Eulerian Number - Identities

Identities

The Eulerian numbers are involved in the generating function for the sequence of nth powers

Worpitzky's identity expresses xn as the linear combination of Eulerian numbers with binomial coefficients:

It follows from Worpitzky's identity that

Another interesting identity is given by the following manipulation:

So for we have that the terms on the right side are positive, so we may switch the sum. The terms on the left make a geometric series, and we know that converges. After all of that, we use the above identity to finish the manipulation:

Finally, for we get

Notice that the sum on the right-hand side is the sum of the Eulerian polynomials shown at the top of this page.