  The analysis of the spectral features of a Toeplitz matrix-sequence
$\left\{T_{n}(f)\right\}_{n\in\mathbb N}$, generated by a symbol $f\in
L^1([-\pi,\pi])$, real-valued almost everywhere (a.e.), has been provided in
great detail in the last century, as well as the study of the conditioning,
when $f$ is nonnegative a.e. Here we consider a novel type of problem arising
in the numerical approximation of distributed-order fractional differential
equations (FDEs), where the matrices under consideration take the form \[
\mathcal{T}_{n}=c_0T_{n}(f_0)+c_{1} h^h T_{n}(f_{1})+c_{2} h^{2h}
T_{n}(f_{2})+\cdots+c_{n-1} h^{(n-1)h}T_{n}(f_{n-1}), \] $c_0,c_{1},\ldots,
c_{n-1} \in [c_*,c^*]$, $c^*\ge c_*>0$, independent of $n$, $h=\frac{1}{n}$,
$f_j\sim g_j$, $g_j=|\theta|^{2-jh}$, $j=0,\ldots,n-1$. Since the resulting
generating function depends on $n$, the standard theory cannot be applied and
the analysis has to be performed using new ideas. Few selected numerical
experiments are presented, also in connection with matrices that come from
distributed-order FDE problems, and the adherence with the theoretical analysis
is discussed together with open questions and future investigations.
