Explicit formulas and recurrence relations for higher order Eulerian polynomials (Q2012577)

The higher order Eulerian polynomials \(A_n^{(\alpha)}(t)\) are defined by the exponential generating function \[ \left(\frac{1-t}{e^{x(t-1)}-t}\right)^\alpha=\sum_{n=0}^\infty A_n^{(\alpha)}(t)\frac{x^n}{n!}. \] In this paper the authors prove a simple representation for these numbers in terms of the Euler gamma function and Stirling numbers, find a nonlinear ordinary differential equation for the above generating function (with respect to the variable \(x\)), and find some recurrences for \(A_n^{(\alpha)}(t)\). Among other things, it is proven that the well known relation \[ A_n(t)=\sum_{k=0}^nk!S(n,k)(t-1)^{n-k} \] generalizes to \[ A_n^{(\alpha)}(t)=\frac{1}{\Gamma(\alpha)}\sum_{k=0}^n\Gamma(\alpha+k)S(n,k)(t-1)^{n-k}. \]