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Some identities related to Eulerian polynomials and involving the Stirling numbers

Qi Feng (Institute of Mathematics Henan Polytechnic University Jiaozuo, Henan, China + College of Mathematics Inner Mongolia University for Nationalities Tongliao, Inner Mongolia, China + Department of Mathematics College of Science Tianjin Polytechnic University )
Lim Dongkyu (Department of Mathematics Sungkyunkwan University Suwon, South Korea)
Guo Bai-Ni (School of Mathematics and Informatics Henan Polytechnic University Jiaozuo, Henan, China)

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, present two identities for Eulerian polynomials and higher order Eulerian polynomials, and 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.

Keywords: identity, generating function, Eulerian polynomial, higher order Eulerian polynomial, Stirling number, open problem