  The Fr\'{e}chet (resp.\ (LB)) sequence spaces $ces(p+) := \cap_{r > p}
ces(r), 1 \leq p < \infty $ (resp.\ $ ces (p-) := \cup_{ 1 < r < p} ces (r), 1
< p \leq \infty),$ are known to be very different to the classical sequence
spaces $ \ell_ {p+} $ (resp., $ \ell_{p_{-}}).$ Both of these classes of
non-normable spaces $ ces (p+), ces (p-)$ are defined via the family of
reflexive Banach sequence spaces $ ces (p), 1 < p < \infty .$ The dual Banach
spaces $ d (q), 1 < q < \infty ,$ of the discrete Ces\`{a}ro spaces $ ces (p),
1 < p < \infty,$ were studied by G.\ Bennett, A.\ Jagers and others. Our aim is
to investigate in detail the corresponding sequence spaces $ d (p+) $ and $ d
(p-),$ which have not been considered before. Some of their properties have
similarities with those of $ ces (p+), ces (p-)$ but, they also exhibit
differences. For instance, $ ces (p+)$ is isomorphic to a power series
Fr\'{e}chet space of order 1, whereas $ d (p+) $ is isomorphic to such a space
of infinite order. Every space $ ces (p+), ces (p-) $ admits an absolute basis
but, none of the spaces $ d (p+), d (p-)$ have any absolute basis.
