Sequence of differential equation solutions
Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i In mathematics , the Laguerre polynomials , named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation:
x
y
″
+
(
1
−
x
)
y
′
+
n
y
=
0
,
y
=
y
(
x
)
{\displaystyle xy''+(1-x)y'+ny=0,\ y=y(x)}
which is a second-order
nonsingular solutions
only if
n is a non-negative integer.
Sometimes the name Laguerre polynomials is used for solutions of
x
y
″
+
(
α
+
1
−
x
)
y
′
+
n
y
=
0
.
{\displaystyle xy''+(\alpha +1-x)y'+ny=0~.}
where
n is still a non-negative integer.
Then they are also named
generalized Laguerre polynomials , as will be done here (alternatively
associated Laguerre polynomials or, rarely,
Sonine polynomials , after their inventor
[1] Nikolay Yakovlevich Sonin ).
More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer.
The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form
∫
0
∞
f
(
x
)
e
−
x
d
x
.
{\displaystyle \int _{0}^{\infty }f(x)e^{-x}\,dx.}
These polynomials, usually denoted L 0 L 1
Rodrigues formula
,
L
n
(
x
)
=
e
x
n
!
d
n
d
x
n
(
e
−
x
x
n
)
=
1
n
!
(
d
d
x
−
1
)
n
x
n
,
{\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right)={\frac {1}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n},}
reducing to the closed form of a following section.
They are
inner product
⟨
f
,
g
⟩
=
∫
0
∞
f
(
x
)
g
(
x
)
e
−
x
d
x
.
{\displaystyle \langle f,g\rangle =\int _{0}^{\infty }f(x)g(x)e^{-x}\,dx.}
The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials .
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the
.
Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n ! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
The first few polynomials These are the first few Laguerre polynomials:
n
L
n
(
x
)
{\displaystyle L_{n}(x)\,}
0
1
{\displaystyle 1\,}
1
−
x
+
1
{\displaystyle -x+1\,}
2
1
2
(
x
2
−
4
x
+
2
)
{\displaystyle {\tfrac {1}{2}}(x^{2}-4x+2)\,}
3
1
6
(
−
x
3
+
9
x
2
−
18
x
+
6
)
{\displaystyle {\tfrac {1}{6}}(-x^{3}+9x^{2}-18x+6)\,}
4
1
24
(
x
4
−
16
x
3
+
72
x
2
−
96
x
+
24
)
{\displaystyle {\tfrac {1}{24}}(x^{4}-16x^{3}+72x^{2}-96x+24)\,}
5
1
120
(
−
x
5
+
25
x
4
−
200
x
3
+
600
x
2
−
600
x
+
120
)
{\displaystyle {\tfrac {1}{120}}(-x^{5}+25x^{4}-200x^{3}+600x^{2}-600x+120)\,}
6
1
720
(
x
6
−
36
x
5
+
450
x
4
−
2400
x
3
+
5400
x
2
−
4320
x
+
720
)
{\displaystyle {\tfrac {1}{720}}(x^{6}-36x^{5}+450x^{4}-2400x^{3}+5400x^{2}-4320x+720)\,}
n
1
n
!
(
(
−
x
)
n
+
n
2
(
−
x
)
n
−
1
+
⋯
+
n
(
n
!
)
(
−
x
)
+
n
!
)
{\displaystyle {\tfrac {1}{n!}}((-x)^{n}+n^{2}(-x)^{n-1}+\dots +n({n!})(-x)+n!)\,}
The first six Laguerre polynomials. Recursive definition, closed form, and generating function One can also define the Laguerre polynomials recursively, defining the first two polynomials as
L
0
(
x
)
=
1
{\displaystyle L_{0}(x)=1}
L
1
(
x
)
=
1
−
x
{\displaystyle L_{1}(x)=1-x}
and then using the following
recurrence relation for any
k ≥ 1:
L
k
+
1
(
x
)
=
(
2
k
+
1
−
x
)
L
k
(
x
)
−
k
L
k
−
1
(
x
)
k
+
1
.
