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Asymptotics and zeros of a special family of Jacobi polynomials

John Lopez Santander, Kenneth D. T-R McLaughlin, Victor H. Moll

TL;DR

This work analyzes the non-classical Jacobi polynomials $P_m(z)=P_m^{(m+\frac{1}{2},-m-\frac{1}{2})}(z)$ that arise in Boros–Moll integrals, by formulating a complex-contour orthogonality and a corresponding Riemann–Hilbert problem. Using a nonlinear steepest-descent analysis, the authors derive global asymptotics away from the origin and construct a local parametrix near $z=0$ based on parabolic cylinder functions, yielding uniform asymptotics for the polynomials. They prove that the zeros split into two regimes: away from the origin, zeros accumulate on the lemniscate $|1-z^2|=1$, and near the origin, zeros are governed by a local equation involving $D_{-1/2}$, with precise near-field behavior established via Rouché's theorem and RH analysis. These results confirm conjectures on zero distribution, connect the local zeros to special functions, and extend prior work on non-classical Jacobi polynomials and their zeros, providing a robust framework for further generalizations.

Abstract

In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form $α_n=n+1/2$ and $β_n=-n-1/2$. We obtain global asymptotics for these polynomials, and use this to establish results on the location their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by Kuijlaars, et al. This family of polynomials arise in the symbolic evaluation integrals in the work of Boros and Moll and corresponds to a limitting case, which is not considered in the works of Kuijlaars, et al. A remarkable feature in the analyisis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole.

Asymptotics and zeros of a special family of Jacobi polynomials

TL;DR

This work analyzes the non-classical Jacobi polynomials that arise in Boros–Moll integrals, by formulating a complex-contour orthogonality and a corresponding Riemann–Hilbert problem. Using a nonlinear steepest-descent analysis, the authors derive global asymptotics away from the origin and construct a local parametrix near based on parabolic cylinder functions, yielding uniform asymptotics for the polynomials. They prove that the zeros split into two regimes: away from the origin, zeros accumulate on the lemniscate , and near the origin, zeros are governed by a local equation involving , with precise near-field behavior established via Rouché's theorem and RH analysis. These results confirm conjectures on zero distribution, connect the local zeros to special functions, and extend prior work on non-classical Jacobi polynomials and their zeros, providing a robust framework for further generalizations.

Abstract

In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form and . We obtain global asymptotics for these polynomials, and use this to establish results on the location their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by Kuijlaars, et al. This family of polynomials arise in the symbolic evaluation integrals in the work of Boros and Moll and corresponds to a limitting case, which is not considered in the works of Kuijlaars, et al. A remarkable feature in the analyisis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole.

Paper Structure

This paper contains 16 sections, 7 theorems, 166 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Orthogonality and Riemann Hilbert formulation
  3. Orthogonality in the complex plane
  4. Riemann Hilbert Problem for Orthogonal Polynomials
  5. Riemann Hilbert Analysis
  6. Solution away from the origin
  7. The parabolic cylinder RHP
  8. Local solution to RHP 3.2 near the origin
  9. Global solution
  10. Adjustment of the local solution
  11. Addressing the issue of the pole introduced at $z=0$.
  12. Location of the zeros
  13. Zeros away from the origin
  14. Zeros near the origin
  15. Conclusions
  16. ...and 1 more sections

Key Result

Proposition 1.1

Figures (16)

  • Figure 1: Zeros of $P_{50}(z)$.
  • Figure 2: Contour $\Sigma_U$.
  • Figure 3: Original contour $\Gamma$.
  • Figure 4: Contour $\Sigma$ and zeros of $P_{50}(z)=P_{50}^{\left(50+\frac{1}{2},-50-\frac{1}{2}\right)}(z)$.
  • Figure 5: Contour $\Sigma_Z$ and jump matrices $J_Z(\eta)$.
  • ...and 11 more figures

Theorems & Definitions (16)

  • Proposition 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 6 more