Recall that a [$q$-Pochhammer symbol][1] is defined as
$$
(x)_n = (x;q)_n := \prod_{l=0}^{n-1}(1-q^l x).
$$

I found the following curious $q$-series identity that seems to hold for any $n\geq 0$:
$$
(-1)^{n+1}q^{\frac{(n+1)(3n+2)}{2}}\sum_{j\geq 0}q^{j}(q^{j+1})_{n}(q^{j+2n+2})_{\infty} \overset{?}{=} \sum_{\substack{k\in \mathbb{Z}\\ |k| > n}}(-1)^{k}q^{\frac{k(3k-1)}{2}}.
$$
Note, the right-hand side is a truncated version of the [Euler function][2]
$$
\phi(q) := (q)_{\infty} = \sum_{k\in \mathbb{Z}}(-1)^{k}q^{\frac{k(3k-1)}{2}}.
$$

How can we prove the above identity? Any suggestions/ideas would be greatly appreciated! 


  [1]: https://en.wikipedia.org/wiki/Q-Pochhammer_symbol
  [2]: https://en.wikipedia.org/wiki/Euler_function