Eulerian numbers with fractional order parameters

Summary

The aim of this paper is to generalize the well-known “Eulerian numbers”, defined by the recursion relationE(n, k) = (k + 1)E(n − 1, k) + (n − k)E(n − 1, k − 1), to the case thatn ∈ ℕ is replaced by α ∈ ℝ. It is shown that these “Eulerian functions”E(α, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(α, k) satisfy recursion formulae, they are monotone ink and, as functions of α, are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(α, k), α > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx ^α in terms of a sum involving theE(α, k) and a hypergeometric function.