Differential equations for Daehee polynomials and their applications


Dongkyu Lim - School of Mathematical Sciences, Nankai University, Tianjin Ciy, 300071, China.


Recently, differential equations for Changhee polynomials and their applications were introduced by Kim et al. and by using their differential equations, they derived some new identities on Changhee polynomials. Specially, they presented Changhee polynomials \(Ch_{n+N}(x)\) by sums of lower terms of Changhee polynomials \(Ch_{n}(x)\). Compare to the result, Kim et al. described Changhee polynomials \(Ch_{n+N}(x)\) via lower term of higher order Chaghee polynomials by using non-linear differential equations arising from generating function of Changhee polynomials. In the first part of this paper, the author uses the idea of Kim et al. to apply to generating function for Daehee polynomials. From differential equations associated with the generating function of those polynomials, we derive some formulae and combinatorial identities. Also, Kwon et al. developed the method of differential equations from the generating function of Daehee numbers and investigated new explicit identities of Daehee numbers. In the second part of the present paper, the author applies their methods to generating function of Daehee polynomials, and get the explicit representations of Daehee polynomials. And specially we put \(x = 0\) in our results, we can get new representations of Daehee numbers compare to the above results.


Daehee polynomial, Daehee number, differential equations.


[1] Y.-K. Cho, T. Kim, T. Mansour, S.-H. Rim,/ On a (r, s)-analogue of Changhee and Daehee numbers and polynomials,/ Kyungpook Math. J.,/ 55 (2015), 225–232.
[2] B. S. El-Desouky, A. Mustafa,/ New results on higher-order Daehee and Bernoulli numbers and polynomials,/ Adv. Difference Equ.,/ 2016 (2016), 21 pages.
[3] T. Kim,/ Identities involving Frobenius-Euler polynomials arising from non-linear differential equations,/ J. Number Theory,/ 132 (2012), 2854–2865.
[4] T. Kim, D. V. Dolgy, D. S. Kim, J. J. Seo,/ Differential equations for Changhee polynomials and their applications,/ J. Nonlinear Sci. Appl.,/ 9 (2016), 2857–2864.
[5] D. S. Kim, T. Kim,/ Daehee numbers and polynomials,/ Appl. Math. Sci. (Ruse),/ 7 (2013), 5969–5976.
[6] D. S. Kim, T. Kim,/ Identities arising from higher-order Daehee polynomial bases,/ Open Math.,/ 13 (2015), 196–208.
[7] D. S. Kim, T. Kim,/ Some identities for Bernoulli numbers of the second kind arising from a non-linear differential equation,/ Bull. Korean Math. Soc.,/ 52 (2015), 2001–2010.
[8] T. Kim, D. S. Kim,/ A note on nonlinear Changhee differential equations,/ Russ. J. Math. Phys.,/ 23 (2016), 88–92.
[9] T. Kim, D. S. Kim, T. Komatsu, S.-H. Lee,/ Higher-order Daehee of the second kind and poly-Cauchy of the second kind mixed-type polynomials,/ J. Nonlinear Convex Anal.,/ 16 (2015), 1993–2015.
[10] H. I. Kwon, T. Kim, J. J. Seo,/ A note on Daehee numbers arising from differential equations,/ Glob. J. Pure Appl. Math.,/ 12 (2016), 2349–2354.
[11] E.-J. Moon, J.-W. Park, S.-H. Rim,/ A note on the generalized q-Daehee numbers of higher order,/ Proc. Jangjeon Math. Soc.,/ 17 (2014), 557–565.
[12] J. J. Seo, S. H. Rim, T. Kim, S. H. Lee,/ Sums products of generalized Daehee numbers,/ Proc. Jangjeon Math. Soc.,/ 17 (2014), 1–9.
[13] Y. Simsek,/ Apostol type Daehee numbers and polynomials,/ Adv. Stud. Contemp. Math.,/ 26 (2016), 555–566.


XML export