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