Title: The mth-order Eulerian Numbers
Speaker: Tian-Xiao He (Illinois Wesleyan University)
Abstract: We use a combinatorial interpretation to define Eulerian numbers of order m. The recurrence relations of m-order Eulerian numbers, the row generating functions and the row sums of m-order Eulerian triangles are given. We also define Eulerian fractions of order $m$ and their alternative forms. Some properties of Eulerian fractions of order m are expressed using differentials and integrals. The inversion relationships between second-order Eulerian numbers and Stirling numbers of the second and first kinds are given. We also give exact expressions for Eulerian values of order m. The Eulerian matrices are discussed from the perspective of Riordan array theory.
