The generalized r-Whitney numbers

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 are given.


Introduction
The Whitney numbers of the first and second kind were introduced, respectively, by Dowling [9] and Benoumhani [1] as m n (x) n = n k=0 w m,r (n, k)(mx + 1) k , and (mx + 1) n = n k=0 W m,r (n, k)m k (x) k .
The r-Whitney numbers of the first and second kind were introduced, respectively, by Mező [15] as m n (x) n = n k=0 w m,r (n, k)(mx + r) k , and (mx + r) n = n k=0 W m,r (n, k)m k (x) k .
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 obtained the generalized Whitney numbers of the first and second kind by setting r = 1. We investigate some relations between the generalized r-Whitney numbers and Stirling numbers and generalized harmonic numbers in Section 5. Finally, we obtained a matrix representation for these relations.

Setting
Sun [17] defined p-Stirling numbers of the first kind as Equating the coefficient of x i on both sides, we get where i p = (0 p , 1 p , · · · , (n − 1) p ).

Setting
El-Desouky and Gomaa [12] defined the generalized q-Stirling numbers of the first kind by hence, we get Equating the coefficient of [x] i q on both sides, we get 3 The generalized r-Whitney numbers of the second kind Definition 3. The generalized r-Whitney numbers of the second kind W m,r (n, k; α) with parameter α = (α 0 , α 1 , · · · , α n−1 ) are defined by where W m,r;α (0, 0) = 1 and W m,r;α (n, k) = 0 for k > n.
Theorem 5. The generalized r-Whitney numbers of the second kind have the exponential generating function Proof. The exponential generating function of W m,r;α (n, k) is defined by where W m,r;α (n, k) = 0 for n < k. If k = 0 we have Differentiating both sides of Eq. (19) with respect to t. We get and from (17) we havè Solving this difference-differential equation we obtain Eq. (18).
Theorem 6. The generalized r-Whitney numbers of the second kind have the explicit formula Proof. The exponential generating of W m,r;α (n, k) is Equating the coefficient of t n on both sides, we get Eq. (21).
Special cases: Equating the coefficient of x k on both sides, we get n k r n−k = W m,r;0 (n, k), where W m,r;0 (n, k) denotes the generalized Pascal numbers, for more details see [4], [16].

The generalized Whitney numbers
When r = 1, the generalized r-Whitney numbers of the first w m,1;α (n, k) and second kind W m, 1; α(n, k) are reduced, respectively, to numbers which we call the generalized Whitney numbers of the first and second kind which briefly are denoted byw m;α (n, k) andW m;α (n, k), respectively.
Proof. The proof follows directly by setting r = 1 in Eq. (17).
Corollary 11. The generalized Whitney numbers of the second kind have the exponential generating function . (41) Proof. The proof follows directly by setting r = 1 in Eq. (18).

Remark:
Setting α i = 0 and r = 1 in Eq. (18) we obtain the exponential generating function of Whitney numbers of the second kind see [9].

Relations between Whitney numbers and some types of numbers
This section is devoted to drive many important relation between The generalized r-Whitney numbers and different types of Stirling numbers of the first and second kind and the generalized harmonic numbers.

5.1
The relation between the generalized Stirling numbers of the first kind and the generalized r-Whitney numbers of the first kind Comtet [7], [8] defined the generalized Stirling numbers of the first kind as

5.2
The relation between the generalized Stirling numbers of the second kind and the generalized r-Whitney numbers of the second kind Comtet [7] defined the generalized Stirling numbers of the second kind as from Equations (16) from Eq. (16) we have Equating the coefficient of (x) k on both sides, we get Setting r = 1, we get