Appell polynomial sequences with respect to some differential operators

Abstract

We present a study of a specific kind of lowering operator, herein called $\Lambda$, which is defined as a finite sum of lowering operators and might be presented by various configurations. 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 $a_{2} \ne 0$. The case where $a_{2}=0$ and $a_{1} \ne 0$ is also naturally recaptured.