Английская Википедия:Angelescu polynomials
In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Шаблон:Harvtxt. The polynomials can be given by the generating function<math display="block">\phi\left(\frac t{1-t}\right)\exp\left(-\frac{xt}{1-t}\right)=\sum_{n=0}^\infty\pi_n(x)t^n. </math>Шаблон:Harvtxt
They can also be defined by the equation <math display="block">\pi_{n}(x) := e^x D^n[e^{-x}A_n(x)],</math>where <math>\frac{A_n(x)}{n!}</math> is an Appell set of polynomialsШаблон:Which (see Шаблон:Harvtxt).
Properties
Addition and recurrence relations
The Angelescu polynomials satisfy the following addition theorem:
<math display="block">(-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)!},</math>where <math>L^{(n)}_{m+n-r}</math> is a generalized Laguerre polynomial.
A particularly notable special case of this is when <math>n=0</math>, in which case the formula simplifies to<math display="block">\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!}.</math>Шаблон:HarvtxtШаблон:Clarify
The polynomials also satisfy the recurrence relation
<math display="block">\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)],</math>
Шаблон:Verify source which simplifies when <math>n=0</math> to <math>\pi'_{s+1}(x) = (s+1)[\pi'_s(x) - \pi_s(x)]</math>. (Шаблон:Harvtxt) This can be generalized to the following:
<math display="block">-\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),</math>
Шаблон:Verify source a special case of which is the formula <math>\frac{d^{m+n}}{dx^m dy^n}\pi_{m+n}(x+y) = (-1)^{m+n} (m+n)! a_0</math>. Шаблон:Harvtxt
Integrals
The Angelescu polynomials satisfy the following integral formulae:
<math display="block">\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}</math>
<math display="block">\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}</math>
(Here, <math>L_m^{(1)}(x)</math> is a Laguerre polynomial.)
Further generalization
We can define a q-analog of the Angelescu polynomials as <math>\pi_{n, q}(x) := e_q(xq^n) D_q^n[E_q(-x)P_n(x)]</math>, where <math>e_q</math> and <math>E_q</math> are the q-exponential functions <math>e_q(x) := \Pi_{n=0}^{\infty} (1 - q^n x)^{-1} = \Sigma_{k=0}^{\infty}\frac{x^k}{[k]!}</math> and <math>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]!}</math>Шаблон:Verify source, <math>D_q</math> is the q-derivative, and <math>P_n</math> is a "q-Appell set" (satisfying the property <math>D_q P_n(x) = [n]P_{n-1}(x)</math>). Шаблон:Harvtxt
This q-analog can also be given as a generating function as well:
<math display="block">\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}},</math>where we employ the notation <math>(a;k) := (1 - q^a)\dots (1 - q^{a+k-1})</math> and <math>[a+b]_n = \sum_{k=0}^n\begin{bmatrix}n\\k\end{bmatrix}a^{n-k}b^k</math>. Шаблон:HarvtxtШаблон:Verify source
References
