  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.
