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PreprintarXiv licencearXiv

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

  • Per Åhag
  • Rafał Czyż
  • Per-Håkan Lundow
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Abstract

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 n3/4, 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 n3/4log w remains bounded.

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Cite

BibTeXRIS
BibTeX
@misc{ahag2026quartic,
  title = {{Quartic Fourier-Laplace limits for $(p,q)$-Rogers-Szegő polynomials}},
  author = {Per Åhag and Rafał Czyż and Per-Håkan Lundow},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.09150},
  eprint = {2607.09150},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.09150},
}
RIS
TY  - UNPB
TI  - Quartic Fourier-Laplace limits for (p,q)-Rogers-Szegő polynomials
AU  - Per Åhag
AU  - Rafał Czyż
AU  - Per-Håkan Lundow
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2607.09150
UR  - https://arxiv.org/abs/2607.09150
AB  - 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^3/4log w remains bounded.
ER  -