Simplifying an expression involving complex exponentials

[jgens]

Homework Statement

Simplify the following expression:
$$\sum_{ \alpha_1 + \cdots + \alpha_n = k} e^{i(\alpha_1 \theta_1 + \cdots + \alpha_n \theta_n)}$$

Homework Equations

$$\alpha_n = k - \sum_{j=1}^{n-1} \alpha_j$$
$$0 = \sum_{j=1}^{n} \theta_j$$

The Attempt at a Solution

I am trying to simplify the expression above into something a little more manageable. This expression comes from determining the characters of irreps of SU(n), so if that gives anyone ideas on how to simplify this, then any advice would be much appreciated.

 

[clamtrox]

I guess the first step I would do is to expand the sum of states over all allowed values of {α₁,α₂,...}. So instead of having 1 sum, try writing it in the "traditional" form with n-1 sums;
$$\sum_{\alpha_1 = 0}^k \sum_{\alpha_2 = ...} ... \sum_{\alpha_{n-1} = ...}$$
There's no sum over the last alpha because you already know it's value.

Then "all" you need to do is to do geometric series over and over again.