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ISRN Discrete Mathematics
Volume 2012 (2012), Article ID 592818, 18 pages
http://dx.doi.org/10.5402/2012/592818
Research Article

A -Analogue of Rucinski-Voigt Numbers

Department of Mathematics, Mindanao State University, Marawi City 9700, Philippines

Received 1 August 2012; Accepted 19 September 2012

Academic Editors: L. Ji and W. F. Klostermeyer

Copyright © 2012 Roberto B. Corcino and Charles B. Montero. 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

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 [24] 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 Mező [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 [79].

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 generalization of Whitney numbers of the first kind in Benoumhani's papers [79].

In this paper, we establish a -analogue of and obtain some properties including recurrence relations, explicit formulas, generating functions, and the orthogonality and inverse relations.

2. Definition and Some Recurrence Relations

It is known that a given polynomial is a -analogue of an integer if

For example, the polynomials are the -analogues of the integers , , and , respectively, since

The last two polynomials in (2.2) are called the -factorial and -binomial coefficients, respectively. With these in mind, it is interesting also that, for a given property of an integer , we can find an analogous property for the polynomial . For example, the binomial coefficients satisfy the known inversion formula and Vandermondes identity while the -binomial coefficients satisfy the -binomial inversion formula [3] and -Vandermondes identity [11]

Carlitz [12] defined a -Stirling number of the second kind in terms of a recurrence relation in connection with a problem in abelian groups, such that when , this gives the triangular recurrence relation for the classical Stirling numbers of the second kind

This motivates the authors to define a -analogue of the as follows.

Definition 2.1. For nonnegative integers and and complex numbers and , a -analogue of is defined by where is the sequence , , and for or .

The numbers may be considered as a -analogue of since, when , and, hence, the recurrence relation in (2.10) will give the recurrence relation in for where is the sequence . This fact will also be verified in Section 3 (Remark 3.4).

The above triangular recurrence relation for the -Stirling numbers of the second kind can easily be deduced from (2.10) by taking and .

Clearly, using the initial conditions of , we can have

By repeated application of (2.10), we obtain the following theorem.

Theorem 2.2. For nonnegative integers and and complex numbers and , the -analogue satisfies the following vertical recurrence relation: with initial conditions and for all .

Using the following notation we can now state the horizontal recurrence relation for .

Theorem 2.3. For nonnegative integers and and complex numbers and , the -analogue satisfies the following horizontal recurrence relation: with initial condition and for all .

Proof. To prove (2.15), we simply evaluate its right-hand side using (2.10) and obtain .

It will be shown in Section 3 that

By taking and , (2.13) and (2.15) yield which are exactly the recurrence relations obtained in [13]. When , these further give the Hockey Stick identities.

3. Explicit Formulas and Generating Functions

The next theorem is analogous to that relation in (1.1). This is necessary in obtaining one of the explicit formulas for and the orthogonality and inverse relations of and .

Theorem 3.1. For nonnegative integers and and complex numbers and , the -analogue satisfies the following relation:

Proof. We proceed by induction on . Clearly, (3.1) is true for . Assume that it is true for . Then using Definition 2.1,

The new -analogue of Newton's Interpolation Formula in [14] states that, for we have where , such that when , this can be simplified as

Using (3.1) with , we get which can be expressed further as

Applying the above Newton's Interpolation Formula and the identity in [14] we get

With , we obtain the following explicit formula.

Theorem 3.2. For nonnegative integers and and complex numbers and , the -analogue is equal to

Remark 3.3. We can also prove Theorem 3.2 using the -binomial inversion formula in (2.6). That is, by taking , (3.1) gives
Applying (2.6), we obtain
This is precisely the explicit formula in Theorem 3.2.

Remark 3.4. Note that , , and as . Thus, using property , as . This implies that is a proper -analogue of .
Now, using the explicit formula in Theorem 3.2, we obtain
Applying Cauchy's formula for the product of two power series [3], we get Thus,

Applying the above identity for to the function defined by we can further express the above generating function in terms of a -difference operator. More precisely,

This is a kind of exponential generating function for which is included in the next theorem. Together with this, a rational generating function for is also stated in the theorem that will be used to derive another explicit formula for in homogeneous symmetric function form.

