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

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.

The following conditions on polynomial sequences can easily be seen to be equivalent:

Suppose

where the last equality is taken to define the linear operator ${\displaystyle S}$ on the space of polynomials in ${\displaystyle x}$. Let

be the inverse operator, the coefficients ${\displaystyle a_{k}}$ being those of the usual reciprocal of a formal power series, so that

In the conventions of the umbral calculus, one often treats this formal power series ${\displaystyle T}$ as representing the Appell sequence ${\displaystyle p_{n}}$. One can define