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We consider random walk polynomial sequences$(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrencerelations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$$n\in\mathbb{N}$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every$k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and$c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansionsin the Chebyshev basis $\{T_n(x)\colon n\in\mathbb{N}_0\}$. As an application,we obtain explicit expansions for the sieved ultraspherical polynomials.Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of theAskey-Wilson operator $\mathcal{D}_q$. It is motivated by the sievedultraspherical polynomials, a generalization of the classical derivative andobtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity.However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has aninfinite-dimensional kernel (in contrast to its ancestor), which leads toadditional degrees of freedom and characterization results for $k$-sievedrandom walk polynomials. Similar characterizations are obtained for a sievedaveraging operator $\mathrm{A}_k$.

Expansions and Characterizations of Sieved Random Walk Polynomials