I was fiddling around with a family of probabilistic models and came across two "identities", which appear to be linked to generalized Frobenius partitions (more on this below). I would be grateful if anyone with expertise in this area can answer the following questions:

1. Are these known "identities"? If so where I find them?
2. Are these equivalent to known "identities" by some non-trivial means? If so then how?
3. Can the sums $S_{\text{even}}(q,t)$ and $S_{\text{odd}}(q,t)$ be written as products?

The identities:

$2\sum_{m\in \mathbb{Z}}S_{\text{even}}(q,t) q^{m^2+m}z^{2m} = \prod_{n\ge 1}(1+tzq^n+z^2q^{2n})(1+tz^{-1}q^{l-1}+z^{-2}q^{2(l-1)}) + \prod_{n\ge 1}(1-tzq^n+z^2q^{2n})(1-tz^{-1}q^{l-1}+z^{-2}q^{2(l-1)})$

$2t\sum_{m\in \mathbb{Z}}S_{\text{odd}}(q,t) q^{(m+1)^2}z^{2m+1} = \prod_{n\ge 1}(1+tzq^n+z^2q^{2n})(1+tz^{-1}q^{l-1}+z^{-2}q^{2(l-1)}) - \prod_{n\ge 1}(1-tzq^n+z^2q^{2n})(1-tz^{-1}q^{l-1}+z^{-2}q^{2(l-1)})$

Where:

$S_{\text{even}}(q,t) = \sum_{w \in E}q^{(\sum_{i \,\text{odd}}iw_i + \sum_{i \,\text{even}}i(w_i-1))}t^{2(\sum_{i \,\text{odd}}\textbf{1}(w_i \ge 1) + \sum_{i \,\text{even}}\textbf{1}(w_i = 0))}$

$S_{\text{odd}}(q,t) = \sum_{w \in O}q^{(\sum_{i \,\text{even}}iw_i + \sum_{i \,\text{odd}}i(w_i-1))}t^{2(\sum_{i \,\text{even}}\textbf{1}(w_i \ge 1) + \sum_{i \,\text{odd}}\textbf{1}(w_i = 0))}$

$E$ is the set of sequences of non-negative integers which agree with $(0,1,0,1,0,1,...)$ far enough to the right and have no two consecutive $0$'s.

$O$ is the set of sequences of non-negative integers which agree with $(1,0,1,0,1,0,...)$ far enough to the right and have no two consecutive $0$'s.

What I know so far:

In a similar vein to the "general principle" for generalised Frobenius partitions (found in Andrews' book) the product $\prod_{n\ge 1}(1+tzq^n+z^2q^{2n})(1+tz^{-1}q^{l-1}+z^{-2}q^{2(l-1)})$ has a Laurent series expansion in $z$ and the constant term should be counting generalised Frobenius partitions of $n$ (the $q$ exponent) allowing at most 1 repetition in each row and a fixed total number of non-repeats (the $t$ exponent). However it is not obvious (at least not to me) that there is any relationship between this and the sums $S_{\text{even}}(q,t)$ and $S_{\text{odd}}(q,t)$.

Indeed if we specialise by letting $t=1$ (which makes perfect sense probabilistically) then the first few terms can be computed and we get $S_{\text{even}}(q,t) = 1+ q + 3q^2 + 5q^3 + 9q^4 + 14q^5 + 24q^6 + ...$, which appears to agree with Andrews' $\phi_2(q)$, measuring GFP's that have at most $1$ repetition in each row (https://oeis.org/A053993). This is expressible as a product, as in Andrews' book. The other sum, after specialisation, doesn't seem to have a similar interpretation but seems to be expressible as an infinite product too (https://oeis.org/A201077). So, assuming this interpretation of the sum can be proved, we then find that the specialised identities are equivalent to two $2$-variable Jacobi-style identities (which I also haven't been able to identify).

The two $3$-variable identities above seem to be isolating the even/odd $z$ terms of the Laurent expansion in $z$. This is the only purpose of the $\pm$ signs on the RHS. Because of this I expect that these identities will follow by taking the "sign transform" of a simpler identity.

Another observation is that if we let $t=x+x^{-1}$ and $q = x^4$ (another natural thing to do probabilistically) then the sum $S_{\text{even}}(q,t)$ appears to collapse to the usual partition function $\prod_{n \ge 1}(1-x^{2n})^{-1}$ and $S_{\text{odd}}(q,t) = \frac{S_{\text{even}}(q,t)}{(1+x^2)}$. Assuming that this can be proved, we find that the corresponding specialised identities are equivalent to two $2$-variable Jacobi-style identities...and in fact are the even/odd parts of the Jacobi triple product.