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Journal of Applied Mathematics
Volume 2012, Article ID 351935, 17 pages
http://dx.doi.org/10.1155/2012/351935
Research Article

A Newton Interpolation Approach to Generalized Stirling Numbers

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Received 16 November 2011; Revised 8 December 2011; Accepted 20 December 2011

Academic Editor: Carlos J. S. Alves

Copyright © 2012 Aimin Xu. 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

We employ the generalized factorials to define a Stirling-type pair which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

1. Introduction

Throughout this paper the following notations will be used. We denote by the set of real numbers and by the set of complex numbers. Let be a vector. If , we denote the vector by . We further denote by and by . Moreover, let us denote the generalized th falling factorial of with increment by . Particularly, if , we write .

In mathematics, Stirling numbers of the first and second kind, which are named after James Stirling, arise in a variety of combinatorics problems. They have played important roles in combinatorics. Stirling numbers of the first kind are the coefficients in the expansion , and Stirling numbers of the second kind are characterized by .

Over the past few decades, there has been an interest in generalizing and extending the Stirling numbers in mathematics literature. By starting with transformations between generalized factorial involving three arbitrary parameters , and , Hsu and Shiue [1] introduced the generalized numbers and unified those generalizations of the Stirling numbers due to Riordan [2], Carlitz [3, 4], Howard [5], Charalambides-Koutras [6], Gould-Hopper [7], Tsylova [8], and others. They define a Stirling-type pair by They systematically investigated many basic properties including orthogonality relations, recurrence relations, generating function, and the Dobinski identity for their Stirling numbers. Recently, Comtet [9] defines and , the generalized Stirling numbers of the first kind and second kind associate with , by El-Desouky [10] modified the noncentral Stirling numbers of the first and second kind. He defined the multiparameter noncentral Stirling numbers of the first kind and second kind as follows: The recurrence relations, generating functions, and explicit forms for El-Desouky's Stirling numbers are obtained.

In another direction, Stirling numbers and their generalizations were investigated via differential operators. Carlitz and Klamkin [11] defined the Stirling numbers of the second kind by where is a differential operator . Actually, this can be traced back at least to Scherk [12]. In the physical literature, Katriel [13] discovered (1.5) was in connection with the normal ordering expressions in the boson creation operator and annihilation , satisfying the commutation relation of the Weyl algebra. Recently, Lang [14, 15] generalized the stirling numbers of the second kind by the following operator identity: where is a nonnegative integer. He further obtained many properties of these numbers. More recently, Blasiak et al. [16] defined , the generalized Stirling numbers of the second kind arising in the solution of the general normal ordering problem for a boson string, as follows These numbers were firstly defined by Carlitz [17]. More generally, given two sequences of nonnegative integers and , Blasiak [18] generalized this formula by where . He gave an explicit formula for the generalized Stirling numbers . In [19], a different explicit expression for these numbers was presented.

By considering powers of the noncommuting variables , satisfying , Mansour and Schork [20] introduced a new family of generalized Stirling numbers as which reduced to the conventional Stirling numbers of second kind and Bell numbers in the case . As mentioned in [21], this type of generalized Stirling numbers is not a special case of Howard's degenerate weight Stirling numbers although they look very similar.

Moreover, for any sequence of real numbers and a sequence of nonnegative integers , by using operational identity [22, 23] El-Desouky and Cakić [24] defined a generalized multiparameter noncentral Stirling numbers of the second kind by where . These numbers reduced to the multiparameter noncentral Stirling numbers of the second kind in (1.3) if all .

As a useful tool, the Newton interpolation with divided differences was utilized to obtain closed formulas for Dickson-Stirling numbers in the paper [25] provided by the referee. In this paper, we make use of the generalized factorials to define a Stirling-type pair which unifies various Stirling-type numbers investigated by previous authors. By using the Newton interpolation and divided differences, we obtain the basic properties including recurrence relations, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated. This paper is organized as follows. In Section 2, we introduce the Newton interpolation and divided differences. Several important properties of divided differences are presented. In Section 3, the definitions of a new family of Stirling numbers are given. According to the definitions, the recurrence relation as well as an explicit formula is derived. Moreover, we also investigate the generating function for our generalized Stirling numbers. In views of our results, we rediscover many interesting special cases which are introduced in the above. Finally, in Section 4, the associated generalized Bell numbers and Bell polynomials are presented. Furthermore, a generalized Dobinski's formula is derived.

2. Divided Differences and Newton Interpolation

For a sequence of points and all or , we define Let be the Newton interpolating polynomial of degree at most that interpolates a function at the point ; then this polynomial is given as in where is the divided difference of the th order of the function . As is well known, for the distinct points , the divided differences of the function are defined recursively by the following formula:

Divided differences as the coefficients of the Newton interpolating polynomial have played an important role in numerical analysis, especially in interpolation and approximation by polynomials and in spline theory; see [26] for a recent survey. They also have many applications in combinatorics [27, 28]. The divided differences can be expressed by the explicit formula From the above expression it is not difficult to find the divided differences are symmetric functions of their arguments. In particular, taking we have where is the difference operator with step size .

