Angelescu polynomials

Template:Short description 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$$\varphi \left(\frac{t}{1-t}\right)\exp (-\frac{xt}{1-t})=\sum _{n=0}^{\mathrm{\infty }}{\pi }_n(x)t^n.$$Template:SfnpTemplate:Sfnp

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 polynomialsTemplate:Which.Template:Sfnp

Properties

Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:

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

A particularly notable special case of this is when $n=0$, in which case the formula simplifies to$$\frac{{\pi }_m(x+y)}{m!}=\sum _{r=0}^m\frac{L_{m-r}(x){\pi }_r(y)}{(m-r)!r!}-\sum _{r=0}^{m-1}\frac{L_{m-r-1}(x){\pi }_r(y)}{(m-r-1)!r!}.$$Template:ClarifyTemplate:Sfnp

The polynomials also satisfy the recurrence relation

$${\pi }_s(x)=\sum _{r=0}^n(-1{)}^{n+r}(\genfrac{}{}{0ex}{}{n}{r})\frac{s!}{(n+s-r)!}\frac{d^n}{d{x}^n}[{\pi }_{n+s-r}(x)],$$

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

$$-\sum _{r=0}^s\frac{1}{(m+n-r-1)!}{L}_{m+n-r-1}^{(m+n-1)}(x)\frac{{\pi }_{r-s}(y)}{(s-r)!}=\frac{1}{(m+n+s)!}\frac{d^{m+n}}{d{x}^{m}d{y}^n}{\pi }_{m+n+s}(x+y),$$

a special case of which is the formula $\frac{d^{m+n}}{d{x}^{m}d{y}^n}{\pi }_{m+n}(x+y)=(-1{)}^{m+n}(m+n)!a_0$.Template:Sfnp

Integrals

The Angelescu polynomials satisfy the following integral formulae:

$$\begin{array}{cc}{\int }_0^{\mathrm{\infty }}\frac{e^{-x/2}}{x}[{\pi }_n(x)-{\pi }_n(0)]dx& =\sum _{r=0}^{n-1}(-1{)}^{n-r+1}\frac{n!}{r!}{\pi }_r(0){\int }_0^{\mathrm{\infty }}[\frac{1}{1/2+p}-1{]}^{n-r-1}d[\frac{1}{1/2+p}]\\ & =\sum _{r=0}^{n-1}(-1{)}^{n-r+1}\frac{n!}{r!}\frac{{\pi }_r(0)}{n-r}[1+(-1{)}^{n-r-1}]\end{array}$$

$${\int }_0^{\mathrm{\infty }}{e}^{-x}[{\pi }_n(x)-{\pi }_n(0)]L_m^{(1)}(x)dx=\{\begin{array}{c}0\text{ if }m\ge n\\ \frac{n!}{(n-m-1)!}{\pi }_{n-m-1}(0)\text{ if }0\le m\le n-1\end{array}$$Template:Sfnp

(Here, $L_m^{(1)}(x)$ is a Laguerre polynomial.)

Further generalization

We can define a q-analog of the Angelescu polynomials as ${\pi }_{n,q}(x):=e_q(x{q}^n)D_q^n[E_q(-x)P_n(x)]$, where $e_q$ and $E_q$ are the q-exponential functions $e_q(x):={\mathrm{\Pi }}_{n=0}^{\mathrm{\infty }}(1-q^{n}x{)}^{-1}={\mathrm{\Sigma }}_{k=0}^{\mathrm{\infty }}\frac{x^k}{[k]!}$ and $E_q(x):={\mathrm{\Pi }}_{n=0}^{\mathrm{\infty }}(1+q^{n}x)={\mathrm{\Sigma }}_{k=0}^{\mathrm{\infty }}\frac{q^{\frac{k(k-1)}{2}}{x}^k}{[k]!}$Template:Verify source, $D_q$ is the q-derivative, and $P_n$ is a "q-Appell set" (satisfying the property $D_q{P}_n(x)=[n]P_{n-1}(x)$).Template:Sfnp

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

$$\sum _{n=0}^{\mathrm{\infty }}\frac{{\pi }_{n,q}(x)t^n}{(1;n)}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{q}^{\frac{n(n-1)}{2}}{t}^n{P}_n(x)}{(1;n)[1-t{]}_{n+1}},$$where we employ the notation $(a;k):=(1-q^a)\dots (1-q^{a+k-1})$ and $[a+b{]}_n=\sum _{k=0}^n\left[\begin{array}{c}n\\ k\end{array}\right]a^{n-k}{b}^k$.Template:SfnpTemplate:Verify source

References

• Template:Citation
• Template:Citation
• Template:Cite journal
• Template:Cite journal