Some combinatorial identities of the r-Whitney-Eulerian numbers

In this paper, we study further properties of a recently introduced generalized Eulerian number, denoted by Am,r(n, k), which reduces to the classical Eulerian number when m = 1 and r = 0. Among our results is a generalization of an earlier symmetric Eulerian number identity of Chung, Graham and Knuth. Using the row generating function for Am,r(n, k) for a fixed n, we introduce the r-Whitney-Euler-Frobenius fractions, which generalize the Euler-Frobenius fractions. Finally, we consider a further four-parameter combinatorial generalization of Am,r(n, k) and find a formula for its exponential generating function in terms of the Lambert-W function.