Divided differences can be extended to the cases with repeated points. From the recursive formula (2.4), it is clear that if the following holds: If repetitions are permitted in the arguments and the function is smooth enough, then This gives the definition of first-order divided differences with repeated points In general, let . Then the divided differences with repeated points obey the following recursive formula: It is evident that divided differences can be viewed as a discrete analogue of derivatives. If , then all the points are the same. In this case, in (2.2) is the Taylor polynomial of the function at the point . More generally, if and where are distinct, we define with . Recall that the cycle index of symmetric group is one of the essential tools in enumerative combinatorics [29]. Using the cycle index of symmetric group, the divided differences with repeated points can be expressed by the following explicit formula [30] (see also [31]): where

It is well known that the Leibniz formula for higher derivatives is basic and important in calculus. A divided difference form of this formula given by [32] is stated as below. Let . If and are sufficiently smooth functions, then for arbitrary points , This formula is called the Steffensen formula which is a generalization of the Leibniz formula. If , then the Leibniz formula holds, namely,

3. Generalized Stirling Numbers

Let and be two vectors. We define two kinds of Stirling-type numbers as where are called the generalized Stirling numbers of the second kind with the parameters , , and , and are called the generalized Stirling numbers of the first kind. It is obvious that . In particular, is the conventional Stirling number of the second kind, and is of the first kind.

In this section, making use of divided difference operator and the Newton interpolation in Section 2, we will investigate orthogonality relations, recurrences, explicit expressions, and generating functions for the generalized Stirling numbers.

3.1. Basic Properties of the Generalized Stirling Numbers

Firstly, let us consider orthogonality relations of the two kinds of the generalized Stirling numbers. By substituting (3.1) into (3.2) and (3.2) into (3.1), one may easily get the following orthogonality relations respectively, where the Kronecker symbol is defined by if , and if . As a consequence, the inverse relations are immediately obtained:

Next, from the definition (3.1), one may see that can be viewed as the coefficients of the Newton interpolation of the function at the points . Thus, we immediately have the following theorem.

Theorem 3.1. For arbitrary parameters , , and , there holds In particular, if are distinct, we have If , then

This theorem gives the explicit expressions for the generalized Stirling numbers. We can similarly get . By (3.6), we can further get the recurrence relations as follows.

Theorem 3.2. For arbitrary parameters , , and , there holds In particular, we have

Proof. According to (3.6), we have By using the Steffensen formula for divided differences and the basic facts we have This leads to (3.9), and the proof is complete.

Finally, let us consider the generating function of the Stirling numbers denoted by . Assume that is of the form: where is a reference sequence. In this way we treat at the same time the case of ordinary coefficients of and the case of Taylor coefficients . Let . Making use of (3.6), we get the following: This formula is essential and important for getting the generating function of the generalized Stirling numbers. If we get the analytic expression of by choosing special , the analytic expression of is obtained as well.

3.2. Special Cases

Because the parameters , and are arbitrary, our results contain many interesting special cases. In this part we will investigate these special cases. Some results have been derived and some are new.

Let and all be distinct. According to Theorems 3.1 and 3.2, we have the explicit expressions and the recurrence relations for new generalized Stirling numbers .

Corollary 3.3. The numbers have the following explicit expression

Corollary 3.4. The numbers satisfy the following recurrence relation

For and , if we have and if we have Thus, by (3.15) one easily obtain the following theorem.

Theorem 3.5. The sequence has the following exponential generating function:

Our generalized Stirling numbers include the Stirling numbers due to Hsu and Shiue [1], EI-Desouky's multiparameter noncentral Stirling numbers [10], and the so-called Comtet numbers [9] as special cases. Now, let us discuss these special cases as follows.

