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
ϕ
exp
[1][2] They can also be defined by the equation
is an Appell set of polynomials[which?].
[3] The Angelescu polynomials satisfy the following addition theorem:
}}=\sum _{r=0}^{m}(-1)^{r}{\binom {-n-1}{r}}{\frac {\pi _{n-r}(x+y)}{(m-r)!
is a generalized Laguerre polynomial.
A particularly notable special case of this is when
, in which case the formula simplifies to
}}=\sum _{r=0}^{m}{\frac {L_{m-r}(x)\pi _{r}(y)}{(m-r)!r!
[clarification needed][4] The polynomials also satisfy the recurrence relation
[verification needed] which simplifies when
}}L_{m+n-r-1}^{(m+n-1)}(x){\frac {\pi _{r-s}(y)}{(s-r)!
[verification needed] a special case of which is the formula
[4] 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!
}}{\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}}}
is a Laguerre polynomial.)
We can define a q-analog of the Angelescu polynomials as
are the q-exponential functions
[verification needed],
is a "q-Appell set" (satisfying the property
[3] This q-analog can also be given as a generating function as well:
where we employ the notation
{\displaystyle [a+b]_{n}=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}a^{n-k}b^{k}}
[3][verification needed]