Let’s start with the geometric series Then differentiating and multiplying by gives

etc. An easy induction over shows that

where is a polynomial of degree in

**Definition**. In the coefficients of are called *Eulerian numbers* and are denoted by So We define for

**Problem**. Show that for all

**Solution**. We have from that and thus

It is clear now that, on the left-hand side of the above equality, the coefficient of is

**Example**. So, by the above problem, Also

etc.

**Exercise 1**. Given evaluate

**Exercise 2**. Show that

**Exercise 3**. Given can you find the value of