Abstract
We derive a new matrix representation for higher-order Daehee numbers and polynomials, higher-order λ-Daehee numbers and polynomials, and twisted λ-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 obtain recurrence relations, explicit formulas, and some new results for these numbers and polynomials. Furthermore, we investigate the relation between these numbers and polynomials and Stirling, Norlund, and Bernoulli numbers of higher-order. Some numerical results and program are introduced using Mathcad for generating higher-order Daehee numbers and polynomials. The results of this article generalize the results derived very recently by El-Desouky and Mustafa (Appl. Math. Sci. 9(73):3593-3610, 2015).
Highlights
For α ∈ N, the Bernoulli polynomials of order α are defined by t et – α ext = ∞ B(nα) (x) tn n! ( ) n=When x =, B(nα) = B(nα)( ) are the Bernoulli numbers of order α defined by t et – α =The Daehee polynomials are defined bylog( + t) t ( + t)x =
We introduce the matrix representation of some results for higher-order Daehee numbers and polynomials obtained by Kim et al [ ] in terms of Stirling numbers, Nörlund numbers, and Bernoulli numbers of higher order and give simple and short proofs of these results
The λ-Daehee polynomials of the first kind of order k can be defined by the generating function λ log( + t) ( + t)λ
Summary
Remark Using the matrix form ( ), we derive a short proof of Theorem in Kim et al [ ] Multiplying both sides by the Stirling number of second kind, we get. Remark We can prove Theorem in Kim et al [ ] by using the matrix form ( ) as follows Multiplying both sides of ( ) by the Stirling number of second kind, we have. Kim et al [ ] defined the Daehee numbers of the second kind of order k by the generating function as follows: Dkn (k). Multiplying both sides of ( ) by the matrix of sign Stirling numbers of second kind S , we have.
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