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 ${P{m,n}(z)}{m,n\\in\\mathbb{N}\_0}$ 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 conditions $P{0,0}(z)=1$ and $P{-m,-n}(z)=0$ for all $m,n\\in\\mathbb{N}$. 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.