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The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials

  • Qi-Feng Bai
  • Yu-Tian Li
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Abstract

Let pn(α,β) denote the Jacobi polynomial orthonormal for the weight (1−x)α(1+x)β on [−1,1], where α,β≥−1/2, and put S=α+β+1. We prove the uniform degree--parameter estimate (1−x)α+1/2(1+x)β+1/2 |pn(α,β)(x)|2 ≤ Cmax{1,S1/3,S1/2(n+1)−1/6}. This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erdélyi--Magnus--Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor S1/3 in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss--Jacobi quadrature weights.

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Cite

BibTeXRIS
BibTeX
@misc{bai2026erdélyimagnusnevai,
  title = {{The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials}},
  author = {Qi-Feng Bai and Yu-Tian Li},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30304},
  eprint = {2608.30304},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30304},
}
RIS
TY  - UNPB
TI  - The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials
AU  - Qi-Feng Bai
AU  - Yu-Tian Li
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30304
UR  - https://arxiv.org/abs/2608.30304
AB  - Let p_n^(α,β) denote the Jacobi polynomial orthonormal for the weight (1−x)^α(1+x)^β on [−1,1], where α,β≥−1/2, and put S=α+β+1. We prove the uniform degree--parameter estimate (1−x)^α+1/2(1+x)^β+1/2 |p_n^(α,β)(x)|^2 ≤ Cmax{1,S^1/3,S^1/2(n+1)^−1/6}. This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erdélyi--Magnus--Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor S^1/3 in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss--Jacobi quadrature weights.
ER  -