Example 3.6. Let and . This implies the points are equally spaced with step size . By Corollaries 3.3 and 3.4, we immediately get the explicit expression for the generalized Stirling numbers and the recurrence relation For , the following holds In a similar manner, we can also get the generating function for . Thus, we have
Here is equivalent to in [1]. As mentioned in [1], the generalized Stirling numbers contain serval special cases, for example, two kinds of the classical Stirling numbers, the binomial coefficients, the Lah numbers, Carlitz's two kinds of weighted Stirling numbers [4], Carlitz's two kinds of degenerate Stirling numbers [3], Howard's weighted degenerate Stirling numbers [5], Gould-Hopper's noncentral Lah numbers [7], Riordan's noncentral Stirling numbers [2], the noncentral numbers extensively studied by Charalambides and Koutras [6], Tsylova's numbers [8], Todorov's numbers [33], Nandi and Dutta's associated Lah numbers [34], and the -Stirling numbers of the first kind fully developed by Broder [35]. Hsu and Shiue obtained the recurrence relation for the generalized Stirling numbers , and they also found the generating function by solving a difference-differential equation. However, the formula (3.21) was new and not given by [1]. Obviously, in the present paper we follow a very different approach to rediscover the recurrence relation and the generating function. In the case , one may also refer to [36].
It is remarkable that Mansour and Schork [20] recently considered to generalize the commutation relation . They defined generalized Stirling numbers by (1.9). The explicit expressions of these generalized Stirling numbers are given by [20] (see also [21]), and they are very closely related to the numbers considered by Lang [14]. In [21], the authors exploited many properties of these generalized Stirling numbers. It is interesting that observing our generalized Stirling numbers by and , we find that the Stirling numbers due to Mansour and Schork are actually a special case of ours and Hsu-Shiue's, and they are equivalent to the numbers due to [36]. Thus, by (3.20) we get the exponential generating function of the generalized Stirling numbers due to Mansour and Schork: In [21], the authors gave the generating function for .

Example 3.7. Let , and be arbitrary but distinct, and one can get the multiparameter noncentral Stirling numbers of the first kind introduced by El-Desouky [10]. Here we denote the numbers by . Using Corollaries 3.3 and 3.4 and Theorem 3.5, we rediscover the explicit expression, recurrence relation, and the generating function, namely,

Example 3.8. Let us consider the case . In this case, there holds which is equivalent to Especially, for the Comtet numbers [9] (see also [10]) are defined associated with the sequence by This implies . Thus, it is not difficult to obtain and the exponential generating function It is worth noting that the Comtet numbers can be rewritten as an alternate form which is really the complete symmetric function of th order with respect to the variables , , , . By (3.15), they have the ordinary generating function:
Moreover, if we let , then we get the noncentral Stirling numbers of the second kind defined by Koutras [37] (see also [24]). For more details one refers to [37].

What has been discussed above in this subsection is relevant to the generalized Stirling numbers with equidistance parameters . However, we are also interested in the other cases. In a recent year, many authors [14, 16, 18, 19, 24] were devoted to the generalized Stirling numbers by differential operator. We here rediscover these generalized Stirling numbers by the Newton interpolation.

Example 3.9. Our generalized Stirling numbers also contain the numbers due to Blasiak [18] as a special case. Here we let , and let and for . Moreover, we let where . By using (3.7) we immediately have the explicit expression of as follows which is in accordance with the generalized Stirling numbers introduced by Blasiak [18]. Blasiak got this explicit formula by using the operator to act on . His proof is very different from ours. Recently, El-Desouky et al. [19] found a new expression by successive application of Leibniz formula. The special case and is investigated by Blasiak et al. [16], and they gave us Lang's result [14] as a special case for .

Example 3.10. By operating with (1.10) on and using Cauchy rule of multiplication of series, El-Desouky and Cakić [24] obtain the explicit formula In fact, let . It is not difficult to find holds. In particular, setting , , , , and in , we get the explicit expression of the number of partitions of into nonempty parts such that the distance of any two members in the same part differs from denoted by ; see [38].

4. Generalized Bell Polynomials and Dobinski-Type Formulas

Recall that the Bell numbers and the exponential polynomials are defined, respectively, by the sums The Bell polynomials have the generating function They also satisfy the following remarkable Dobinski-type formula which reduces to the Dobinski formula when . It is worth noting that is represented as an infinite series in .

As we know, the Dobinski-type formulas have been the subject of much combinatorial interest. Thus, it is worth looking for a general Dobinski-type formula.

In this section, we define a generalized Bell polynomials by where and . Naturally, one get an extended definition of generalized Bell numbers as follow: Note that and . We can make use of (3.6) to obtain the following Dobinski-type formula.

Theorem 4.1. For and arbitrary , we have the Dobinski-type formula where is defined by (2.1).

Proof. By (4.4) we have Replacing by in (3.6) yields By equating the coefficient of within the first and last expressions, we arrive at Using the Cauchy product rule gives This implies (4.6) is true and completes the proof.

Letting we directly obtain the generalized Dobinski formula.

Corollary 4.2. For and arbitrary , we have where is defined by (2.1).

It is clear that (4.3) is a special case of (4.6) with .

Let . It is worth noting that the formula (4.6) can be used to obtain a closed sum formula for this type of infinite series. As mentioned in [1], such a type of series cannot be summed by using the hypergeometric series method. Let ; then according to (4.6) we have

Example 4.3. Letting we immediately obtain the Dobinski-type formula due to Hsu and Shiue [1] as follows:

Example 4.4. Letting we have the following Dobinski-type formula due to Blasiak [18]:

Acknowledgments

The author thanks the anonymous referees for their valuable suggestions and comments. This work was supported by the Zhejiang Province Natural Science Foundation (Grant nos. Y6110310 and Y6100021) and the Ningbo Natural Science Foundation.

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