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.

[1][2]

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[which?].[3]

Properties

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!}.

[clarification needed][4]

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)],

[verification needed]

which simplifies when n=0 to \pi'_{s+1}(x) = (s+1)[\pi'_s(x) - \pi_s(x)].[4] 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),

[verification needed]

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.[4]

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}

[4]

(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]!}[verification needed], 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)).[3]

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.[3][verification needed]

References

  1. ^ Angelescu (1938).
  2. ^ Boas & Buck (1958), p. 41.
  3. ^ Shukla (1981).
  4. ^ Shastri (1940).