Expansions and Characterizations of Sieved Random Walk Polynomials

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2023
• Автор: Kahler, Stefan
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2023
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/212028
• 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

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.