Appell sequence

In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence ${\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }}$ satisfying the identity

${\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),}$

and in which ${\displaystyle p_{0}(x)}$ is a non-zero constant.

Among the most notable Appell sequences besides the trivial example ${\displaystyle \{x^{n}\}}$ are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.