# 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

## 2 Answers

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
• Actually, I propose homogeneous versions of all of the classical polynomials such as Bernoulli, Chebyshev, Fibonacci, Hermite, Legendre, Laguerre, Lucas, and so on. – Somos Feb 28 at 22:32

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

• that's great but I was thinking something more like {n \choose k} or similar. – Bob Vance Feb 28 at 21:25
• @BobVance The expression doesn't include binomial coefficients. Or do you mean you want a compact symbol as elegant as that? If so, see my answer. – J.G. Feb 28 at 21:25
• @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
• @J.G. that's exactly what I was looking for: something as elegant as the compact symbol for "n choose k". – Bob Vance Feb 28 at 21:33