# Notation for newton-like expansion

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!

• It's a finite geometric series (with ratio $r=b/a$), so use the geometric series formula. – Mike Earnest Feb 28 at 21:27

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 $$b^n[n+1]_{a/b}$$.
• The nice feature of expression in question is that it is homogeneous in $\,a,b\,$ and symmetric. I propose $\,[n\!+\!1]_{a,b}\,$ whose special cases are $\,[n]_{q,1} = [n]_{1,q} = [n]_q.$ – Somos Feb 28 at 21:52
• @Somos Maybe this can be a starting point for the invention of $q$-analogs with $d$-dimensional $q$. – J.G. Feb 28 at 22:26
Using sigma notation, you can write it as $$\sum_{k=0}^{n}a^{n-k}b^k \, .$$
• @BobVance: I don't know of any specific notation to refer to this series, but you could say 'let $S=\sum_{k=0}^{n}a^{n-k}b^k$' and then whenever you want to refer to the series you can reference $S$. – Joe Feb 28 at 21:29