Applied Mathematics
Vol.08 No.01(2017), Article ID:73870,16 pages
10.4236/am.2017.81010

The Generalized r-Whitney Numbers

B. S. El-Desouky, F. A. Shiha, Ethar M. Shokr

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Copyright © 2017 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

Received: September 12, 2016; Accepted: January 22, 2017; Published: January 26, 2017

ABSTRACT

In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given.

Keywords:

Whitney Numbers, r-Whitney Numbers, p-Stirling Numbers, Generalized q-Stirling Numbers, Generalized Stirling Numbers

1. Introduction

The r-Whitney numbers of the first and second kind were introduced, respec- tively, by Mezö [1] as

(1)

(2)

Many properties of these numbers and their combinatorial interpretations can be seen in Mezö [2] and Cheon [3] . At the r-Whitney numbers are reduced to the Whitney numbers of Dowling lattice introduced by Dowling [4] and Benoumhani [5] .

In this paper we use the following notations ( see [6] [7] [8] ):

Let where are real numbers.

(3)

(4)

where,

This paper is organized as follows:

In Sections 2 and 3 we derive the generalized r-Whitney numbers of the first and second kind.The recurrence relations and the generating functions of these numbers are derived. Furthermore, some interesting special cases of these numbers are given. In Section 4 we obtain the generalized Whitney numbers of the first and second kind by setting. We investigate some relations between the generalized r-Whitney numbers and Stirling numbers and genera- lized harmonic numbers in Section 5. Finally, we obtain a matrix represen- tation for these relations in Section 6.

2. The Generalized r-Whitney Numbers of the First Kind

Definition 1. The generalized r-Whitney numbers of the first kind with parameter

are defined by

(5)

where and for.

Theorem 2. The generalized r-Whitney numbers of the first kind satisfy the recurrence relation

(6)

for and.

Proof. Since, we have

Equating the coefficients of on both sides, we get Equation (6).

Using Equation (6) it is easy to prove that .

Special cases:

1. Setting for, hence Equation (5) is reduced to

(7)

Thus

(8)

hence

(9)

where and is Kronecker’s delta.

2. Setting for hence Equation (5) is reduced to

(10)

therefore we have

(11)

Equating the coefficient of on both sides, we get

(12)

where.

3. Setting for hence Equation (5) is reduced to

(13)

where and are the r-Whitney numbers of the first kind.

4. Setting for and hence are the noncentral Whitney numbers of the first kind, see [9] .

5. Setting for and, hence Equation (5) is reduced to

(14)

where and are the translated Whitney numbers of the first kind defined by Belbachir and Bousbaa [10] .

6. Setting for hence Equation (5) is reduced to

(15)

Sun [11] defined p-Stirling numbers of the first kind as

therefore, we have

Equating the coefficient of on both sides, we get

(16)

where.

7. Setting and for Equation (5) is reduced to

(17)

where.

El-Desouky and Gomaa [12] defined the generalized q-Stirling numbers of the first kind by

(18)

hence, we get

thus we have

(19)

Equating the coefficient of on both sides, we get

(20)

3. The Generalized r-Whitney Numbers of the Second Kind

Definition 3. The generalized r-Whitney numbers of the second kind with parameter are defined by

(21)

where and for.

Theorem 4. The generalized r-Whitney numbers of the second kind satisfy the recurrence relation

(22)

for, and.

Proof. Since we have

Equating the coefficient of on both sides, we get Equation (22).

From Equation (22) it is easy to prove that.

Theorem 5. The generalized r-Whitney numbers of the second kind have the exponential generating function

(23)

Proof. The exponential generating function of is defined by

(24)

where for. If we have

Differentiating both sides of Equation (24) with respect to t, we get

(25)

and from Equation (22) we have

The solution of this difference-differential equation is

(26)

where

(27)

Setting in Equation (26) and Equation (27), we get

(28)

if then substituting in Equation (28), we get

(29)

Similarly at we get

(30)

and

(31)

by iteration we get Equation (23).

Theorem 6. The generalized r-Whitney numbers of the second kind have the explicit formula

(32)

Proof. From Equation (23), we get

Equating the coefficient of on both sides, we get Equation (32).

Special cases:

1. Setting for, hence Equation (21) is reduced to

(33)

Equating the coefficients of on both sides, we get

(34)

where denotes the generalized Pascal numbers, for more details see [13] , [14] .

2. Setting for, hence Equation (21) is reduced to

(35)

hence we have

(36)

Equating the coefficients of on both sides, we get

(37)

3. Setting for, hence Equation (21) is reduced to

(38)

where are the r-Whitney numbers of the second kind.

Remark 7 Setting in Equation (23) and using the identity given by Gould [15] , we obtain the

exponential generating function of r-Whitney numbers of the second kind, see [1] , [3] .

