Angelescu polynomials

In mathematics, the Angelescu polynomials π_n(x) are a series of polynomials generalizing the Laguerre polynomials introduced by (Angelescu 1938). The polynomials can be given by the generating function$${\displaystyle \varphi \left(\frac{t}{1-t}\right)\exp (-\frac{xt}{1-t})={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\pi }_n(x)t^n.}$$(Boas Buck)

They can also be defined by the equation $${\displaystyle {\pi }_n(x):=e^x{D}^n[e^{-x}{A}_n(x)],}$$where ${\displaystyle \frac{A_n(x)}{n!}}$ is an Appell set of polynomials^[which?] (see (Shukla 1981)).

The Angelescu polynomials satisfy the following addition theorem:

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

A particularly notable special case of this is when ${\displaystyle n=0}$, in which case the formula simplifies to$${\displaystyle \frac{{\pi }_m(x+y)}{m!}={\displaystyle \sum _{r=0}^m}\frac{L_{m-r}(x){\pi }_r(y)}{(m-r)!r!}-{\displaystyle \sum _{r=0}^{m-1}}\frac{L_{m-r-1}(x){\pi }_r(y)}{(m-r-1)!r!}.}$$(Shastri 1940)^{[clarification needed]}

The polynomials also satisfy the recurrence relation

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

which simplifies when ${\displaystyle n=0}$ to ${\displaystyle \pi {\text{'}}_{s+1}(x)=(s+1)[\pi {\text{'}}_s(x)-{\pi }_s(x)]}$. ((Shastri 1940)) This can be generalized to the following:

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

a special case of which is the formula ${\displaystyle \frac{d^{m+n}}{d{x}^{m}d{y}^n}{\pi }_{m+n}(x+y)=(-1{)}^{m+n}(m+n)!a_0}$. (Shastri 1940)

The Angelescu polynomials satisfy the following integral formulae:

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

$${\displaystyle {\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{e}^{-x}[{\pi }_n(x)-{\pi }_n(0)]L_m^{(1)}(x)dx=\{\begin{array}{l}0\text{ if }m\ge n\\ \frac{n!}{(n-m-1)!}{\pi }_{n-m-1}(0)\text{ if }0\le m\le n-1\end{array}}$$

(Shastri 1940)

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

We can define a q-analog of the Angelescu polynomials as ${\displaystyle {\pi }_{n,q}(x):=e_q(x{q}^n)D_q^n[E_q(-x)P_n(x)]}$, where ${\displaystyle e_q}$ and ${\displaystyle E_q}$ are the q-exponential functions ${\displaystyle e_q(x):={\mathrm{\Pi }}_{n=0}^{\mathrm{\infty }}(1-q^{n}x{)}^{-1}={\mathrm{\Sigma }}_{k=0}^{\mathrm{\infty }}\frac{x^k}{[k]!}}$ and ${\displaystyle E_q(x):={\mathrm{\Pi }}_{n=0}^{\mathrm{\infty }}(1+q^{n}x)={\mathrm{\Sigma }}_{k=0}^{\mathrm{\infty }}\frac{q^{\frac{k(k-1)}{2}}{x}^k}{[k]!}}$^{[verification needed]}, ${\displaystyle D_q}$ is the q-derivative, and ${\displaystyle P_n}$ is a "q-Appell set" (satisfying the property ${\displaystyle D_q{P}_n(x)=[n]P_{n-1}(x)}$). (Shukla 1981)

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

$${\displaystyle {\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\frac{{\pi }_{n,q}(x)t^n}{(1;n)}={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\frac{(-1{)}^n{q}^{\frac{n(n-1)}{2}}{t}^n{P}_n(x)}{(1;n)[1-t{]}_{n+1}},}$$where we employ the notation ${\displaystyle (a;k):=(1-q^a)\dots (1-q^{a+k-1})}$ and ${\displaystyle [a+b{]}_n={\displaystyle \sum _{k=0}^n}\left[\begin{array}{c}n\\ k\end{array}\right]a^{n-k}{b}^k}$. (Shukla 1981)^{[verification needed]}

• Angelescu, A. (1938), "Sur certains polynomes généralisant les polynomes de Laguerre." (in French), C. R. Acad. Sci. Roumanie 2: 199–201 
• Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge., 19, Berlin, New York: Springer-Verlag, ISBN 9783540031239, https://books.google.com/books?id=eihMuwkh4DsC 
• Shukla, D. P. (1981). "q-Angelescu polynomials". Publications de l'Institut Mathématique 43: 205–213. http://elib.mi.sanu.ac.rs/files/journals/publ/49/n043p205.pdf. 
• Shastri, N. A. (1940). "On Angelescu's polynomial πn (x)". Proceedings of the Indian Academy of Sciences, Section A 11 (4): 312–317. doi:10.1007/BF03051347. https://www.ias.ac.in/article/fulltext/seca/011/04/0312-0317. 

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