Theorem 3.5. For nonnegative integers and and complex numbers and , the -analogue satisfies the exponential generating function and the rational generating function

Proof. We are done with the proof of the first generating function. We are left to prove the second one and we are going to prove this by induction on . For , we have With and using Definition 2.1, we obtain
Hence, which gives

The rational generating function in Theorem 3.5 can then be expressed as

This sum may be written further as follows.

Theorem 3.6. For nonnegative integers and and complex numbers and , the explicit formula for in homogeneous symmetric function form is given by

This explicit formula is necessary in giving combinatorial interpretation of in the context of 0-1 tableau. Note that when and , Theorem 3.6 yields the -Stirling numbers of the second kind [12]. Moreover, taking and , Theorem 3.6 reduces to

Using the representation given in [15] for the -binomial coefficients, we have

This is the identity that we used in Section 2.

4. Combinatorial Interpretation of

Definition 4.1 (see [15]). A 0-1 tableau is a pair , where is a partition of an integer and is a “filling” of the cells of corresponding Ferrers diagram of the shape with 0's and 1's, such that there is exactly one “1” in each column.

Using the partition , we can construct 60 distinct 0-1 tableaux. Figure 1 below shows one of these tableaux with , elsewhere such that .

592818.fig.001
Figure 1: The 0-1 tableau .

Definition 4.2 (see [15]). An -tableau is a list of column of a Ferrer's diagram of a partition (by decreasing order of length) such that the lengths are part of the sequence , a strictly increasing sequence of nonnegative integers.
Let be a function from the set of nonnegative integers to a ring K. Suppose is an -tableau with columns of lengths . Then, we set

Note that might contain a finite number of columns whose lengths are zero since and if .

From this point onward, whenever an -tableau is mentioned, it is always associated with the sequence .

We are now ready to mention the following theorem.

Theorem 4.3. Let denote a function from to a ring (column weights according to length) which is defined by where and are complex numbers, and is the length of column of an -tableau in . Then

Proof. This can easily be proved using Definition 4.2 and Theorem 3.6.

Now, we demonstrate simple combinatorics of 0-1 tableaux to obtain certain relation for . To start with, we have, from Theorem 4.3, where

Substituting , we obtain

Let where and for some numbers and . Then, with , we have

Now, we are going to count the number of tableaux with columns such that columns are of weight and columns are of weight , . Note that there are tableaux with columns whose lengths are taken from the lengths of the columns of . Since there is a one-to-one correspondence between weights and -tableaux, the number of -tableaux in is equal to the number of possible multisets with in . That is,

Thus, for all , we can generate tableaux with columns whose weights are , . However, there are only

distinct tableaux with columns whose lengths are in . Hence, every distinct tableau with columns, of which are of weight other than , appears times in the collection. Thus,

where denotes the set of all tableaux having columns of weights . Reindexing the double sum, we get where is the set of all tableaux with columns of weights for each . Clearly, . Therefore,

Applying Theorem 4.3 completes the proof of the following theorem.

Theorem 4.4. For nonnegative integers and and complex numbers and , the -analogue satisfies the following identity: where .

Taking , , and , Theorem 4.4 gives

Using (2.16) and (3.26), we obtain the Carlitz identity in [12]. Hence, we can consider the identity in Theorem 4.4 as a generalization of the above Carlitz identity.

5. Orthogonality and Inverse Relations

We notice that (1.4) can be written as

Using (1.1), it can easily be shown that where is the Kronecker delta defined by if , and if . Moreover, the following inverse relations hold:

Relation (5.2) is exactly the orthogonality relation for -Whitney numbers that appeared in [6]. Consequently, the generating functions in and can be transformed, respectively, using (5.4) into the following identities: which will reduce to the following interesting identities for when :

Note that the number can be expressed in terms of the unified generalization of Stirling numbers by Hsu  and  Shiue   [16] as . Hence, the identity in coincides with the identity in [17, Theorem 9] by taking .

Parallel to (5.2), (5.3), and (5.4), we will establish in this section the orthogonality and inverse relations of and .

