Zeros of a table of polynomials satisfying a four-term contiguous relation

TL;DR

This paper investigates the zero distribution of a two-parameter family of polynomials satisfying a specific four-term recurrence relation, showing that their zeros lie on an explicitly described curve.

Contribution

It provides a detailed analysis of the zero distribution for polynomials satisfying a four-term contiguous relation, including explicit characterization of the zero locus.

Findings

Zeros lie on a specific algebraic curve.

Explicit formula for the zero locus in terms of recurrence coefficients.

Extension to cases with general initial conditions.

Abstract

For any $A(z),B(z),C(z)\in\mathbb{C}[z]$, we study the zero distribution of a table of polynomials $\left\{ P_{m,n}(z)\right\} _{m,n\in\mathbb{N}_{0}}$ satisfying the recurrence relation \[ P_{m,n}(z)=A(z)P_{m-1,n}(z)+B(z)P_{m,n-1}(z)+C(z)P_{m-1,n-1}(z) \] with the initial condition $P_{0,0}(z)=1$ and $P_{-m,-n}(z)=0$ $\forall m,n\in\mathbb{N}$. We show that the zeros of $P_{m,n}(z)$ lie on a curve whose equation is given explicitly in terms of $A(z),B(z)$, and $C(z)$. We also study the zero distribution of a case with a general initial condition.