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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.

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

Many properties of these numbers and their combinatorial interpretations can be seen in Mezö [

In this paper we use the following notations ( see [

Let

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

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

where

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

for

Proof. Since

Equating the coefficients of

Using Equation (6) it is easy to prove that

Special cases:

1. Setting

Thus

hence

where

2. Setting

therefore we have

Equating the coefficient of

where

3. Setting

where

4. Setting

5. Setting

where

6. Setting

Sun [

therefore, we have

Equating the coefficient of

where

7. Setting

where

El-Desouky and Gomaa [

hence, we get

thus we have

Equating the coefficient of

Definition 3. The generalized r-Whitney numbers of the second kind

where

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

for

Proof. Since

Equating the coefficient of

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

Proof. The exponential generating function of

where

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

and from Equation (22) we have

The solution of this difference-differential equation is

where

Setting

if

Similarly at

and

by iteration we get Equation (23).

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

Proof. From Equation (23), we get

Equating the coefficient of

Special cases:

1. Setting

Equating the coefficients of

where

2. Setting

hence we have

Equating the coefficients of

3. Setting

where

Remark 7 Setting

exponential generating function of r-Whitney numbers of the second kind, see [

4. Setting

5. Setting

where

6. Setting

Sun [

hence we have

Equating the coefficients of

where

7. Setting

El-Desouky and Gomaa [

therefore we have

Equating the coefficient of

When

Definition 8. The generalized Whitney numbers of the first kind

where

Corollary 1. The generalized Whitney numbers of the first kind

for

Proof. The proof follows directly by setting

Special cases:

1. Setting

2. Setting

3. Setting

where

4. Setting

5. Setting

Definition 9. The generalized Whitney numbers of the second kind

where

Corollary 2. The generalized Whitney numbers of the second kind

for

Proof. The proof follows directly by setting

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

Proof. The proof follows directly by setting

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

Proof. The proof follows directly by setting

Special cases:

1. Setting

where

2. Setting

3. Setting

where

Remark 10. Setting

4. Setting

5. Setting

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 [

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

Equating the coefficients of

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

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

Equating the coefficients of

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

3. El-Desouky [

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

from Equation (63) we get

Equating the coefficients of

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

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

Equating the coefficients of

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

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

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

6. Cakić [

From Eq (5), we have

Also,

using Cauchy rule product, this lead to

therefore, we get

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

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

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

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

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

where

For example if

where

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

where

For example if

where

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

For example if

where

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