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 Pn(ωr+1 z) = ωr+1n Pn(z), where ωr+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 nγ 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.
Related
- Time-and-band limiting for symmetric classical Dunkl orthogonal polynomials
- Positivity and Asymptotics for Chenevier's Orthogonal Polynomials
- Asymptotics of the coefficients of polynomials, asymptotic zero distribution and free probability
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
BibTeX
@misc{loureiro2026ratio,
title = {{Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials}},
author = {Ana Loureiro and Walter Van Assche},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02801},
eprint = {2609.02801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02801},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials
AU - Ana Loureiro
AU - Walter Van Assche
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02801
UR - https://arxiv.org/abs/2609.02801
AB - 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.
ER -