The $m$th-order Eulerian Numbers

We define the $m$th-order Eulerian numbers with a combinatorial interpretation. The recurrence relation of the $m$th-order Eulerian numbers, the row generating function and the row sums of the $m$th-order Eulerian triangle are presented. We also define the $m$th-order Eulerian fraction and its alternative form. Some properties of the $m$th-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 $m$th-order Eulerian numbers.