Closed form for the expansion of a polynomial expression with an arbitrary number of terms

Question

I'm working on a problem that requires me to find the generating function of a sequence. I've figured out that the generating function of this sequence will be $(1 + x + x^2 + ... + x^b)^a$, where a and b are arbitrary non-negative integers. However, I'm struggling to find an equivalent closed-form expression. How would I go about finding one? Thanks!

Answer 1

You're probably looking for the identity $$1 + x + \cdots + x^n = \frac{1 - x^{n + 1}}{1 - x}$$ (valid in the realm of generating functions).

Thus your answer simplifies to $$\left(\frac{1 - x^{b + 1}}{1 - x}\right)^a$$