To derive the orthogonality relation for and , we need to rewrite first (1.5) and (3.1). By taking , (1.5) gives and, by replacing with , (3.1) yields

Using (5.8), (5.7) can be expressed as Thus

Theorem 5.1. For nonnegative integers , , and and complex numbers and , the following orthogonality relation holds: where .

Remark 5.2. It can easily be shown that as . This implies that as . Since as , (5.11) yields (5.2) easily.

Remark 5.3. Let and be two matrices whose entries are and , respectively. That is, and . Then using Theorem 5.1, , the identity matrix of order . This implies that and are orthogonal matrices.
Using the orthogonality relation in Theorem 5.1, we can easily prove the following inverse relation.

Theorem 5.4. For nonnegative integers , , and , and complex numbers and , the following inverse relation holds: where .

Proof. Given , we have

The converse can be shown similarly.

One can easily prove the following inverse relation.

Theorem 5.5. For nonnegative integers , , and and complex numbers and , the following inverse relation holds: where .

Remark 5.6. The exponential and rational generating functions in Theorem 3.5 can be transformed into the following identities for the -analogue of : when , will exactly give , respectively.

Acknowledgments

The authors wish to thank the referees for reading and evaluating the paper. This research was partially funded by the Commission on Higher Education-Philippines and Mindanao State University-Main Campus, Marawi City, Philippines.

References

  1. A. Rucinski and B. Voigt, “A local limit theorem for generalized Stirling numbers,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 35, no. 2, pp. 161–172, 1990. View at Zentralblatt MATH
  2. D. Branson, “Stirling numbers and Bell numbers: their role in combinatorics and probability,” The Mathematical Scientist, vol. 25, no. 1, pp. 1–31, 2000. View at Zentralblatt MATH
  3. L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, The Netherlands, 1974.
  4. Ch. A. Charalambides and J. Singh, “A review of the Stirling numbers, their generalizations and statistical applications,” Communications in Statistics, vol. 17, no. 8, pp. 2533–2595, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. A. Z. Broder, “The r-Stirling numbers,” Discrete Mathematics, vol. 49, no. 3, pp. 241–259, 1984. View at Publisher · View at Google Scholar
  6. I. Mező, “A new formula for the Bernoulli polynomials,” Results in Mathematics, vol. 58, no. 3-4, pp. 329–335, 2010. View at Publisher · View at Google Scholar
  7. M. Benoumhani, “On Whitney numbers of Dowling lattices,” Discrete Mathematics, vol. 159, no. 1–3, pp. 13–33, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Benoumhani, “On some numbers related to Whitney numbers of Dowling lattices,” Advances in Applied Mathematics, vol. 19, no. 1, pp. 106–116, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. M. Benoumhani, “Log-concavity of Whitney numbers of Dowling lattices,” Advances in Applied Mathematics, vol. 22, no. 2, pp. 186–189, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. R. B. Corcino and M. Hererra, “The limit of the differences of the generalized factorial, its q- and p,q-analogue,” Utilitas Mathematica, vol. 72, pp. 33–49, 2007.
  11. G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 96 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. View at Publisher · View at Google Scholar
  12. L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. R. B. Corcino, “On p,q-binomial coefficients,” Integers, vol. 8, no. 1, article A29, 2008.
  14. M.-S. Kim and J.-W. Son, “A note on q-difference operators,” Korean Mathematical Society. Communications, vol. 17, no. 3, pp. 423–430, 2002. View at Publisher · View at Google Scholar
  15. A. de Médicis and P. Leroux, “Generalized Stirling numbers, convolution formulae and p,q-analogues,” Canadian Journal of Mathematics, vol. 47, no. 3, pp. 474–499, 1995. View at Publisher · View at Google Scholar
  16. L. C. Hsu and P. J.-S. Shiue, “A unified approach to generalized Stirling numbers,” Advances in Applied Mathematics, vol. 20, no. 3, pp. 366–384, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. R. B. Corcino, “Some theorems on generalized Stirling numbers,” Ars Combinatoria, vol. 60, pp. 273–286, 2001. View at Zentralblatt MATH