Angelescu polynomials

Polynomial sequence

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 ϕ ( t 1 − t ) exp ⁡ ( − x t 1 − t ) = ∑ n = 0 ∞ π n ( x ) t n . {\displaystyle \phi \left({\frac {t}{1-t}}\right)\exp \left(-{\frac {xt}{1-t}}\right)=\sum _{n=0}^{\infty }\pi _{n}(x)t^{n}.} [1][2]

They can also be defined by the equation π n ( x ) := e x D n [ e − x A n ( x ) ] , {\displaystyle \pi _{n}(x):=e^{x}D^{n}[e^{-x}A_{n}(x)],} where A n ( x ) n ! {\displaystyle {\frac {A_{n}(x)}{n!}}} is an Appell set of polynomials[which?].[3]

Properties

Addition and recurrence relations

The Angelescu polynomials satisfy the following addition theorem:

( − 1 ) n ∑ r = 0 m L m + n − r ( n ) ( x ) π r ( y ) ( n + m − r ) ! r ! = ∑ r = 0 m ( − 1 ) r ( − n − 1 r ) π n − r ( x + y ) ( m − r ) ! , {\displaystyle (-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 ) {\displaystyle L_{m+n-r}^{(n)}} is a generalized Laguerre polynomial.

A particularly notable special case of this is when n = 0 {\displaystyle n=0} , in which case the formula simplifies to π m ( x + y ) m ! = ∑ r = 0 m L m − r ( x ) π r ( y ) ( m − r ) ! r ! − ∑ r = 0 m − 1 L m − r − 1 ( x ) π r ( y ) ( m − r − 1 ) ! r ! . {\displaystyle {\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!}}.} [clarification needed][4]

The polynomials also satisfy the recurrence relation

π s ( x ) = ∑ r = 0 n ( − 1 ) n + r ( n r ) s ! ( n + s − r ) ! d n d x n [ π n + s − r ( x ) ] , {\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)],} [verification needed]

which simplifies when n = 0 {\displaystyle n=0} to π s + 1 ′ ( x ) = ( s + 1 ) [ π s ′ ( x ) − π s ( x ) ] {\displaystyle \pi '_{s+1}(x)=(s+1)[\pi '_{s}(x)-\pi _{s}(x)]} .[4] This can be generalized to the following:

− ∑ r = 0 s 1 ( m + n − r − 1 ) ! L m + n − r − 1 ( m + n − 1 ) ( x ) π r − s ( y ) ( s − r ) ! = 1 ( m + n + s ) ! d m + n d x m d y n π m + n + s ( x + y ) , {\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}}{dx^{m}dy^{n}}}\pi _{m+n+s}(x+y),} [verification needed]

a special case of which is the formula d m + n d x m d y n π m + n ( x + y ) = ( − 1 ) m + n ( m + n ) ! a 0 {\displaystyle {\frac {d^{m+n}}{dx^{m}dy^{n}}}\pi _{m+n}(x+y)=(-1)^{m+n}(m+n)!a_{0}} .[4]

Integrals

The Angelescu polynomials satisfy the following integral formulae:

∫ 0 ∞ e − x / 2 x [ π n ( x ) − π n ( 0 ) ] d x = ∑ r = 0 n − 1 ( − 1 ) n − r + 1 n ! r ! π r ( 0 ) ∫ 0 ∞ [ 1 1 / 2 + p − 1 ] n − r − 1 d [ 1 1 / 2 + p ] = ∑ r = 0 n − 1 ( − 1 ) n − r + 1 n ! r ! π r ( 0 ) n − r [ 1 + ( − 1 ) n − r − 1 ] {\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}}}

∫ 0 ∞ e − x [ π n ( x ) − π n ( 0 ) ] L m ( 1 ) ( x ) d x = { 0  if  m ≥ n n ! ( n − m − 1 ) ! π n − m − 1 ( 0 )  if  0 ≤ m ≤ n − 1 {\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}}} [4]

(Here, L m ( 1 ) ( x ) {\displaystyle L_{m}^{(1)}(x)} is a Laguerre polynomial.)

Further generalization

We can define a q-analog of the Angelescu polynomials as π n , q ( x ) := e q ( x q n ) D q n [ E q ( − x ) P n ( x ) ] {\displaystyle \pi _{n,q}(x):=e_{q}(xq^{n})D_{q}^{n}[E_{q}(-x)P_{n}(x)]} , where e q {\displaystyle e_{q}} and E q {\displaystyle E_{q}} are the q-exponential functions e q ( x ) := Π n = 0 ∞ ( 1 − q n x ) − 1 = Σ k = 0 ∞ x k [ k ] ! {\displaystyle e_{q}(x):=\Pi _{n=0}^{\infty }(1-q^{n}x)^{-1}=\Sigma _{k=0}^{\infty }{\frac {x^{k}}{[k]!}}} and E q ( x ) := Π n = 0 ∞ ( 1 + q n x ) = Σ k = 0 ∞ q k ( k − 1 ) 2 x k [ k ] ! {\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]!}}} [verification needed], D q {\displaystyle D_{q}} is the q-derivative, and P n {\displaystyle P_{n}} is a "q-Appell set" (satisfying the property D q P n ( x ) = [ n ] P n − 1 ( x ) {\displaystyle D_{q}P_{n}(x)=[n]P_{n-1}(x)} ).[3]

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

∑ n = 0 ∞ π n , q ( x ) t n ( 1 ; n ) = ∑ n = 0 ∞ ( − 1 ) n q n ( n − 1 ) 2 t n P n ( x ) ( 1 ; n ) [ 1 − t ] n + 1 , {\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}}},} where we employ the notation ( a ; k ) := ( 1 − q a ) … ( 1 − q a + k − 1 ) {\displaystyle (a;k):=(1-q^{a})\dots (1-q^{a+k-1})} and [ a + b ] n = ∑ k = 0 n [ n k ] a n − k b k {\displaystyle [a+b]_{n}=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}a^{n-k}b^{k}} .[3][verification needed]

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.