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

$$\phi \left({\frac {t}{1-t}}\right)\exp \left(-{\frac {xt}{1-t}}\right)=\sum _{n=0}^{\infty }\pi _{n}(x)t^{n}.$$

Boas & Buck (1958, p.41)

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 (see Shukla (1981)).

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 $L_{m+n-r}^{(n)}$ is a generalized Laguerre polynomial.

A particularly notable special case of this is when $n=0$, in which case the formula simplifies to

$${\frac {\pi _{m}(x+y)}{m!}}=\sum _{r=0}^{m}{\frac {L_{m-r}(x)\pi _{r}(y)}{(m-r)!r!}}-\sum _{r=0}^{m-1}{\frac {L_{m-r-1}(x)\pi _{r}(y)}{(m-r-1)!r!}}.$$

Shastri (1940)

The polynomials also satisfy the recurrence relation

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

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

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

a special case of which is the formula ${\frac {d^{m+n}}{dx^{m}dy^{n}}}\pi _{m+n}(x+y)=(-1)^{m+n}(m+n)!a_{0}$. Shastri (1940)

Integrals

The Angelescu polynomials satisfy the following integral formulae:

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

$$\int _{0}^{\infty }e^{-x}[\pi _{n}(x)-\pi _{n}(0)]L_{m}^{(1)}(x)dx={\begin{cases}0{\text{ if }}m\geq n\\{\frac {n!}{(n-m-1)!}}\pi _{n-m-1}(0){\text{ if }}0\leq m\leq n-1\end{cases}}$$

Shastri (1940)

(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 $E_{q}(x):=\Pi _{n=0}^{\infty }(1+q^{n}x)=\Sigma _{k=0}^{\infty }{\frac {q^{\frac {k(k-1)}{2}}x^{k}}{[k]!}}$, $D_{q}$ is the q-derivative, and $P_{n}$ is a "q-Appell set" (satisfying the property $D_{q}P_{n}(x)=[n]P_{n-1}(x)$). Shukla (1981)

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 $[a+b]_{n}=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}a^{n-k}b^{k}$. Shukla (1981)

References

• Angelescu, A. (1938), "Sur certains polynomes généralisant les polynomes de Laguerre.", C. R. Acad. Sci. Roumanie (in French), 2: 199–201, JFM 64.0328.01
• Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge., vol. 19, Berlin, New York: Springer-Verlag, ISBN 9783540031239, MR 0094466
• Shukla, D. P. (1981). "q-Angelescu polynomials" (PDF). Publications de l'Institut Mathématique. 43: 205–213.
• 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. S2CID 125446896.