Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials

Abstract

We investigate the ratio asymptotics and the asymptotic zero distribution of a sequence of polynomials that satisfy a recurrence relation of order $r+1$ with all recurrence coefficients, except the last one, equal to zero. Such a sequence is part of a system of multiple orthogonal polynomials and it satisfies the symmetry property $P_n(ω_{r+1} z) = ω_{r+1}^n P_n(z)$, where $ω_{r+1}$ is the primitive $(r+1)$th root of unity. We consider the unbounded regime in which the recurrence coefficients exhibit algebraic growth and, after division by $n^γ$ become asymptotically periodic and bounded. After the appropriate scaling, we establish ratio asymptotics and characterize the limiting ratio as the distinguished solution of an algebraic equation. We then determine the limiting zero distribution through its Stieltjes transform and investigate the associated \(\St\)-transform, which in several cases yields connections with hypergeometric polynomial sequences and distributions arising in free probability. The recurrence is represented by a two-diagonal non-self-adjoint Hessenberg operator, so that the limiting zero measure also admits a natural interpretation as a limiting empirical spectral distribution of its rescaled finite sections. Our analysis is based solely on the positivity and asymptotic behavior of the recurrence coefficients and requires no explicit knowledge of the underlying orthogonality measures.