{\displaystyle L_{k+1}(x)={\frac {(2k+1-x)L_{k}(x)-kL_{k-1}(x)}{k+1}}.}
Furthermore,
x
L
n
′
(
x
)
=
n
L
n
(
x
)
−
n
L
n
−
1
(
x
)
.
{\displaystyle xL'_{n}(x)=nL_{n}(x)-nL_{n-1}(x).}
In solution of some boundary value problems, the characteristic values can be useful:
L
k
(
0
)
=
1
,
L
k
′
(
0
)
=
−
k
.
{\displaystyle L_{k}(0)=1,L_{k}'(0)=-k.}
The closed form is
L
n
(
x
)
=
∑
k
=
0
n
(
n
k
)
(
−
1
)
k
k
!
x
k
.
{\displaystyle L_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{k!}}x^{k}.}
The generating function for them likewise follows,
∑
n
=
0
∞
t
n
L
n
(
x
)
=
1
1
−
t
e
−
t
x
/
(
1
−
t
)
.
{\displaystyle \sum _{n=0}^{\infty }t^{n}L_{n}(x)={\frac {1}{1-t}}e^{-tx/(1-t)}.}
The operator form is
L
n
(
x
)
=
1
n
!
e
x
d
n
d
x
n
(
x
n
e
−
x
)
{\displaystyle L_{n}(x)={\frac {1}{n!}}e^{x}{\frac {d^{n}}{dx^{n}}}(x^{n}e^{-x})}
Polynomials of negative index can be expressed using the ones with positive index:
L
−
n
(
x
)
=
e
x
L
n
−
1
(
−
x
)
.
{\displaystyle L_{-n}(x)=e^{x}L_{n-1}(-x).}
Generalized Laguerre polynomials For arbitrary real α the polynomial solutions of the differential equation[2]
x
y
″
+
(
α
+
1
−
x
)
y
′
+
n
y
=
0
{\displaystyle x\,y''+\left(\alpha +1-x\right)y'+n\,y=0}
are called
generalized Laguerre polynomials , or
associated Laguerre polynomials .
One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as
L
0
(
α
)
(
x
)
=
1
{\displaystyle L_{0}^{(\alpha )}(x)=1}
L
1
(
α
)
(
x
)
=
1
+
α
−
x
{\displaystyle L_{1}^{(\alpha )}(x)=1+\alpha -x}
and then using the following recurrence relation for any k ≥ 1
L
k
+
1
(
α
)
(
x
)
=
(
2
k
+
1
+
α
−
x
)
L
k
(
α
)
(
x
)
−
(
k
+
α
)
L
k
−
1
(
α
)
(
x
)
k
+
1
.
{\displaystyle L_{k+1}^{(\alpha )}(x)={\frac {(2k+1+\alpha -x)L_{k}^{(\alpha )}(x)-(k+\alpha )L_{k-1}^{(\alpha )}(x)}{k+1}}.}
The simple Laguerre polynomials are the special case α = 0
L
n
(
0
)
(
x
)
=
L
n
(
x
)
.
{\displaystyle L_{n}^{(0)}(x)=L_{n}(x).}
The
Rodrigues formula
for them is
L
n
(
α
)
(
x
)
=
x
−
α
e
x
n
!
d
n
d
x
n
(
e
−
x
x
n
+
α
)
=
x
−
α
n
!
(
d
d
x
−
1
)
n
x
n
+
α
.
{\displaystyle L_{n}^{(\alpha )}(x)={x^{-\alpha }e^{x} \over n!}{d^{n} \over dx^{n}}\left(e^{-x}x^{n+\alpha }\right)={\frac {x^{-\alpha }}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n+\alpha }.}
The generating function for them is
∑
n
=
0
∞
t
n
L
n
(
α
)
(
x
)
=
1
(
1
−
t
)
α
+
1
e
−
t
x
/
(
1
−
t
)
.
