Expansions and Characterizations of Sieved Random Walk Polynomials

Abstract

We consider random walk polynomial sequences (𝑃ₙ(𝑥))ₙ∈ℕ₀ ⊆ ℝ[𝑥] given by recurrence relations 𝑃₀(𝑥) = 1, 𝑃₁(𝑥) = 𝑥, 𝑥𝑃ₙ(𝑥) = (1−cₙ)𝑃ₙ₊₁(𝑥)+cₙ𝑃ₙ₋₁(𝑥), 𝑛 ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every 𝑘 ∈ ℕ, the 𝑘-sieved polynomials (𝑃ₙ(𝑥; 𝑘))ₙ∈ℕ₀ arise from the recurrence coefficients c(𝑛; 𝑘):= cₙ/ₖ if 𝑘|𝑛 and c(𝑛; 𝑘):= 1/2 otherwise. A main objective of this paper is to study expansions in the Chebyshev basis {Tₙ(𝑥): 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 𝒟𝑞. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative, and obtained from 𝒟𝑞 by letting 𝑞 approach a 𝑘-th root of unity. However, for 𝑘 ≥ 2, the new operator Dₖ on ℝ[𝑥] has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for 𝑘-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator Aₖ.