Jacobi matrices generated by ratios of hypergeometric functions

TL;DR

This paper explores the connection between ratios of hypergeometric functions and Jacobi matrices, using spectral theory of non-Hermitian matrices to analyze the zeros of hypergeometric functions.

Contribution

It introduces a novel approach linking hypergeometric function ratios to Jacobi matrices via spectral theory, providing new insights into their zeros.

Findings

Ratios of hypergeometric functions can be represented as m-functions of Jacobi matrices.

Spectral theory of non-Hermitian Jacobi matrices offers new tools for analyzing hypergeometric zeros.

Revisiting classical problems with modern spectral methods yields fresh perspectives.

Abstract

A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.