{\displaystyle \sum _{n=0}^{\infty }t^{n}L_{n}^{(\alpha )}(x)={\frac {1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.}
The first few generalized Laguerre polynomials, Ln (k ) (x ) Explicit examples and properties of the generalized Laguerre polynomials As a contour integral Given the generating function specified above, the polynomials may be expressed in terms of a
contour integral
L
n
(
α
)
(
x
)
=
1
2
π
i
∮
C
e
−
x
t
/
(
1
−
t
)
(
1
−
t
)
α
+
1
t
n
+
1
d
t
,
{\displaystyle L_{n}^{(\alpha )}(x)={\frac {1}{2\pi i}}\oint _{C}{\frac {e^{-xt/(1-t)}}{(1-t)^{\alpha +1}\,t^{n+1}}}\;dt,}
where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1
Recurrence relations The addition formula for Laguerre polynomials:[8]
L
n
(
α
+
β
+
1
)
(
x
+
y
)
=
∑
i
=
0
n
L
i
(
α
)
(
x
)
L
n
−
i
(
β
)
(
y
)
.
{\displaystyle L_{n}^{(\alpha +\beta +1)}(x+y)=\sum _{i=0}^{n}L_{i}^{(\alpha )}(x)L_{n-i}^{(\beta )}(y).}
Laguerre's polynomials satisfy the recurrence relations
L
n
(
α
)
(
x
)
=
∑
i
=
0
n
L
n
−
i
(
α
+
i
)
(
y
)
(
y
−
x
)
i
i
!
,
{\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}L_{n-i}^{(\alpha +i)}(y){\frac {(y-x)^{i}}{i!}},}
in particular
L
n
(
α
+
1
)
(
x
)
=
∑
i
=
0
n
L
i
(
α
)
(
x
)
{\displaystyle L_{n}^{(\alpha +1)}(x)=\sum _{i=0}^{n}L_{i}^{(\alpha )}(x)}
and
L
n
(
α
)
(
x
)
=
∑
i
=
0
n
(
α
−
β
+
n
−
i
−
1
n
−
i
)
L
i
(
β
)
(
x
)
,
{\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}{\alpha -\beta +n-i-1 \choose n-i}L_{i}^{(\beta )}(x),}
or
L
n
(
α
)
(
x
)
=
∑
i
=
0
n
(
α
−
β
+
n
n
−
i
)
L
i
(
β
−
i
)
(
x
)
;
{\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}{\alpha -\beta +n \choose n-i}L_{i}^{(\beta -i)}(x);}
moreover
L
n
(
α
)
(
x
)
−
∑
j
=
0
Δ
−
1
(
n
+
α
n
−
j
)
(
−
1
)
j
x
j
j
!
=
(
−
1
)
Δ
x
Δ
(
Δ
−
1
)
!
∑
i
=
0
n
−
Δ
(
n
+
α
n
−
Δ
−
i
)
(
n
−
i
)
(
n
i
)
L
i
(
α
+
Δ
)
(
x
)
=
(
−
1
)
Δ
x
Δ
(
Δ
−
1
)
!
∑
i
=
0
n
−
Δ
(
n
+
α
−
i
−
1
n
−
Δ
−
i
)
(
n
−
i
)
(
n
i
)
L
i
(
n
+
α
+
Δ
−
i
)
(
x
)
{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)-\sum _{j=0}^{\Delta -1}{n+\alpha \choose n-j}(-1)^{j}{\frac {x^{j}}{j!}}&=(-1)^{\Delta }{\frac {x^{\Delta }}{(\Delta -1)!}}\sum _{i=0}^{n-\Delta }{\frac {n+\alpha \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(\alpha +\Delta )}(x)\\[6pt]&=(-1)^{\Delta }{\frac {x^{\Delta }}{(\Delta -1)!}}\sum _{i=0}^{n-\Delta }{\frac {n+\alpha -i-1 \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(n+\alpha +\Delta -i)}(x)\end{aligned}}}
They can be used to derive the four 3-point-rules
L
n
(
α
)
(
x
)
=
L
n
(
α
+
1
)
(
x
)
−
L
n
−
1
(
α
+
1
)
(
x
)
=
∑
j
=
0
k
(
k
j
)
L
n
−
j
(
α
+
k
)
(
x
)
,
n
L
n
(
α
)
(
x
)
=
(
n
+
α
)
L
n
−
1
(
α
)
(
x
)
−
x
L
n
−
1
(
α
+
1
)
(
x
)
,
or
x
k
k
!
