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Angelescu polynomials

From Wikipedia, the free encyclopedia

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]

Properties

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Addition and recurrence relations

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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

[verification needed]

which simplifies when to .[4] This can be generalized to the following:

[verification needed]

a special case of which is the formula .[4]

Integrals

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The Angelescu polynomials satisfy the following integral formulae:

[4]

(Here, is a Laguerre polynomial.)

Further generalization

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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]

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

where we employ the notation and .[3][verification needed]

References

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  • 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.