New Results on Higher-Order Daehee and Bernoulli Numbers and Polynomials

We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and short proofs of many previous results on higher order Daehee numbers and polynomials. Moreover, we obtained recurrence relation, explicit formulas and some new results for these numbers and polynomials. Furthermore, we investigated the relation between these numbers and polynomials and Stirling numbers, Norlund and Bernoulli numbers of higher order. The results of this article gives a generalization of the results derived very recently by El-Desouky and Mustafa [6].


Higher order Daehee Numbers and Polynomials
In this section, we derive an explicit formulas and recurrence relations for the higher order Daehee numbers and polynomials of the first and second kinds. Also the relation between these numbers and Nörlund numbers are given. Furthermore, we introduce the matrix representation of some results for higher order Daehee numbers and polynomial obtained by Kim et al. [8] in terms of Stirling numbers, Nörlund numbers and Bernoulli numbers of higher order and investigate a simple and short proofs of these results. Kim et al. [8] defined the Daehee numbers of the first kind of order k, by the following generating function ∞ n=0 D (k) n t n n! = log (1 + t) t k . (2.1) Next, an explicit formula for D (k) n is given by the following theorem. Using Cauchy rule of product of series, we obtain ∞ n=0 D (k) n t n+k n! = ∞ r=k ∞ l 1 +l 2 +···+l k =r (−1) r−k l 1 l 2 · · · l k t r , let r − k = n, in the right hand side, we have ∞ n=0 D (k) n t n+k n! = ∞ n=0 ∞ l 1 +l 2 +···+l k =n+k (−1) n l 1 l 2 · · · l k t n+k .
Equating the coefficients of t n+k on both sides yields (2.2). This completes the proof.
Kim et al. [8, 2014, Theorem 1] proved that, see [16], for n ∈ Z, k ∈ N, we have We can represent the Daehee numbers of the first kind of order k, by (n+1)×(k+1) matrix , 0 ≤ k ≤ n, as follows Kim et al. [8,Theorem 4], proved the following result. For n ∈ Z, k ∈ N, we have where D (k) is (n + 1) × (k + 1), 0 ≤ k ≤ n, matrix for the Daehee numbers of the first kind of order k and S 2 is (n + 1) × (n + 1) lower triangular matrix for the Strirling numbers of the second kind and B (k) is (n + 1) × (k + 1), 0 ≤ k ≤ n, matrix for the Bernoulli numbers of order k.
Kim et al. [8,Theorem 3] introduced the following result. For n ∈ Z, k ∈ N, we have We can write this relation in the matrix form as follows where S 1 is (n + 1) × (n + 1) lower triangular matrix for the Strirling numbers of the first kind. For example, if setting 0 ≤ n ≤ 3, 0 ≤ k ≤ n , in (2.7), we have Remark 2.4. Using the matrix form (2.7), we easily derive a short proof of Theorem 4 in Kim et al. [8]. Multiplying both sides by the Striling number of second kind as follows.
where I is the identity matrix of order (n + 1).
Kim et al. [8] defined the Daehee polynomials of order k by the generating function as follows.
Liu and Srivastava [14] define the Nörlund numbers of the second kind b (x) n as follows. (2.9) Next, we find the relation between the Daehee polynomials of order k and the Nörlund numbers of the second kind b (2.10) Proof. From Eq. (2.9), by multiplying both sides by (1 + t) z , we have (2.14) The relation between the Bernoulli numbers and Bernoulli polynomials of order k are given by Kimura [13], as follows.
Therefore, we can represent (2.15) in the matrix form Bernoulli polynomials of order k as follows where the column k represents the Bernoulli polynomials of order k, B (k) is (n + 1) × (k + 1) matrix, 0 ≤ k ≤ n for Bernoulli numbers of order k and the matrix P(x), the Pascal matrix, is (n + 1) × (n + 1) lower triangular matrix defined by For example if setting 0 ≤ n ≤ 3, 0 ≤ k ≤ n in (2.16), we have Kim et al. [8,Theorem 5] introduced the following result. For n ∈ Z, k ∈ N, We can write this relation in the matrix form as follows where D (k) (x) is (n + 1) × (k + 1) matrix for the Daehee polynomials of the first kind with order k and Kim et al. [8,Theorem 7] introduced the following result. For n ∈ Z, k ∈ N, We can write Eq. (2.19) in the matrix form as follows Remark 2.6. We can prove Theorem 7 in Kim et al. [8] Kim et al. [8] defined the Daehee numbers of the second kind of order k by the generating function as follows. (2.21) Kim et al. [8,Theorem 8] introduced the following result. For n ∈ Z, k ∈ N, where n l = (−1) n−l s 1 (n, l) = |s 1 (n, k)| = s(n, k), where s(n, k) is the signless Stirling numbers of the first kind, see [2] and [4,5].
We can write this theorem in the matrix form as followŝ whereD (k) is (n + 1) × (k + 1) matrix of Daehee numbers of the second kind with order k and S is (n + 1) × (n + 1) lower triangular matrix for the signless Stirling numbers of first kind.
Kim et al. [8,Theorem 9] introduced the following result. For n ∈ Z, k ∈ N, we have We can write Eq. (2.24) in the matrix form as follows whereS 2 is (n + 1) × (n + 1) lower triangular matrix for signed Stirling numbers of the second kind defined by Remark 2.7. We can prove Theorem 9 in Kim et al. [8] by using the matrix form (2.23) as follows.
Multiplying both sides of (2.23) by the matrix of sign Striling numbers of second kindS 2 we havẽ we obtain Eq. (2.25), where we used the identity,S 2 S = I.
Kim et al. [8] defined the Daehee polynomials of the second kind of order k by the generating function as follows.

