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A proof of the multi-component $q$-Baker--Forrester conjecture

Yue Zhou

TL;DR

The paper proves the $(p+1)$-component $q$-Baker–Forrester conjecture, which generalizes the $q$-Morris identity and its connection to the Selberg integral, resolving a 26-year-old problem. The authors develop a multi-faceted approach combining a splitting formula, the Gessel–Xin constant-term method, and a careful root analysis: they show $D_n((n_0, ots,n_p);a,b,c)$ is a polynomial in $q^a$ of degree at most $nb$, identify large-$c$ root sets, establish a rationality extension to all $c$, and derive a recursion for $D_n(0)$ that propagates to a closed-form expression. The core ideas include a detailed splitting of the associated rational function, extraction of vanishing coefficients, and a recursive framework that reduces higher-dimensional cases to lower ones. The results not only settle the conjecture but also enrich the theory of constant-term identities and their connections to $q$-Dyson, $q$-Morris, and Macdonald–Cherednik structures, with implications for multivariable orthogonal polynomials and related integrals.

Abstract

The Selberg integral, an $n$-dimensional generalization of the Euler beta integral, plays a central role in random matrix theory, Calogero--Sutherland quantum many body systems, Knizhnik--Zamolodchikov equations, and multivariable orthogonal polynomial theory. The Selberg integral is known to be equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a $(p+1)$-component generalization of the $q$-Morris identity. It in turn yields a generalization of the Selberg integral. The $p=1$ case of Baker and Forrester's conjecture was proved by Károlyi, Nagy, Petrov and Volkov in 2015. In this paper, we give a proof of the $(p+1)$-component $q$-Baker--Forrester conjecture, thereby settling this 26-year-old conjecture.

A proof of the multi-component $q$-Baker--Forrester conjecture

TL;DR

The paper proves the -component -Baker–Forrester conjecture, which generalizes the -Morris identity and its connection to the Selberg integral, resolving a 26-year-old problem. The authors develop a multi-faceted approach combining a splitting formula, the Gessel–Xin constant-term method, and a careful root analysis: they show is a polynomial in of degree at most , identify large- root sets, establish a rationality extension to all , and derive a recursion for that propagates to a closed-form expression. The core ideas include a detailed splitting of the associated rational function, extraction of vanishing coefficients, and a recursive framework that reduces higher-dimensional cases to lower ones. The results not only settle the conjecture but also enrich the theory of constant-term identities and their connections to -Dyson, -Morris, and Macdonald–Cherednik structures, with implications for multivariable orthogonal polynomials and related integrals.

Abstract

The Selberg integral, an -dimensional generalization of the Euler beta integral, plays a central role in random matrix theory, Calogero--Sutherland quantum many body systems, Knizhnik--Zamolodchikov equations, and multivariable orthogonal polynomial theory. The Selberg integral is known to be equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a -component generalization of the -Morris identity. It in turn yields a generalization of the Selberg integral. The case of Baker and Forrester's conjecture was proved by Károlyi, Nagy, Petrov and Volkov in 2015. In this paper, we give a proof of the -component -Baker--Forrester conjecture, thereby settling this 26-year-old conjecture.

Paper Structure

This paper contains 19 sections, 26 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. A sketch of the proof of Theorem \ref{['thm-1']}
  3. The main steps
  4. The polynomiality
  5. The roots
  6. A rationality result
  7. A recursion for $D_n(0)$
  8. A proof of Theorem \ref{['thm-1']}
  9. The splitting formula and consequences
  10. The splitting formula
  11. The recursion for $D_n(0)$
  12. A family of vanishing coefficients
  13. The Gessel--Xin method to evaluate constant terms
  14. The roots
  15. Preliminaries
  16. ...and 4 more sections

Key Result

Theorem 1.1

For nonnegative integers $a_1,\dots,a_n$, where $\mathop{\mathrm{CT}}\limits_{\boldsymbol{x}}f$ denotes taking the constant term of the Laurent polynomial $f$ with respect to $\boldsymbol{x}:=(x_1,\dots,x_n)$.

Theorems & Definitions (41)

  • Theorem 1.1: Dyson's ex-conjecture
  • Theorem 1.2: Andrews' ex-conjecture
  • Theorem 1.3: Kadell's ex-conjecture
  • Theorem 1.4: Macdonald's ex-conjecture
  • Theorem 1.5: Morris' ex-conjecture
  • Theorem 1.6: Askey's ex-conjecture
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Corollary 2.2
  • ...and 31 more