Polynomial sequence
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 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}.}
Boas & Buck (1958 , p.41)
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? (see
Shukla (1981) ).
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}
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!}}.}
Shastri (1940) [clarification needed
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)]}
. (
Shastri (1940) ) 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}}
.
Shastri (1940)
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}}}
Shastri (1940)
(Here,
L
m
(
1
)
(
x
)
{\displaystyle L_{m}^{(1)}(x)}
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)]}
e
q
{\displaystyle e_{q}}
E
q
{\displaystyle E_{q}}
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]!}}}
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}}
q-derivative , and
P
n
{\displaystyle P_{n}}
D
q
P
n
(
x
)
=
[
n
]
P
n
−
1
(
x
)
{\displaystyle 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:
∑
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}}
.
Shukla (1981) [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 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 .
![{\displaystyle \pi _{n}(x):=e^{x}D^{n}[e^{-x}A_{n}(x)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ead5cb06218f42d4112e0accd34ad073243acd4)
![{\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)
![{\displaystyle \pi '_{s+1}(x)=(s+1)[\pi '_{s}(x)-\pi _{s}(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce0ef476fff5235aeb3d3f65b158ce8c2efa0d3)
![{\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)
![{\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)
![{\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)
