Gabor Frames for a Rational Window with Symmetric Poles

We investigate the Gabor frame problem for a rational window $$g(x)=\frac{x}{(x^2+1)(x^2+4)}\mathrm{.}$$ This window falls outside the classes covered by existing frame set characterizations for rational Herglotz functions and ratios of exponential polynomials. For rational densities ($\alpha \beta =p\mathrm{/}q$), we leverage the Zak transform to map the Gabor frame condition directly to a finite-dimensional equivalence criterion, which is completely determined by the rank of an associated polynomial matrix. Applying this framework, we establish new frame results for every rational density $$\alpha \beta =\frac{p}{q}\in (0,\frac{1}{2})\cup (\frac{1}{2},\frac{2}{3})\cup (\frac{2}{3},\frac{5}{7}]\mathrm{.}$$