发表机构
University of Kent; KU Leuven（肯特大学; 荷语鲁汶大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究具有对称性的多重正交多项式序列的比值渐近性与极限零点分布，基于递推系数性质推导相关结果，揭示其与自由概率论等领域的关联。
AI 中文摘要
我们研究满足r+1阶递推关系的多项式序列的比值渐近性与极限零点分布，该递推关系中除最后一项外的所有递推系数均为零。该序列是多重正交多项式系统的一部分，且满足对称性性质$P_n(\omega_{r+1} z) = \omega_{r+1}^n P_n(z)$，其中$\omega_{r+1}$是本原(r+1)次单位根。我们考虑无界情形，此时递推系数呈现代数增长，除以$n^\gamma$后成为渐近周期且有界的量。经过适当缩放后，我们建立了比值渐近性，并将极限比值表征为某个代数方程的特解。随后，我们通过Stieltjes变换确定了极限零点分布，并研究了相关的St-变换，该变换在若干情形下与超几何多项式序列及自由概率论中出现的分布存在关联。该递推关系由一个非自伴的双对角Hessenberg算子表示，因此极限零点测度也可自然地解释为其缩放有限截面的极限经验谱分布。我们的分析仅依赖于递推系数的正性与渐近行为，无需明确了解 underlying 正交测度。
英文摘要
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.
Comments35 pages, 8 figures
