Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. **My question is whether this identity is true, known, and whether it has a direct proof.** It might also happen that this identity is simpler than I think.

Let $d$ be a natural number. For $0\leq k\leq d$ let $e_k(z_1\dots,z_d)$ denote the $k$th elementary symmetric polynomial in $d$ variables, namely $$e_k(z_1\dots,z_d)=\sum_{1\leq i_1<\dots <i_k\leq d}z_{i_1}\dots z_{i_k}.$$

Let $x$ be a variable. **The relevant identity is
\begin{eqnarray*}
\sum_{k=0}^d (-1)^k(x-d)^{2k}\cdot e_{d-k}((x-d+1)^2,\dots,(x-1)^2,x^2)=\\
d!\cdot (2x-2d+1)\dots (2x-d-1)
(2x-d).\end{eqnarray*}**

Notice that the right hand side is a polynomial in $x$ of degree $d$, while the left hand side has a priori degree at most $2d$ only.