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The generalized Rogers-Ramanujan identity has the following form

$$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots (x)_{k_{r-1}-k_r}(x)_{k_r}}=\prod_{j>0,\, j\not\equiv 0,\pm i\pmod{2r+1}}\frac{1}{1-x^j},$$

where $r\geq 1, 1\leq i\leq r+1$ and $(x)_n:= (1-x)(1-x^2)\cdots (1-x^n)$ .

I am looking for a Rogers-Ramanujan type identity, where in the RHS, the product is over all $j\geq u+1$ (for a given $u\in\mathbb{N}$), if we let $r\to\infty$. For example, the RHS could be

$$\prod_{j>0,\, j\not\equiv 0,\pm 1,\ldots ,\pm u\pmod{2r+1}}\frac{1}{1-x^j},$$ what could be the LHS in this case? Are such identities there in the literature? Clearly, for $u=1$, the the above mentioned Rogers-Ramanujan identity with $i=1$ is the answer.

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