L
n
(
α
)
(
x
)
=
∑
i
=
0
k
(
−
1
)
i
(
n
+
i
i
)
(
n
+
α
k
−
i
)
L
n
+
i
(
α
−
k
)
(
x
)
,
n
L
n
(
α
+
1
)
(
x
)
=
(
n
−
x
)
L
n
−
1
(
α
+
1
)
(
x
)
+
(
n
+
α
)
L
n
−
1
(
α
)
(
x
)
x
L
n
(
α
+
1
)
(
x
)
=
(
n
+
α
)
L
n
−
1
(
α
)
(
x
)
−
(
n
−
x
)
L
n
(
α
)
(
x
)
;
{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)&=L_{n}^{(\alpha +1)}(x)-L_{n-1}^{(\alpha +1)}(x)=\sum _{j=0}^{k}{k \choose j}L_{n-j}^{(\alpha +k)}(x),\\[10pt]nL_{n}^{(\alpha )}(x)&=(n+\alpha )L_{n-1}^{(\alpha )}(x)-xL_{n-1}^{(\alpha +1)}(x),\\[10pt]&{\text{or }}\\{\frac {x^{k}}{k!}}L_{n}^{(\alpha )}(x)&=\sum _{i=0}^{k}(-1)^{i}{n+i \choose i}{n+\alpha \choose k-i}L_{n+i}^{(\alpha -k)}(x),\\[10pt]nL_{n}^{(\alpha +1)}(x)&=(n-x)L_{n-1}^{(\alpha +1)}(x)+(n+\alpha )L_{n-1}^{(\alpha )}(x)\\[10pt]xL_{n}^{(\alpha +1)}(x)&=(n+\alpha )L_{n-1}^{(\alpha )}(x)-(n-x)L_{n}^{(\alpha )}(x);\end{aligned}}}
combined they give this additional, useful recurrence relations
L
n
(
α
)
(
x
)
=
(
2
+
α
−
1
−
x
n
)
L
n
−
1
(
α
)
(
x
)
−
(
1
+
α
−
1
n
)
L
n
−
2
(
α
)
(
x
)
=
α
+
1
−
x
n
L
n
−
1
(
α
+
1
)
(
x
)
−
x
n
L
n
−
2
(
α
+
2
)
(
x
)
{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)&=\left(2+{\frac {\alpha -1-x}{n}}\right)L_{n-1}^{(\alpha )}(x)-\left(1+{\frac {\alpha -1}{n}}\right)L_{n-2}^{(\alpha )}(x)\\[10pt]&={\frac {\alpha +1-x}{n}}L_{n-1}^{(\alpha +1)}(x)-{\frac {x}{n}}L_{n-2}^{(\alpha +2)}(x)\end{aligned}}}
Since
L
n
(
α
)
(
x
)
{\displaystyle L_{n}^{(\alpha )}(x)}
n
{\displaystyle n}
α
{\displaystyle \alpha }
partial fraction decomposition
n
!
L
n
(
α
)
(
x
)
(
α
+
1
)
n
=
1
−
∑
j
=
1
n
(
−
1
)
j
j
α
+
j
(
n
j
)
L
n
(
−
j
)
(
x
)
=
1
−
∑
j
=
1
n
x
j
α
+
j
L
n
−
j
(
j
)
(
x
)
(
j
−
1
)
!
=
1
−
x
∑
i
=
1
n
L
n
−
i
(
−
α
)
(
x
)
L
i
−
1
(
α
+
1
)
(
−
x
)
α
+
i
.
