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Approximation of Askey-Wilson roots

Jan Felipe van Diejen, Andrés Soledispa, Adrián Vidal

TL;DR

This work addresses locating the $n$ ordered roots $\theta_{j,n}$ of the Askey-Wilson polynomial $p_n(\cos\theta; a_1,a_2,a_3,a_4|q)$ by recasting the root equations as the global minimum of a strictly convex Morse function. It develops a fixed-point iteration based on the auxiliary functions $v_{\epsilon}$ and $\nu_{\epsilon}$ to obtain a practical first-order root approximation with a provable error bound, and derives refined root bounds that fix previous limitations for large $\theta$. Establishes a contraction regime with $|q|<1/3$ and $|a_r|<1/3$, ensuring convergence and enabling efficient computation, with numerical illustrations showing rapid error decay as $n$ grows. The results yield a reliable, implementable scheme for Askey-Wilson root computation and tighter bounds, with potential impact on orthogonal-polynomial theory and quantum integrable systems via Bethe-Ansatz connections.

Abstract

This note presents a fixed-point formula designed to approximate the roots of Askey-Wilson poynomials for small parameter values.

Approximation of Askey-Wilson roots

TL;DR

This work addresses locating the ordered roots of the Askey-Wilson polynomial by recasting the root equations as the global minimum of a strictly convex Morse function. It develops a fixed-point iteration based on the auxiliary functions and to obtain a practical first-order root approximation with a provable error bound, and derives refined root bounds that fix previous limitations for large . Establishes a contraction regime with and , ensuring convergence and enabling efficient computation, with numerical illustrations showing rapid error decay as grows. The results yield a reliable, implementable scheme for Askey-Wilson root computation and tighter bounds, with potential impact on orthogonal-polynomial theory and quantum integrable systems via Bethe-Ansatz connections.

Abstract

This note presents a fixed-point formula designed to approximate the roots of Askey-Wilson poynomials for small parameter values.

Paper Structure

This paper contains 7 sections, 4 theorems, 47 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Approximation of Askey-Wilson roots
  3. Bounds for the Askey-Wilson Roots
  4. Fixed-point Iteration towards the Askey-Wilson Roots
  5. Proofs
  6. Proof of Theorem \ref{['bounds:thm']}
  7. Proof of Theorem \ref{['fixed-point-error:thm']}

Key Result

Theorem 1

For $-1<q<1$ and $a_1,a_2,a_3,a_4$ taken inside the open unit disk $\mathbb{U}$U with possible non-real parameters arising in complex conjugate pairs, one has that the $n$ ordered roots $\theta_{j,n}$roots of the Askey-Wilson polynomial $\mathrm{p}_n(\cos\theta ; a_1,a_2,a_3,a_4|q)$ provide the coor

Theorems & Definitions (6)

  • Theorem 1: die:gradientdie-ems:solutions
  • Theorem 2: Bounds for $\theta_{j,n}$
  • Remark 3
  • Theorem 4: Error of Approximation
  • Corollary 5: Convergence
  • Remark 6