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PreprintCC BY 4.0arXiv

Asymptotics of the coefficients of polynomials, asymptotic zero distribution and free probability

  • Walter Van Assche
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Abstract

The asymptotic behavior of the coefficients of a sequence of polynomials is given under the conditions that all the zeros are real and positive and the zeros have an asymptotic zero distribution on (0,∞). We show how the limit for the coefficients is related to the Stieltjes transform of the asymptotic zero distribution and make a connection with the S-transform in free probability. We illustrate with some examples involving orthogonal and multiple orthogonal polynomials.

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Cite

BibTeXRIS
BibTeX
@misc{assche2026asymptotics,
  title = {{Asymptotics of the coefficients of polynomials, asymptotic zero distribution and free probability}},
  author = {Walter Van Assche},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12717},
  eprint = {2609.12717},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12717},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotics of the coefficients of polynomials, asymptotic zero distribution and free probability
AU  - Walter Van Assche
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12717
UR  - https://arxiv.org/abs/2609.12717
AB  - The asymptotic behavior of the coefficients of a sequence of polynomials is given under the conditions that all the zeros are real and positive and the zeros have an asymptotic zero distribution on (0,∞). We show how the limit for the coefficients is related to the Stieltjes transform of the asymptotic zero distribution and make a connection with the S-transform in free probability. We illustrate with some examples involving orthogonal and multiple orthogonal polynomials.
ER  -