I encountered the following problem which I believe contains a mistake.
Denote $E(t) = \sum \limits s_kt^k$ and $H(t) = \sum h_kt^k$, where $s_i$ are the elementary symmetric functions of $x_1, \ldots, x_n$ and $h_k$ I don't know what.
1) $\prod \limits_{i = 1}^n (1 + x_it) = E(t)$ -- it's easy.
2) $\prod \limits_{i = 1}^n (1 + x_it)^{-1} = H(t)$ -- it clearly can't be true.
I think that $H(t)$ is actually a formal power series and not a polynomial and there should be a minus sign inside the brackets ( $1 - x_it)$. And I think that $h_k = \sum \limits_{i_1 + i_2 \ldots i_n = k} x_1^{i_1} \ldots x_n^{i_n}$
Then follows
3) $E(-t)H(t) = 1$ which confirms my suspicion about the minus sign.
All in all, my questions are:
1) Does what I wrote make sense?
2) Does this $H(t)$ thing have a name?
3) If it does and it is some standard thing, does my calculation coincide with its definition?
You are right about the minus sign. $h_k$ is the complete homogeneous symmetric polynomial of degree $k$, which as you wrote is the sum of all monomials of degree $k$ in the variables. $H(t)$ is a formal power series because there are infinitely many nonzero $h_k$.
$s_k$ is a very disturbing choice of notation for the elementary symmetric polynomial in my opinion. It makes me wonder what they call Schur polynomials (though they may just not come up).