From continuous to discrete: weak limit of normalized Askey–Wilson measure

Dan Dai; Mourad E. H. Ismail; Xiang-Sheng Wang

doi:10.1007/s11139-023-00820-3

From continuous to discrete: weak limit of normalized Askey–Wilson measure

Dan Dai^*
, Mourad E. H. Ismail
, Xiang-Sheng Wang

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

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Abstract

In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches -1 from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

Original language	English
Pages (from-to)	227-252
Journal	Ramanujan Journal
Volume	64
Issue number	1
Online published	18 Jan 2024
DOIs	https://doi.org/10.1007/s11139-023-00820-3
Publication status	Published - May 2024

Funding

Dan Dai was partially supported by a grant from the City University of Hong Kong (Project No. 7005597), and grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11300520 and CityU 11311622).

Research Keywords

Askey–Wilson polynomials
Orthogonality measures
Weak limit

Publisher's Copyright Statement

COPYRIGHT TERMS OF DEPOSITED POSTPRINT FILE: This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s11139-023-00820-3.

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title = "From continuous to discrete: weak limit of normalized Askey–Wilson measure",

abstract = "In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches -1 from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity. {\textcopyright} The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.",

keywords = "Askey–Wilson polynomials, Orthogonality measures, Weak limit",

author = "Dan Dai and Ismail, \{Mourad E. H.\} and Xiang-Sheng Wang",

year = "2024",

month = may,

doi = "10.1007/s11139-023-00820-3",

language = "English",

volume = "64",

pages = "227--252",

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T2 - weak limit of normalized Askey–Wilson measure

AU - Dai, Dan

AU - Ismail, Mourad E. H.

AU - Wang, Xiang-Sheng

PY - 2024/5

Y1 - 2024/5

N2 - In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches -1 from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

AB - In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches -1 from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

KW - Askey–Wilson polynomials

KW - Orthogonality measures

KW - Weak limit

UR - https://www.scopus.com/pages/publications/85182694393

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85182694393&origin=recordpage

U2 - 10.1007/s11139-023-00820-3

DO - 10.1007/s11139-023-00820-3

M3 - RGC 21 - Publication in refereed journal

SN - 1382-4090

VL - 64

SP - 227

EP - 252

JO - Ramanujan Journal

JF - Ramanujan Journal

IS - 1

ER -

From continuous to discrete: weak limit of normalized Askey–Wilson measure

Abstract

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Research Keywords

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GRF: Eigenvalue Rigidity of Random Unitary Matrices

GRF: Spacing Problems in Random Matrix Theory

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From continuous to discrete: weak limit of normalized Askey–Wilson measure

Abstract

Funding

Research Keywords

Publisher's Copyright Statement

RGC Funding Information

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GRF: Eigenvalue Rigidity of Random Unitary Matrices

GRF: Spacing Problems in Random Matrix Theory

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