{\displaystyle {\begin{aligned}{\frac {n!\,L_{n}^{(\alpha )}(x)}{(\alpha +1)_{n}}}&=1-\sum _{j=1}^{n}(-1)^{j}{\frac {j}{\alpha +j}}{n \choose j}L_{n}^{(-j)}(x)\\&=1-\sum _{j=1}^{n}{\frac {x^{j}}{\alpha +j}}\,\,{\frac {L_{n-j}^{(j)}(x)}{(j-1)!}}\\&=1-x\sum _{i=1}^{n}{\frac {L_{n-i}^{(-\alpha )}(x)L_{i-1}^{(\alpha +1)}(-x)}{\alpha +i}}.\end{aligned}}}
The second equality follows by the following identity, valid for integer
i and
n and immediate from the expression of
L
n
(
α
)
(
x
)
{\displaystyle L_{n}^{(\alpha )}(x)}
in terms of
Charlier polynomials :
(
−
x
)
i
i
!
L
n
(
i
−
n
)
(
x
)
=
(
−
x
)
n
n
!
L
i
(
n
−
i
)
(
x
)
.
{\displaystyle {\frac {(-x)^{i}}{i!}}L_{n}^{(i-n)}(x)={\frac {(-x)^{n}}{n!}}L_{i}^{(n-i)}(x).}
For the third equality apply the fourth and fifth identities of this section.
Derivatives of generalized Laguerre polynomials Differentiating the power series representation of a generalized Laguerre polynomial k times leads to
d
k
d
x
k
L
n
(
α
)
(
x
)
=
{
(
−
1
)
k
L
n
−
k
(
α
+
k
)
(
x
)
if
k
≤
n
,
0
otherwise.
{\displaystyle {\frac {d^{k}}{dx^{k}}}L_{n}^{(\alpha )}(x)={\begin{cases}(-1)^{k}L_{n-k}^{(\alpha +k)}(x)&{\text{if }}k\leq n,\\0&{\text{otherwise.}}\end{cases}}}
This points to a special case (α = 0α = k
L
n
(
k
)
(
x
)
=
(
−
1
)
k
d
k
L
n
+
k
(
x
)
d
x
k
,
{\displaystyle L_{n}^{(k)}(x)=(-1)^{k}{\frac {d^{k}L_{n+k}(x)}{dx^{k}}},}
the shift by
k sometimes causing confusion with the usual parenthesis notation for a derivative.
Moreover, the following equation holds:
1
k
!
d
k
d
x
k
x
α
L
n
(
α
)
(
x
)
=
(
n
+
α
k
)
x
α
−
k
L
n
(
α
−
k
)
(
x
)
,
{\displaystyle {\frac {1}{k!}}{\frac {d^{k}}{dx^{k}}}x^{\alpha }L_{n}^{(\alpha )}(x)={n+\alpha \choose k}x^{\alpha -k}L_{n}^{(\alpha -k)}(x),}
which generalizes with
Cauchy's formula to
L
n
(
α
′
)
(
x
)
=
(
α
′
−
α
)
(
α
′
+
n
α
′
−
α
)
∫
0
x
t
α
(
x
−
t
)
α
′
−
α
−
1
x
α
′
L
n
(
α
)
(
t
)
d
t
.
{\displaystyle L_{n}^{(\alpha ')}(x)=(\alpha '-\alpha ){\alpha '+n \choose \alpha '-\alpha }\int _{0}^{x}{\frac {t^{\alpha }(x-t)^{\alpha '-\alpha -1}}{x^{\alpha '}}}L_{n}^{(\alpha )}(t)\,dt.}
The derivative with respect to the second variable α has the form,[9]
d
d
α
L
n
(
α
)
(
x
)
=
∑
i
=
0
n
−
1
L
i
(
α
)
(
x
)
n
−
i
.
{\displaystyle {\frac {d}{d\alpha }}L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n-1}{\frac {L_{i}^{(\alpha )}(x)}{n-i}}.}
This is evident from the contour integral representation below.
