A note on "Higher order linear differential equations for unitary matrix integrals: applications and generalisations"

TL;DR

This note studies higher order linear differential equations satisfied by Hankel and Toeplitz determinants involving $I$-Bessel functions, recast as unitary matrix integrals, and places them in a broader integrable-systems context. By rewriting the Hankel determinant as a unitary matrix integral and exploiting the Selberg-class structure, the authors derive an ($l+1$)×($l+1$) matrix differential system that yields a higher-order scalar differential equation, extendable to the circular $\beta$-ensemble, and connect it to a $\sigma$-Painlevé III' transcendent from prior work. They show how these differential structures provide efficient recursions for the Taylor coefficients of the relevant determinants and related combinatorial quantities, enabling computation of $a_{h,l}(n_1,n_2)$ and $b_{h,l}(n_1,n_2)$ for large $h,l$. The paper also links these analytic structures to applications in number theory, notably large gaps between zeros of derivatives of Hardy's $Z$-function under a joint moments conjecture, and provides conditional lower bounds on $\Theta^{(m)}$ that improve upon existing results. Overall, the work furnishes a unified framework connecting unitary matrix integrals, integrable differential equations, and number-theoretic zero-spacing problems, with concrete computational tools for moments and Taylor coefficients.

Abstract

In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied by certain Hankel and Toeplitz determinants involving I-Bessel functions, or equivalently certain unitary matrix integrals, and moreover puts this property in a broader context. We also investigate large gaps between zeros of the derivatives of the Hardy $\mathsf{Z}$-function, assuming the validity of a certain joint moments conjecture in random matrix theory.

Theorems & Definitions (10)

• theorem 1
• theorem 2
• lemma 1
• lemma 2
• proof
• lemma 3
• proof
• proof : Proof of Theorem \ref{['largegap1']}
• proof : Proof of Theorem \ref{['largegap2']}
• remark 1