Symmetry fermionic \(p\)-adic \(q\)-integral on \(\mathbb Z_p\) for Eulerian polynomials (Q456499)

Summary: \textit{D. S. Kim} et al. [Abstr. Appl. Anal. 2012, Article ID 269640, 10 p. (2012; Zbl 1253.11026)] introduced an interesting \(p\)-adic analogue of the Eulerian polynomials. They studied some identities on the Eulerian polynomials in connection with the Genocchi, Euler, and tangent numbers. In this paper, by applying the symmetry of the fermionic \(p\)-adic \(q\)-integral on \(\mathbb Z_p\), defined by \textit{T. Kim} [J. Difference Equ. Appl. 14, No. 12, 1267--1277 (2008; Zbl 1229.11152)], we show a symmetric relation between the \(q\)-extension of the alternating sum of integer powers and the Eulerian polynomials.