthermore, when , a complex number

The Bernoulli polynomial can be expressed using the explicit formula in [16] and the first formula in Theorem 8, with and , as

Moreover, it is known that, for , the Hurwitz zeta function . Note that, as , . By Cauchy's integral formula which further gives, using a result in [17], the following relation:

An asymptotic formula for -Stirling numbers of the second kind was first considered by Corcino et al. in [14]. However, the formula only holds for integral arguments. In this section, we are going to establish an asymptotic formula for -Stirling numbers of the second kind that will hold for real arguments.

Consider the integral representation in (59) where . To see the analysis of Moser-Wyman applies to -Stirling numbers with real arguments and , we deform the path into the following contour: where(1) is the line and , is a small positive number; (2) is the line segment , going from to the circle ; (3) and are the reflections in the real axis of and , respectively; and (4) 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 [11].

The integrals along , , , and are seen to be

It will also be shown that these integrals go to 0 as provided that where . To see this, we consider . For the other contours, the estimate can be seen similarly.

Note that for in , we can choose so that

From this and the assumptions that and , it follows that

Thus,

Consequently, where is the length of . With the horizontal line , the length of is a linear function of the real part of , given by

Hence, we have

The last inequality follows from the fact that . With the condition that , , and the fact that , it follows that the integral along goes to 0 as . Thus, we have

Using the method of Moser and Wyman [3], we obtain the following asymptotic formula.

Theorem 9. The -Stirling numbers of the second kind with real arguments and have the following asymptotic formula: valid for as , provided that with , where and is the unique positive solution to the equation as a function of .

Chelluri et al. [11] has made a modification of Moser and Wyman formula and analysis. Using this analysis of Chelluri, we can restate Theorem 9 as follows.

Theorem 10. The -Stirling numbers of the second kind with real arguments and have the following asymptotic formula: valid for as , provided that , where and is the unique positive solution to the equation as a function of .

As noted in (58), the -Stirling numbers can be obtained using the -Stirling numbers of the second kind for complex arguments , , , and . This implies, using Theorems 9 and 10, the following asymptotic formulas which agree with the formula in (55) for the integral values of and .

Corollary 11. The -Stirling numbers with real arguments and have the following asymptotic formulas: provided that with , and provided that , where and is the unique positive solution to the equation as a function of .

Acknowledgment

The authors would like to thank the referee for reading and evaluating the paper.

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