Properties of the zeros of generalized hypergeometric polynomials

TL;DR

This paper investigates the zeros of generalized hypergeometric polynomials, deriving algebraic equations they satisfy, and constructs a matrix with eigenvalues depending only on certain parameters, revealing isospectral properties and diophantine relations.

Contribution

It introduces a set of algebraic equations for the zeros and constructs a matrix with eigenvalues depending solely on the q parameters, highlighting isospectrality and diophantine properties.

Findings

Zeros satisfy specific nonlinear algebraic equations

Constructed matrix has eigenvalues depending only on q parameters

Eigenvalues are integer or rational when q parameters are integer or rational

Abstract

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex, numbers. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros z_n of this polynomial. We moreover manufacture an NxN matrix L in terms of the 1+p+q parameters N, a_j, b_k characterizing this polynomial, and of its N zeros z_n. We show that the matrix L features N eigenvalues that depend only on the q parameters b_k, implying that this matrix is isospectral for the variations of the p parameters a_j. These eigenvalues are integer (or rational) numbers if the q parameters b_k are themselves integer (or rational) numbers: a nontrivial diophantine property.