4. Setting for and hence Equation (21) is reduced to the noncentral Whitney numbers of the second kind see, [9] .

5. Setting for, hence Equation (21) is reduced to

(39)

where and are the translated Whitney numbers of the second kind defined by Belbachir and Bousbaa [10] .

6. Setting for, hence Equation (21) is reduced to

(40)

Sun [11] defined the p-Stirling numbers of the second kind as

hence we have

Equating the coefficients of on both sides, we get the identity

where.

7. Setting and for, hence Equation (21) is reduced to

(41)

El-Desouky and Gomaa [12] defined the generalized q-Stirling numbers of the second kind as

therefore we have

Equating the coefficient of on both sides we get

4. The Generalized Whitney Numbers

When, the generalized r-Whitney numbers of the first and second kind and, respectively, are reduced to numbers which we call the generalized Whitney numbers of the first and second kind, which briefly are denoted by and.

4.1. The Generalized Whitney Numbers of the First Kind

Definition 8. The generalized Whitney numbers of the first kind with parameter are defined by

(42)

where and for.

Corollary 1. The generalized Whitney numbers of the first kind satisfy the recurrence relation

(43)

for, and.

Proof. The proof follows directly by setting in Equation (6).

Special cases:

1. Setting for in Equation (42), we get

(44)

2. Setting in Equation (42), for, we get,

(45)

3. Setting for in Equation (42), we get

(46)

where are the Whitney numbers of the first kind.

4. Setting for in Equation (42), we get

(47)

5. Setting and for in Equation (42), we get

(48)

4.2. The Generalized Whitney Numbers of the Second Kind

Definition 9. The generalized Whitney numbers of the second kind with parameter are defined by

(49)

where and for.

Corollary 2. The generalized Whitney numbers of the second kind satisfy the recurrence relation

(50)

for, and.

Proof. The proof follows directly by setting in Equation (22).

Corollary 3. The generalized Whitney numbers of the second kind have the exponential generating function

(51)

Proof. The proof follows directly by setting in Equation (23).

Corollary 4. The generalized Whitney numbers of the second kind have the explicit formula

(52)

Proof. The proof follows directly by setting in Equation (32).

Special cases:

1. Setting for, in Equation (49), then we get

(53)

where are the Pascal numbers.

2. Setting for, in Equation (49), then we get

(54)

3. Setting for, in Equation (49), then we get

(55)

where are the Whitney numbers of the second kind.

Remark 10. Setting and in Equation (23) we obtain the exponential generating function of Whitney numbers of the second kind, see [4] .

4. Setting for, in Equation (49), we get

5. Setting and for in Equation (49), we get

(56)

5. Relations between Whitney Numbers and Some Types of Numbers

This section is devoted to drive many important relations between the gene- ralized r-Whitney numbers and different types of Stirling numbers of the first and second kind and the generalized harmonic numbers.

1. Comtet [7] , [16] defined the generalized Stirling numbers of the first and second kind, respectively by,

(57)

(58)

substituting Equation (57) in Equation (5), we obtain

Equating the coefficients of on both sides, we have

(59)

This equation gives the generalized Stirling numbers of the first kind in terms of the generalized r-Whitney numbers of the first kind. Moreover, setting we get

(60)

2. From Equation (21) and Equation (58), we have

Equating the coefficients of on both sides, we have

(61)

which gives the generalized r-Whitney numbers of the second kind in terms of the generalized Stirling numbers of the second kind. Moreover setting we get

(62)

3. El-Desouky [17] defined the multiparameter noncentral Stirling numbers of the first and second kind, respectively by,

(63)

(64)

using Equation (21) and Equation (2), we have

(65)

from Equation (63) we get

Equating the coefficients of on both sides, we have

(66)

This equation gives the generalized r-Whitney numbers of the second kind in terms of r-Whitney numbers of the second kind and the multiparameter noncentral Stirling numbers of the first kind. Moreover setting we get

(67)

4. From Equation (64) and Equation (5), we have

Equating the coefficients of on both sides, we get

(68)

which gives the multiparameter noncentral Stirling numbers of the second kind in terms of the generalized r-Whitney numbers of the first kind and r-Whitney numbers of the second kind. Also, setting we get

(69)

5. Similarly, from Equation (65) and Equation (64), we get

(70)

Equation (70) gives r-Whitney numbers of the second kind in terms of the multiparameter noncentral Stirling numbers and the generalized r-Whitney numbers of the second kind. Setting we have

(71)

6. Cakić [18] defined the generalized harmonic numbers as

From Eq (5), we have

(72)

Also,

using Cauchy rule product, this lead to

therefore, we get

(73)

From Equation (72) and Equation (73) we have the following identity

(74)

From Equation (59) and Equation (74) we have

(75)

this equation gives the generalized Stirling numbers of the first kind in terms of the generalized Harmonic numbers.

6. Matrix Representation

In this section we drive a matrix representation for some given relations.

