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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!

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    $\begingroup$ It's a finite geometric series (with ratio $r=b/a$), so use the geometric series formula. $\endgroup$ – Mike Earnest Feb 28 at 21:27
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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 $b^n[n+1]_{a/b}$.

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  • $\begingroup$ 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 $\,[n]_{q,1} = [n]_{1,q} = [n]_q.$ $\endgroup$ – Somos Feb 28 at 21:52
  • $\begingroup$ @Somos Maybe this can be a starting point for the invention of $q$-analogs with $d$-dimensional $q$. $\endgroup$ – J.G. Feb 28 at 22:26
  • $\begingroup$ Actually, I propose homogeneous versions of all of the classical polynomials such as Bernoulli, Chebyshev, Fibonacci, Hermite, Legendre, Laguerre, Lucas, and so on. $\endgroup$ – Somos Feb 28 at 22:32
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Using sigma notation, you can write it as $$ \sum_{k=0}^{n}a^{n-k}b^k \, . $$

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  • $\begingroup$ that's great but I was thinking something more like {n \choose k} or similar. $\endgroup$ – Bob Vance Feb 28 at 21:25
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    $\begingroup$ @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. $\endgroup$ – J.G. Feb 28 at 21:25
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    $\begingroup$ @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 $S$. $\endgroup$ – Joe Feb 28 at 21:29
  • $\begingroup$ @J.G. that's exactly what I was looking for: something as elegant as the compact symbol for "n choose k". $\endgroup$ – Bob Vance Feb 28 at 21:33

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