Angelescu Polynomials

In mathematics, the Angelescu polynomials π_''n''(''x'') are a series of polynomials generalizing the Laguerre polynomials introduced by . The polynomials can be given by the generating function$\phi\left(\frac t\right)\exp\left(-\frac\right)=\sum_^\infty\pi_n(x)t^n.$

They can also be defined by the equation
\pi_(x) := e^x D^n ^A_n(x)where $\frac$ is an Appell set of polynomials (see ).

Properties

Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:

$(-1)^n\sum_^m\frac = \sum_^m (-1)^r\binom \frac,$where $L^_$ is a generalized Laguerre polynomial.

A particularly notable special case of this is when $n=0$, in which case the formula simplifies to$\frac = \sum_^m \frac - \sum_^ \frac.$

The polynomials also satisfy the recurrence relation

\pi_s(x) = \sum_^n (-1)^\binom\frac\frac pi_(x)

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

$-\sum_^s \fracL^_(x)\frac = \frac\frac\pi_(x+y),$

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

Integrals

The Angelescu polynomials satisfy the following integral formulae:

\begin
\int_0^\frac pi_n(x) - \pi_n(0)x &= \sum_^ (-1)^\frac\pi_r(0)\int_0^ frac - 1 dfrac\
&= \sum_^ (-1)^\frac\frac + (-1)^\end

\int_0^ e^ pi_n(x) - \pi_n(0)_m^(x)dx = \begin
0\textm\geq n\\
\frac\pi_(0)\text0\leq m\leq n-1
\end

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

Further generalization

We can define a q-analog of the Angelescu polynomials as \pi_(x) := e_q(xq^n) D_q^n _q(-x)P_n(x)/math>, where $e_q$ and $E_q$ are the q-exponential functions $e_q(x) := \Pi_^ (1 - q^n x)^ = \Sigma_^\frac$ and $E_q(x) := \Pi_^ (1 + q^n x) = \Sigma_^\frac$, $D_q$ is the q-derivative, and $P_n$ is a "q-Appell set" (satisfying the property D_q P_n(x) = _(x)).

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

$\sum_^\frac = \sum_^\frac,$where we employ the notation $(a;k) := (1 - q^a)\dots (1 - q^)$ and +bn = \sum_^n\beginn\\k\enda^b^k.

References

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*{{Cite journal, last=Shastri, first=N. A., date=1940, title=On Angelescu's polynomial πn (x), url=https://www.ias.ac.in/article/fulltext/seca/011/04/0312-0317, journal=Proceedings of the Indian Academy of Sciences, Section A, volume=11, issue=4, pages=312–317, doi=10.1007/BF03051347, s2cid=125446896

Polynomials