1. Equation (66) can be represented in matrix form as

(76)

where, and and are lower triangle matrices whose entries are, respectively, the r-Whitney numbers of the second kind, the multiparameter noncentral Stirling numbers of the first kind and the generalized r-Whitney numbers of the second kind.

For example if and using matrix representation given in [19] , hence Equation (76) can be written as

where

2. Equation (68) can be represented in a matrix form as

(77)

where and and are lower triangle matrices whose entries are, respectively, the generalized r-Whitney numbers of the first kind and the multiparameter noncentral Stirling numbers of the second kind.

For example if hence Equation (77) can be written as

where

,

,

3. Equation (70) can be represented in a matrix form as

(78)

For example if hence Equation (77) can be written as

where

Cite this paper

El-Desouky, B.S., Shiha, F.A. and Shokr, E.M. (2017) The Generalized r-Whitney Numbers. Applied Mathematics, 8, 117-132. http://dx.doi.org/10.4236/am.2017.81010

References

  1. 1. Mezo, I. (2010) A New Formula for the Bernoulli Polynomials. Results in Mathematics, 58, 329-335.
    https://doi.org/10.1007/s00025-010-0039-z

  2. 2. Mezo, I. and Ramírez, J.L. (2016) Some Identities of the r-Whitney numbers. Aequationes Mathematicae, 90, 393-406.
    https://doi.org/10.1007/s00010-015-0404-9

  3. 3. Cheon, G.-S. and Jung, J.-H. (2012) r-Whitney Numbers of Dowling Lattices. Discrete Math., 312, 2337-2348.
    https://doi.org/10.1016/j.disc.2012.04.001

  4. 4. Dowling, T.A. (1973) A Class of Geometric Lattices Passed on Finite Groups. Journal of Combinatorial Theory, Series B, 14, 61-86.

  5. 5. Benoumhani, M. (1996) On Whitney Numbers of Dowlling Lattices. Discrete Mathematics, 159, 13-33.
    https://doi.org/10.1016/0012-365X(95)00095-E

  6. 6. El-Desouky, B.S. and Cakic, N.P. (2011) Generalized Higher Order Stirling Numbers. Mathematical and Computer Modelling, 54, 2848-2857.
    https://doi.org/10.1016/j.mcm.2011.07.005

  7. 7. Comtet, L. (1972) Numbers de Stirling generaux et fonctions symetriques. Comptes Rendus de l’Académie des Siences Paris, 275(Ser. A), 747-750.

  8. 8. Charalambides, Ch.A. (1996) On the q-Differences of the Generalized q-Factorials. Journal of Statistical Planning and Inference, 54:31-34.
    https://doi.org/10.1016/0378-3758(95)00154-9

  9. 9. Mangontarum, M.M., Cauntongan, O.I. and Macodi-Ringia, A.P. (2016) The Noncentral Version of the Whitney Numbers. International Journal of Mathematics and Mathematical Science, Article ID: 6206207, 16 pages.

  10. 10. Belbachir, H. and Bousbaa, I.E. (2013) Translated Whitney and r-Whitney Numbers: A Combinatorial Approach. Journal of Integer Sequences, 16, Article 13.8.6.

  11. 11. Sun, Y. (2006) Two Classes of p-Stirling Numbers. Discrete Mathematics, 306, 2801-2805.
    https://doi.org/10.1016/j.disc.2006.05.016

  12. 12. El-Desouky, B.S. and Gomaa, R.S. (2011) Q-Comtet and Generalized q-Harmonic Numbers. Journal of Mathematical Sciences: Advances and Applications, 211, 52-71.

  13. 13. Call, G.S. and Vellman, D.J. (1993) Pascal Matrices. American Mathematical Monthly, 100, 372-376.
    https://doi.org/10.2307/2324960

  14. 14. Stanimirovic, S. (2011) A Generalization of the Pascal Matrix and Its Properties. Series Mathematics and Informatic, 26, 17-27.

  15. 15. Gould, H.W. (2010) Combinatorial Numbers and Associated Identities. Unpublished Manuscript.
    http://www.math.wvu.edu/~gould/

  16. 16. Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. D Reidel Publishing Company, Dordrecht.
    https://doi.org/10.1007/978-94-010-2196-8

  17. 17. El-Desouky, B.S. (1994) The Multiparameter Noncentral Stirling Numbers. Fibonacci Quarterly, 32, 218-225.

  18. 18. Cakic, N.P. (1995) The Complete Bell Polynomials and Numbers of Mitrinovic. University of Belgrade, Publikacije Elektrotehni_ckog fakulteta, Serija Matematika, 6, 74-78.

  19. 19. Cakic, N.P., El-Desouky, B.S. and Milovanovic, G.V. (2013) Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers. Mediterranean Journal of Mathematics, 10, 375-385.
    https://doi.org/10.1007/s00009-011-0169-x