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

Per Åhag; Rafał Czyż; Per-Håkan Lundow

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

Classical Analysis and ODEs 2026-07-09 v1 Combinatorics

Authors: Per Åhag , Rafał Czyż , Per-Håkan Lundow

Abstract

We prove local asymptotics for near-central complex $(p,q)$ -binomial coefficient moduli 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 asymptotes for near-central ratios.

Cite

@article{arxiv.2607.08163,
  title  = {On $(p,q)$-binomial coefficient ratios for complex parameters},
  author = {Per Åhag and Rafał Czyż and Per-Håkan Lundow},
  journal= {arXiv preprint arXiv:2607.08163},
  year   = {2026}
}