Angelescu polynomials

In mathematics, the Angelescu polynomials π_n are a series of polynomials generalizing the Laguerre polynomials introduced by Aurel Angelescu. The polynomials can be given by the generating function
They can also be defined by the equation
where is an Appell set of polynomials.

Properties

Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:
where is a generalized Laguerre polynomial.
A particularly notable special case of this is when, in which case the formula simplifies to
The polynomials also satisfy the recurrence relation
which simplifies when to. This can be generalized to the following:
a special case of which is the formula.

Integrals

The Angelescu polynomials satisfy the following integral formulae:

Further generalization

We can define a q-analog of the Angelescu polynomials as, where and are the q-exponential functions and, is the q-derivative, and is a "q-Appell set".
This q-analog can also be given as a generating function as well:
where we employ the notation and.