Angelescu polynomials

In mathematics, the Angelescu polynomials π_n(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Aurel Angelescu. The polynomials can be given by the generating function$${\displaystyle \phi \left({\frac {t}{1-t}}\right)\exp \left(-{\frac {xt}{1-t}}\right)=\sum _{n=0}^{\infty }\pi _{n}(x)t^{n}.}$$^[1][2]

They can also be defined by the equation $${\displaystyle \pi _{n}(x):=e^{x}D^{n}[e^{-x}A_{n}(x)],}$$where ${\displaystyle {\frac {A_{n}(x)}{n!}}}$ is an Appell set of polynomials^[which?].^[3]

Properties

Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:

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

A particularly notable special case of this is when ${\displaystyle n=0}$, in which case the formula simplifies to$${\displaystyle {\frac {\pi _{m}(x+y)}{m!}}=\sum _{r=0}^{m}{\frac {L_{m-r}(x)\pi _{r}(y)}{(m-r)!r!}}-\sum _{r=0}^{m-1}{\frac {L_{m-r-1}(x)\pi _{r}(y)}{(m-r-1)!r!}}.}$$^{[clarification needed][4]}

The polynomials also satisfy the recurrence relation

$${\displaystyle \pi _{s}(x)=\sum _{r=0}^{n}(-1)^{n+r}{\binom {n}{r}}{\frac {s!}{(n+s-r)!}}{\frac {d^{n}}{dx^{n}}}[\pi _{n+s-r}(x)],}$$ ^{[verification needed]}

which simplifies when ${\displaystyle n=0}$ to ${\displaystyle \pi '_{s+1}(x)=(s+1)[\pi '_{s}(x)-\pi _{s}(x)]}$.^[4] This can be generalized to the following:

$${\displaystyle -\sum _{r=0}^{s}{\frac {1}{(m+n-r-1)!}}L_{m+n-r-1}^{(m+n-1)}(x){\frac {\pi _{r-s}(y)}{(s-r)!}}={\frac {1}{(m+n+s)!}}{\frac {d^{m+n}}{dx^{m}dy^{n}}}\pi _{m+n+s}(x+y),}$$ ^{[verification needed]}

a special case of which is the formula ${\displaystyle {\frac {d^{m+n}}{dx^{m}dy^{n}}}\pi _{m+n}(x+y)=(-1)^{m+n}(m+n)!a_{0}}$.^[4]

Integrals

The Angelescu polynomials satisfy the following integral formulae:

$${\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {e^{-x/2}}{x}}[\pi _{n}(x)-\pi _{n}(0)]dx&=\sum _{r=0}^{n-1}(-1)^{n-r+1}{\frac {n!}{r!}}\pi _{r}(0)\int _{0}^{\infty }[{\frac {1}{1/2+p}}-1]^{n-r-1}d[{\frac {1}{1/2+p}}]\\&=\sum _{r=0}^{n-1}(-1)^{n-r+1}{\frac {n!}{r!}}{\frac {\pi _{r}(0)}{n-r}}[1+(-1)^{n-r-1}]\end{aligned}}}$$

$${\displaystyle \int _{0}^{\infty }e^{-x}[\pi _{n}(x)-\pi _{n}(0)]L_{m}^{(1)}(x)dx={\begin{cases}0{\text{ if }}m\geq n\\{\frac {n!}{(n-m-1)!}}\pi _{n-m-1}(0){\text{ if }}0\leq m\leq n-1\end{cases}}}$$^[4]

(Here, ${\displaystyle L_{m}^{(1)}(x)}$ is a Laguerre polynomial.)

Further generalization

We can define a q-analog of the Angelescu polynomials as ${\displaystyle \pi _{n,q}(x):=e_{q}(xq^{n})D_{q}^{n}[E_{q}(-x)P_{n}(x)]}$, where ${\displaystyle e_{q}}$ and ${\displaystyle E_{q}}$ are the q-exponential functions ${\displaystyle e_{q}(x):=\Pi _{n=0}^{\infty }(1-q^{n}x)^{-1}=\Sigma _{k=0}^{\infty }{\frac {x^{k}}{[k]!}}}$ and ${\displaystyle E_{q}(x):=\Pi _{n=0}^{\infty }(1+q^{n}x)=\Sigma _{k=0}^{\infty }{\frac {q^{\frac {k(k-1)}{2}}x^{k}}{[k]!}}}$^{[verification needed]}, ${\displaystyle D_{q}}$ is the q-derivative, and ${\displaystyle P_{n}}$ is a "q-Appell set" (satisfying the property ${\displaystyle D_{q}P_{n}(x)=[n]P_{n-1}(x)}$).^[3]

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

$${\displaystyle \sum _{n=0}^{\infty }{\frac {\pi _{n,q}(x)t^{n}}{(1;n)}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{\frac {n(n-1)}{2}}t^{n}P_{n}(x)}{(1;n)[1-t]_{n+1}}},}$$where we employ the notation ${\displaystyle (a;k):=(1-q^{a})\dots (1-q^{a+k-1})}$ and ${\displaystyle [a+b]_{n}=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}a^{n-k}b^{k}}$.^{[3][verification needed]}