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

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.