Angelescu polynomials

In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Aurel Angelescu. The polynomials can be given by the generating function$$\phi\left(\frac t{1-t}\right)\exp\left(-\frac{xt}{1-t}\right)=\sum_{n=0}^\infty\pi_n(x)t^n. $$

They can also be defined by the equation $$\pi_{n}(x) := e^x D^n[e^{-x}A_n(x)],$$where $$\frac{A_n(x)}{n!}$$ is an Appell set of polynomials.

Addition and recurrence relations
The Angelescu polynomials satisfy the following addition theorem:

$$(-1)^n\sum_{r=0}^m\frac{L_{m+n-r}^{(n)}(x)\pi_r(y)}{(n+m-r)!r!} = \sum_{r=0}^m (-1)^r\binom{-n-1}{r} \frac{\pi_{n-r}(x+y)}{(m-r)!},$$where $$L^{(n)}_{m+n-r}$$ is a generalized Laguerre polynomial.

A particularly notable special case of this is when $$n=0$$, in which case the formula simplifies to$$\frac{\pi_m(x+y)}{m!} = \sum_{r=0}^m \frac{L_{m-r}(x)\pi_r(y)}{(m-r)!r!} - \sum_{r=0}^{m-1} \frac{L_{m-r-1}(x)\pi_r(y)}{(m-r-1)!r!}.$$

The polynomials also satisfy the recurrence relation

$$\pi_s(x) = \sum_{r=0}^n (-1)^{n+r}\binom{n}{r}\frac{s!}{(n+s-r)!}\frac{d^n}{dx^n}[\pi_{n+s-r}(x)],$$

which simplifies when $$n=0$$ to $$\pi'_{s+1}(x) = (s+1)[\pi'_s(x) - \pi_s(x)]$$. This can be generalized to the following:

$$-\sum_{r=0}^s \frac{1}{(m+n-r-1)!}L^{(m+n-1)}_{m+n-r-1}(x)\frac{\pi_{r-s}(y)}{(s-r)!} = \frac{1}{(m+n+s)!}\frac{d^{m+n}}{dx^m dy^n}\pi_{m+n+s}(x+y),$$

a special case of which is the formula $$\frac{d^{m+n}}{dx^m dy^n}\pi_{m+n}(x+y) = (-1)^{m+n} (m+n)! a_0$$.

Integrals
The Angelescu polynomials satisfy the following integral formulae:

$$\begin{align} \int_0^{\infty}\frac{e^{-x/2}}{x}[\pi_n(x) - \pi_n(0)]dx &= \sum_{r=0}^{n-1} (-1)^{n-r+1}\frac{n!}{r!}\pi_r(0)\int_0^{\infty} [\frac{1}{1/2 + p} - 1]^{n-r-1} d[\frac{1}{1/2+p}]\\ &= \sum_{r=0}^{n-1} (-1)^{n-r+1}\frac{n!}{r!}\frac{\pi_r(0)}{n-r}[1 + (-1)^{n-r-1}] \end{align}$$

$$\int_0^{\infty} e^{-x}[\pi_n(x) - \pi_n(0)]L_m^{(1)}(x)dx = \begin{cases} 0\text{ if }m\geq n\\ \frac{n!}{(n-m-1)!}\pi_{n-m-1}(0)\text{ if }0\leq m\leq n-1 \end{cases}$$

(Here, $$L_m^{(1)}(x)$$ is a Laguerre polynomial.)

Further generalization
We can define a q-analog of the Angelescu polynomials as $$\pi_{n, q}(x) := e_q(xq^n) D_q^n[E_q(-x)P_n(x)]$$, where $$e_q$$ and $$E_q$$ are the q-exponential functions $$e_q(x) := \Pi_{n=0}^{\infty} (1 - q^n x)^{-1} = \Sigma_{k=0}^{\infty}\frac{x^k}{[k]!}$$ and $$E_q(x) := \Pi_{n=0}^{\infty} (1 + q^n x) = \Sigma_{k=0}^{\infty}\frac{q^{\frac{k(k-1)}{2}}x^k}{[k]!}$$, $$D_q$$ is the q-derivative, and $$P_n$$ is a "q-Appell set" (satisfying the property $$D_q P_n(x) = [n]P_{n-1}(x)$$).

This q-analog can also be given as a generating function as well:

$$\sum_{n=0}^{\infty}\frac{\pi_{n, q}(x)t^n}{(1;n)} = \sum_{n=0}^{\infty}\frac{(-1)^n q^{\frac{n(n-1)}{2}}t^n P_n(x)}{(1;n)[1-t]_{n+1}},$$where we employ the notation $$(a;k) := (1 - q^a)\dots (1 - q^{a+k-1})$$ and $$[a+b]_n = \sum_{k=0}^n\begin{bmatrix}n\\k\end{bmatrix}a^{n-k}b^k$$.