Quartic Fourier-Laplace limits for $(p,q)$-Rogers-Szegő polynomials

We prove asymptotic formulae for positive real $(p,q)$-Rogers-Szeg\H{o} polynomials when the quadratic term in the expansion about the middle coefficient vanishes. For coefficient indices whose distance from $n/2$ is of order $n^{3/4}$, the coefficient ratios have a quadratic-quartic exponential limit. A uniform bound valid for all indices allows the ratios to be summed and gives locally uniform convergence of the centred generating polynomials to a Fourier--Laplace integral. We also obtain an asymptotic formula for the sum of the coefficients, convergence of all rescaled moments, weak convergence of the normalised coefficient measures, and convergence with multiplicity of the zeros tending to $w=1$.