Notation for newton-like expansion

Question

Is there a compact way of referring to the expression $$a^n + a^{n - 1}b + a^{n - 2}b^2 + \cdots + b^n\:?$$ Maybe some notation I do not know about it.

Thanks!

Answer 1

As @MikeEarnest notes, it's a finite geometric progression with $n+1$ terms, of sum $\frac{a^{n+1}-b^{n+1}}{a-b}$. With classical $q$-analogs it can be written as $a^n[n+1]_{b/a}$, or $b^n[n+1]_{a/b}$.

Answer 2

Using sigma notation, you can write it as $$ \sum_{k=0}^{n}a^{n-k}b^k \, . $$