  We consider random walk polynomial sequences
$(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence
relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$
$n\in\mathbb{N}$ 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 expansions
in 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 the
Askey-Wilson operator $\mathcal{D}_q$. It is motivated by the sieved
ultraspherical polynomials, a generalization of the classical derivative and
obtained 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 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 $\mathrm{A}_k$.