The λ-Daehee Numbers and Polynomials of Higher Order
In this section we introduce the matrix representation for the λ-Daehee numbers and polynomials of higher order given by Kim et al. [9]. Hence, we can derive these results in matrix representation and prove these results simply by using the given matrix forms.
The λ-Daehee polynomials of the first kind with order k can be defined by the generating function n,λ (0) are called the λ-Daehee numbers of order k.

Theorem 3.2.
For m ≥ 0, we have Using Cauchy rule of product of series, we obtain Equating the coefficients of t m on both sides yields (3.7). This completes the proof.
Setting x = 0, in (3.7), we have the following corollary as a special case. Kim et al. [9], defined the λ-Daehee polynomials of the second kind with order k as follows.

Remark 3.4.
We can prove Eq. (3.11) easily by using the matrix form, multiplying Eq.(3.13) by S 2 as follows.

The Twisted λ-Daehee Numbers and Polynomials of Higher Order
Kim et al. [10] defined the twisted λ-Daehee polynomials of the first kind of order k by the generating function λ log (1 + ξt) In the special case, n,ξ (0|λ) are called the twisted λ-Daehee numbers of the first kind of order k.

Remark 4.2.
In fact, it seems that the statement in (4.7) is not correct, the second equation of, Kim et al. [10,Theorem 1]. From (4.9), multiplying both sides by Ξ −1 , we have, then multiplying both sides by S 2 , we have (4.10)

From (4.7) and (4.10), it is clear that there is a contradiction.
In the following theorem we obtained the corrected relation as follows. Proof. From Eq. (4.1), replacing t by (e t − 1)/ξ , we have (4.14) Equating the coefficients of t m on both sides gives (4.11). This completes the proof.
Moreover, we can represent Equation (4.11), in the following matrix form as (4.10).
Kim et al. [10] introduced the twisted λ-Daehee polynomials of the second kind of order k as follows: n,ξ (0|λ), we have the twisted Daehee numbers of second kind of order k. Using Eq. (3.12), we can write (4.18) in the following matrix form.

Remark 4.4.
In fact, it seems that there is something not correct in (4.19), the second equation of Kim et al. [10,Theorem 2]. From (4.20), multiplying both sides by Ξ −1 , we have, multiplying both sides by S 2 , we have From (4.19) and (4.21) there is a contradiction.
We obtained the corrected relation in the following theorem as follows.  Equating the coefficients of t m on both sides gives (4.22). This completes the proof.
Moreover, by using Eq. (3.12), we can represent Equation (4.22), in the following matrix form.