AI 中文总结
研究正实$(p,q)$-罗杰斯-塞戈多项式，在特定条件下证明其系数比的极限及相关渐近公式，通过一致界使生成多项式收敛到傅里叶-拉普拉斯积分，还得到系数和等多项收敛结果。
AI 中文摘要
我们证明了正实$(p,q)$-罗杰斯-塞戈多项式在关于中间系数展开时二次项消失的渐近公式。对于与$n/2$距离为$n^{3/4}$阶的系数指标，系数比有一个二次-四次指数极限。一个对所有指标都有效的一致界允许对这些比值求和，并使中心化生成多项式局部一致收敛到一个傅里叶-拉普拉斯积分。我们还得到了系数和的渐近公式、所有重标矩的收敛性、归一化系数测度的弱收敛性以及零点趋于$w = 1$时的多重收敛性。
英文摘要
We prove asymptotic formulae for positive real $(p,q)$-Rogers-Szegő 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 $w=1$.
