I am trying to find out what is known about polynomials of the form $$ \sum_{i = 0}^a x^{a - i} y^i - 1 = x^a + x^{a-1}y + \dots + y^a - 1. $$ I tried searching for anything regarding the sums of the Veronese embedding into projective $n$-space, binomial expansions where all coefficients are set to one, and terms related to symmetric polynomials but came up blank. Any suggestions for things to read are greatly appreciated! This feels like the sort of polynomial that has been heavily studied.
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You can write it as $\frac{y^{a+1}-x^{a+1}}{y-x} - 1$, not sure if that helps, but basically you are after polynomials of form $\frac{y^n-x^n}{y-x}$ – Sil Dec 30 '22 at 00:22
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@Sil this is where I ended up getting this polynomial. The link you sent mentions some connections to cyclotomic polynomials I'll have to look at further. Thanks for looking. – Lguy Dec 30 '22 at 00:45
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No problem. They also seem to be Complete homogeneous symmetric polynomials in $2$ variables – Sil Dec 30 '22 at 00:49
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You might be interested in this article about the $q$-analog, e.g. quantum integers, etc. – Sammy Black Dec 30 '22 at 01:05