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. 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. 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. 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. Dowling, T.A. (1973) A Class of Geometric Lattices Passed on Finite Groups. Journal of Combinatorial Theory, Series B, 14, 61-86.
- 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. 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. Comtet, L. (1972) Numbers de Stirling generaux et fonctions symetriques. Comptes Rendus de l’Académie des Siences Paris, 275(Ser. A), 747-750.
- 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. 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. 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. 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. 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. Call, G.S. and Vellman, D.J. (1993) Pascal Matrices. American Mathematical Monthly, 100, 372-376. https://doi.org/10.2307/2324960
- 14. Stanimirovic, S. (2011) A Generalization of the Pascal Matrix and Its Properties. Series Mathematics and Informatic, 26, 17-27.
- 15. Gould, H.W. (2010) Combinatorial Numbers and Associated Identities. Unpublished Manuscript. http://www.math.wvu.edu/~gould/
- 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. El-Desouky, B.S. (1994) The Multiparameter Noncentral Stirling Numbers. Fibonacci Quarterly, 32, 218-225.
- 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. 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