Angelescu polynomials

Short description: Polynomial sequence

In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by (Angelescu 1938). The polynomials can be given by the generating functionϕ(t1−t)exp⁡(−xt1−t)=∑n=0∞πn(x)tn.(Boas Buck)

They can also be defined by the equation πn(x):=exDn[e−xAn(x)],where An(x)n! is an Appell set of polynomials[which?] (see (Shukla 1981)).

## Contents

* 1 Properties
* 1.1 Addition and recurrence relations
* 1.2 Integrals
* 2 Further generalization
* 3 References

## Properties

### Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:

(−1)n∑r=0mLm+n−r(n)(x)πr(y)(n+m−r)!r!=∑r=0m(−1)r(−n−1r)πn−r(x+y)(m−r)!,where Lm+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 toπm(x+y)m!=∑r=0mLm−r(x)πr(y)(m−r)!r!−∑r=0m−1Lm−r−1(x)πr(y)(m−r−1)!r!.(Shastri 1940)[clarification needed]

The polynomials also satisfy the recurrence relation

πs(x)=∑r=0n(−1)n+r(nr)s!(n+s−r)!dndxn[πn+s−r(x)],

which simplifies when n=0 to π's+1(x)=(s+1)[π's(x)−πs(x)]. ((Shastri 1940)) This can be generalized to the following:

−∑r=0s1(m+n−r−1)!Lm+n−r−1(m+n−1)(x)πr−s(y)(s−r)!=1(m+n+s)!dm+ndxmdynπm+n+s(x+y),

a special case of which is the formula dm+ndxmdynπm+n(x+y)=(−1)m+n(m+n)!a0. (Shastri 1940)

### Integrals

The Angelescu polynomials satisfy the following integral formulae:

∫0∞e−x/2x[πn(x)−πn(0)]dx=∑r=0n−1(−1)n−r+1n!r!πr(0)∫0∞[11/2+p−1]n−r−1d[11/2+p]=∑r=0n−1(−1)n−r+1n!r!πr(0)n−r[1+(−1)n−r−1]

∫0∞e−x[πn(x)−πn(0)]Lm(1)(x)dx={0 if m≥nn!(n−m−1)!πn−m−1(0) if 0≤m≤n−1

(Shastri 1940)

(Here, Lm(1)(x) is a Laguerre polynomial.)

## Further generalization

We can define a q-analog of the Angelescu polynomials as πn,q(x):=eq(xqn)Dqn[Eq(−x)Pn(x)], where eq and Eq are the q-exponential functions eq(x):=Πn=0∞(1−qnx)−1=Σk=0∞xk[k]! and Eq(x):=Πn=0∞(1+qnx)=Σk=0∞qk(k−1)2xk[k]![verification needed], Dq is the q-derivative, and Pn is a "q-Appell set" (satisfying the property DqPn(x)=[n]Pn−1(x)). (Shukla 1981)

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

∑n=0∞πn,q(x)tn(1;n)=∑n=0∞(−1)nqn(n−1)2tnPn(x)(1;n)[1−t]n+1,where we employ the notation (a;k):=(1−qa)…(1−qa+k−1) and [a+b]n=∑k=0n[nk]an−kbk. (Shukla 1981)[verification needed]

## References

* 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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