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PreprintCC BY 4.0arXiv

The Marvelous Meixner Polynomials: Second Kind, First Class

  • Olga Holtz
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Abstract

Meixner's classification asks which exponential generating functions can generate orthogonal polynomials. We follow the continuous Meixner family from this question to its moments, measures, and continued fractions. A beta integral and a bilinear identity establish positivity and orthogonality; Carlitz's Bernoulli and Euler constructions reveal further connections. Turning the parameter into a variable leads to rational orthogonality and Wall's fraction. We derive its Nevanlinna measure and unite the two spectral directions in a joint law with Cauchy marginals. Even finite pencils produce couplings with these exact marginals. Laguerre limits explain the gamma law governing joint extremes. Along the way, Meixner's scientific life, combinatorics, tangent numbers, and even zeta values illuminate the same structure.

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Cite

BibTeXRIS
BibTeX
@misc{holtz2026marvelous,
  title = {{The Marvelous Meixner Polynomials: Second Kind, First Class}},
  author = {Olga Holtz},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01422},
  eprint = {2611.01422},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01422},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Marvelous Meixner Polynomials: Second Kind, First Class
AU  - Olga Holtz
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01422
UR  - https://arxiv.org/abs/2611.01422
AB  - Meixner's classification asks which exponential generating functions can generate orthogonal polynomials. We follow the continuous Meixner family from this question to its moments, measures, and continued fractions. A beta integral and a bilinear identity establish positivity and orthogonality; Carlitz's Bernoulli and Euler constructions reveal further connections. Turning the parameter into a variable leads to rational orthogonality and Wall's fraction. We derive its Nevanlinna measure and unite the two spectral directions in a joint law with Cauchy marginals. Even finite pencils produce couplings with these exact marginals. Laguerre limits explain the gamma law governing joint extremes. Along the way, Meixner's scientific life, combinatorics, tangent numbers, and even zeta values illuminate the same structure.
ER  -