Some Identities Relating to Eulerian Polynomials and Involving Stirling Numbers

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results. 1. Motivations In [6, 7], Kims stated that Eulerian polynomials An(t) for n ≥ 0 can be generated by 1− t ex(t−1) − t = ∞ ∑ n=0 An(t) x n! , t 6= 1 E-mail addresses: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]. 2010 Mathematics Subject Classification. Primary 11B73; Secondary 05A15, 11B68, 11B83, 11C08, 33B10.