Expansions and Characterizations of Sieved Random Walk Polynomials
We consider random walk polynomial sequences (ₙ())ₙ∈ℕ₀ ⊆ ℝ[] given by recurrence relations ₀() = 1, ₁() = , ₙ() = (1−cₙ)ₙ₊₁()+cₙₙ₋₁(), ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every ∈ ℕ, the -sieved polynomials (ₙ(; ))ₙ∈ℕ₀ arise from the recurrence coefficients c(; ):= cₙ/ₖ if | and c(; ):= 1/2 otherwise. A...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212028 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Expansions and Characterizations of Sieved Random Walk Polynomials. Stefan Kahler. SIGMA 19 (2023), 103, 18 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862719542690578432 |
|---|---|
| author | Kahler, Stefan |
| author_facet | Kahler, Stefan |
| citation_txt | Expansions and Characterizations of Sieved Random Walk Polynomials. Stefan Kahler. SIGMA 19 (2023), 103, 18 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider random walk polynomial sequences (ₙ())ₙ∈ℕ₀ ⊆ ℝ[] given by recurrence relations ₀() = 1, ₁() = , ₙ() = (1−cₙ)ₙ₊₁()+cₙₙ₋₁(), ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every ∈ ℕ, the -sieved polynomials (ₙ(; ))ₙ∈ℕ₀ arise from the recurrence coefficients c(; ):= cₙ/ₖ if | and c(; ):= 1/2 otherwise. A main objective of this paper is to study expansions in the Chebyshev basis {Tₙ(): n ∈ℕ₀}. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version Dₖ of the Askey-Wilson operator . It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative, and obtained from by letting approach a -th root of unity. However, for ≥ 2, the new operator Dₖ on ℝ[] has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for -sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator Aₖ.
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| first_indexed | 2026-03-20T23:59:24Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212028 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T23:59:24Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kahler, Stefan 2026-01-23T10:08:18Z 2023 Expansions and Characterizations of Sieved Random Walk Polynomials. Stefan Kahler. SIGMA 19 (2023), 103, 18 pages 1815-0659 2020 Mathematics Subject Classification: 42C05; 33C47; 42C10 arXiv:2306.16411 https://nasplib.isofts.kiev.ua/handle/123456789/212028 https://doi.org/10.3842/SIGMA.2023.103 We consider random walk polynomial sequences (ₙ())ₙ∈ℕ₀ ⊆ ℝ[] given by recurrence relations ₀() = 1, ₁() = , ₙ() = (1−cₙ)ₙ₊₁()+cₙₙ₋₁(), ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every ∈ ℕ, the -sieved polynomials (ₙ(; ))ₙ∈ℕ₀ arise from the recurrence coefficients c(; ):= cₙ/ₖ if | and c(; ):= 1/2 otherwise. A main objective of this paper is to study expansions in the Chebyshev basis {Tₙ(): n ∈ℕ₀}. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version Dₖ of the Askey-Wilson operator . It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative, and obtained from by letting approach a -th root of unity. However, for ≥ 2, the new operator Dₖ on ℝ[] has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for -sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator Aₖ. The research was begun when the author worked at the Technical University of Munich, and the author gratefully acknowledges support from the graduate program TopMath of the ENB (Elite Network of Bavaria) and the TopMath Graduate Center of TUM Graduate School at the Technical University of Munich. The research was continued and completed at RWTH Aachen University. The author also thanks the referees for carefully reading the manuscript, as well as for their valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Expansions and Characterizations of Sieved Random Walk Polynomials Article published earlier |
| spellingShingle | Expansions and Characterizations of Sieved Random Walk Polynomials Kahler, Stefan |
| title | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_full | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_fullStr | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_full_unstemmed | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_short | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_sort | expansions and characterizations of sieved random walk polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212028 |
| work_keys_str_mv | AT kahlerstefan expansionsandcharacterizationsofsievedrandomwalkpolynomials |