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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!

vitamin d
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2 Answers2

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

Joe
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  • that's great but I was thinking something more like {n \choose k} or similar. –  Feb 28 '21 at 21:25
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    @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 '21 at 21:25
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    @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 '21 at 21:29
  • @J.G. that's exactly what I was looking for: something as elegant as the compact symbol for "n choose k". –  Feb 28 '21 at 21:33
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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 $b^n[n+1]_{a/b}$.

J.G.
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  • 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 '21 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 '21 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 '21 at 22:32