In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Aurel Angelescu. The polynomials can be given by the generating function[1][2]
They can also be defined by the equation where is an Appell set of polynomials[which?].[3]
![{\displaystyle \pi _{n}(x):=e^{x}D^{n}[e^{-x}A_{n}(x)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ead5cb06218f42d4112e0accd34ad073243acd4)
Properties
Addition and recurrence relations
The Angelescu polynomials satisfy the following addition theorem:
where is a generalized Laguerre polynomial.
A particularly notable special case of this is when , in which case the formula simplifies to[clarification needed][4]
The polynomials also satisfy the recurrence relation
![{\displaystyle \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)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/763972055b87a6bd254dc0e39f8302386a444cc5)
which simplifies when to .[4] This can be generalized to the following:
![{\displaystyle \pi '_{s+1}(x)=(s+1)[\pi '_{s}(x)-\pi _{s}(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce0ef476fff5235aeb3d3f65b158ce8c2efa0d3)
a special case of which is the formula .[4]
Integrals
The Angelescu polynomials satisfy the following integral formulae:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ece5bd9062231514f288b13ae1904ac13ccffaa9)
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1270ea2b7598cb5e8962e8c96ddf75c0bc3785)
(Here, is a Laguerre polynomial.)
Further generalization
We can define a q-analog of the Angelescu polynomials as , where and are the q-exponential functions and [verification needed], is the q-derivative, and is a "q-Appell set" (satisfying the property ).[3]
![{\displaystyle \pi _{n,q}(x):=e_{q}(xq^{n})D_{q}^{n}[E_{q}(-x)P_{n}(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6846ed2ed0205854b8ff708d0dad33b3bb7bb880)
![{\displaystyle e_{q}(x):=\Pi _{n=0}^{\infty }(1-q^{n}x)^{-1}=\Sigma _{k=0}^{\infty }{\frac {x^{k}}{[k]!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3765f8e1aff6622fbbd7219bba7658509273cad1)
![{\displaystyle 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]!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/187f455df0e34f7829055c0bf7008c0295fde8bc)
![{\displaystyle D_{q}P_{n}(x)=[n]P_{n-1}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbde7001bd5b800dc33a68e1be143fbfdf00ee56)
This q-analog can also be given as a generating function as well:
where we employ the notation and .[3][verification needed]
![{\displaystyle \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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa55f46a4153e37e04250b22864f6cd5df19c9f)
![{\displaystyle [a+b]_{n}=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}a^{n-k}b^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b88b8b0c6130c2d89070555f4ea33a278b63518)
References
- ^ Angelescu (1938).
- ^ Boas & Buck (1958), p. 41.
- ^ a b c Shukla (1981).
- ^ a b c d Shastri (1940).
- 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.
You must be logged in to post a comment.