Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials
UDC 517.587, 512.643 We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambd...
Збережено в:
| Дата: | 2026 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2026
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/9553 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1871646540285607936 |
|---|---|
| author | Neha, N. Swaminathan, A. Neha, N. Swaminathan, A. |
| author_facet | Neha, N. Swaminathan, A. Neha, N. Swaminathan, A. |
| author_institution_txt_mv | [
{
"author": "N. Neha",
"institution": "Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, India"
},
{
"author": "A. Swaminathan",
"institution": "Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, India"
}
] |
| author_sort | Neha, N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-07-24T15:24:12Z |
| description | UDC 517.587, 512.643
We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambda>0.$ Here, the point $a$ is chosen to guarantee that the spectrum of Meixner polynomials does not intersect the interval $(a, a+1).$ This inner product and its generalization already exist in the literature for various other measures. We consider a specific measure. For the resulting orthonormal Sobolev-type polynomials, we obtain a five-term recurrence relation. The associated pentadiagonal matrix is analyzed in relation to the Jacobi pencil matrix. The behavior of zeros with respect to the parameters $a$ and $\lambda$ is investigated and the Plancherel–Rotach-type asymptotic behavior is obtained. |
| doi_str_mv | 10.3842/umzh.v78i7-8.9553 |
| first_indexed | 2026-07-25T01:00:34Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9553 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-25T01:00:34Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-95532026-07-24T15:24:12Z Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials Neha, N. Swaminathan, A. Neha, N. Swaminathan, A. Orthogonal polynomials; Meixner polynomials; Sobolev-type orthogonality; Zeros; Plancherel-Rotach asymptotics; Linear pencil matrix; Five Term reccurence relation MATHEMATICS UDC 517.587, 512.643 We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambda>0.$ Here, the point $a$ is chosen to guarantee that the spectrum of Meixner polynomials does not intersect the interval $(a, a+1).$ This inner product and its generalization already exist in the literature for various other measures. We consider a specific measure. For the resulting orthonormal Sobolev-type polynomials, we obtain a five-term recurrence relation. The associated pentadiagonal matrix is analyzed in relation to the Jacobi pencil matrix. The behavior of zeros with respect to the parameters $a$ and $\lambda$ is investigated and the Plancherel–Rotach-type asymptotic behavior is obtained. УДК 517.587, 512.643 Нулі та асимптотика певного класу поліномів Мейкснера соболєвського типу Розглянуто відомий з літератури скалярний добуток соболєвського типу. Цей скалярний добуток є модифікацією скалярного добутку для поліномів Мейкснера і містить додатковий член у точці $a\in\mathbb{R},$ що включає оператор прямої різниці $\Delta$ та ваговий параметр $\lambda>0.$ Точку $a$ вибрано так, щоб спектр поліномів Мейкснера не перетинав інтервалу $(a,a+1).$ Зазначений скалярний добуток та його узагальнення вже досліджено в літературі для різних мір. Mи розглядаємо конкретну міру. Для відповідних ортонормованих поліномів соболєвського типу одержано п'ятичленне рекурентне співвідношення. Проаналізовано пов'язану з матрицею олівця Якобі п'ятидіагональну матрицю. Досліджено поведінку нулів залежно від параметрів $a$ та $\lambda,$ а також одержано асимптотичну поведінку типу Планшереля–Ротаха. Institute of Mathematics, NAS of Ukraine 2026-07-24 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9553 10.3842/umzh.v78i7-8.9553 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 7-8 (2026); 609–610 Український математичний журнал; Том 78 № 7-8 (2026); 609–610 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9553/10689 Copyright (c) 2026 N. Neha, A. Swaminathan |
| spellingShingle | Neha, N. Swaminathan, A. Neha, N. Swaminathan, A. Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title | Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title_alt | Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title_full | Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title_fullStr | Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title_full_unstemmed | Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title_short | Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials |
| title_sort | zeros and asymptotics of a certain class of sobolev-type meixner polynomials |
| topic_facet | Orthogonal polynomials Meixner polynomials Sobolev-type orthogonality Zeros Plancherel-Rotach asymptotics Linear pencil matrix Five Term reccurence relation MATHEMATICS |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9553 |
| work_keys_str_mv | AT nehan zerosandasymptoticsofacertainclassofsobolevtypemeixnerpolynomials AT swaminathana zerosandasymptoticsofacertainclassofsobolevtypemeixnerpolynomials AT nehan zerosandasymptoticsofacertainclassofsobolevtypemeixnerpolynomials AT swaminathana zerosandasymptoticsofacertainclassofsobolevtypemeixnerpolynomials |