  We present a study of a specific kind of lowering operator, herein called
$\Lambda$, which is defined as a finite sum of lowering operators, proving that
this configuration can be altered, for instance, by the use of Stirling
numbers. We characterize the polynomial sequences fulfilling an Appell relation
with respect to $\Lambda$, and considering a concrete cubic decomposition of a
simple Appell sequence, we prove that the polynomial component sequences are
$\Lambda$-Appell, with $\Lambda$ defined as previously, although by a three
term sum. Ultimately, we prove the non-existence of orthogonal polynomial
sequences which are also $\Lambda$-Appell, when $\Lambda$ is the lowering
operator $\Lambda=a_{0}D+a_{1}DxD+a_{2}\left(Dx\right)^2D$, where $a_{0}$,
$a_{1}$ and $a_{2}$ are constants and $a_{2} \neq 0$. The case where $a_{2}=0$
and $a_{1} \neq 0$ is also naturally recaptured.
