The mth-order Eulerian Numbers

2023/12/28 by Tian-Xiao He, He, Tian-Xiao · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications

paper · pdf · doi:10.48550/arxiv.2312.17153

Abstract

We define the mth-order Eulerian numbers with a combinatorial interpretation. The recurrence relation of the mth-order Eulerian numbers, the row generating function and the row sums of the mth-order Eulerian triangle are presented. We also define the mth-order Eulerian fraction and its alternative form. Some properties of the mth-order Eulerian fractions are represented by using differentiation and integration. An inversion relationship between second-order Eulerian numbers and Stirling numbers of the second kind is given. Finally, we give the exact expression of the values of the mth-order Eulerian numbers.

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