An application of Kapteyn series to a problem from queueing theory

In this article, we prove weak type $(1,1)$ bounds for the variation and jump operators corresponding to the family of truncations of singular integrals with rough kernels. This resolves an open question raised by Jones, Seeger and Wright (Trans. Amer. Math. Soc. (2008)). Moreover, as an immediate consequence of the variational estimate, we recover the weak type $(1,1)$ boundedness of the maximal truncation operator corresponding to singular integrals with rough kernels.

math.CA↗

arXiv2026

On $(p,q)$-binomial coefficient ratios for complex parameters

We prove local asymptotics for near-central complex $(p,q)$-binomial coefficient modulus ratios allowing an imaginary parameter perturbation of order $n^{-3/4}$ at a $\sqrt{n}$ length scale from the centre. Moreover, we obtain ratio asymptotics for a smaller imaginary perturbation of order $n^{-5/4}$ at the length scale $n^{3/4}$. These results were obtained by reducing the two-parameter coefficients to just one parameter, giving a branch-free logarithmic representation of the second-order ratio and, hence, uniform complex curvature asymptotics for near-central ratios.

math.CA↗

arXiv2026

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

We prove asymptotic formulae for $(p,q)$-Rogers--Szegő polynomials with positive real parameters when the leading quadratic term in the logarithmic 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 and normalised 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 near $w=1$ for which $n^{3/4}\log w$ remains bounded.

math.CA↗