The generalized Laguerre polynomials obey the differential equation
x
L
n
(
α
)
′
′
(
x
)
+
(
α
+
1
−
x
)
L
n
(
α
)
′
(
x
)
+
n
L
n
(
α
)
(
x
)
=
0
,
{\displaystyle xL_{n}^{(\alpha )\prime \prime }(x)+(\alpha +1-x)L_{n}^{(\alpha )\prime }(x)+nL_{n}^{(\alpha )}(x)=0,}
which may be compared with the equation obeyed by the
k th derivative of the ordinary Laguerre polynomial,
x
L
n
[
k
]
′
′
(
x
)
+
(
k
+
1
−
x
)
L
n
[
k
]
′
(
x
)
+
(
n
−
k
)
L
n
[
k
]
(
x
)
=
0
,
{\displaystyle xL_{n}^{[k]\prime \prime }(x)+(k+1-x)L_{n}^{[k]\prime }(x)+(n-k)L_{n}^{[k]}(x)=0,}
where
L
n
[
k
]
(
x
)
≡
d
k
L
n
(
x
)
d
x
k
{\displaystyle L_{n}^{[k]}(x)\equiv {\frac {d^{k}L_{n}(x)}{dx^{k}}}}
for this equation only.
In Sturm–Liouville form the differential equation is
−
(
x
α
+
1
e
−
x
⋅
L
n
(
α
)
(
x
)
′
)
′
=
n
⋅
x
α
e
−
x
⋅
L
n
(
α
)
(
x
)
,
{\displaystyle -\left(x^{\alpha +1}e^{-x}\cdot L_{n}^{(\alpha )}(x)^{\prime }\right)'=n\cdot x^{\alpha }e^{-x}\cdot L_{n}^{(\alpha )}(x),}
which shows that L (α) n n .
Orthogonality The generalized Laguerre polynomials are
weighting function
xα e −x :
[10]
∫
0
∞
x
α
e
−
x
L
n
(
α
)
(
x
)
L
m
(
α
)
(
x
)
d
x
=
Γ
(
n
+
α
+
1
)
n
!
δ
n
,
m
,
{\displaystyle \int _{0}^{\infty }x^{\alpha }e^{-x}L_{n}^{(\alpha )}(x)L_{m}^{(\alpha )}(x)dx={\frac {\Gamma (n+\alpha +1)}{n!}}\delta _{n,m},}
which follows from
∫
0
∞
x
α
′
−
1
e
−
x
L
n
(
α
)
(
x
)
d
x
=
(
α
−
α
′
+
n
n
)
Γ
(
α
′
)
.
{\displaystyle \int _{0}^{\infty }x^{\alpha '-1}e^{-x}L_{n}^{(\alpha )}(x)dx={\alpha -\alpha '+n \choose n}\Gamma (\alpha ').}
If
Γ
(
x
,
α
+
1
,
1
)
{\displaystyle \Gamma (x,\alpha +1,1)}
∫
0
∞
L
n
(
α
)
(
x
)
L
m
(
α
)
(
x
)
Γ
(
x
,
α
+
1
,
1
)
d
x
=
(
n
+
α
n
)
δ
n
,
m
,
{\displaystyle \int _{0}^{\infty }L_{n}^{(\alpha )}(x)L_{m}^{(\alpha )}(x)\Gamma (x,\alpha +1,1)dx={n+\alpha \choose n}\delta _{n,m},}
The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula )[citation needed
K
n
(
α
)
(
x
,
y
)
:=
1
Γ
(
α
+
1
)
∑
i
=
0
n
L
i
(
α
)
(
x
)
L
i
(
α
)
(
y
)
(
α
+
i
i
)
=
1
Γ
(
α
+
1
)
L
n
(
α
)
(
x
)
L
n
+
1
(
α
)
(
y
)
−
L
n
+
1
(
α
)
(
x
)
L
n
(
α
)
(
y
)
x
−
y
n
+
1
(
n
+
α
n
)
=
1
Γ
(
α
+
1
)
∑
i
=
0
n
x
i
i
!
L
n
−
i
(
α
+
i
)
(
x
)
L
n
−
i
(
α
+
i
+
1
)
(
y
)
(
α
+
n
n
)
(
n
i
)
;
{\displaystyle {\begin{aligned}K_{n}^{(\alpha )}(x,y)&:={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {L_{i}^{(\alpha )}(x)L_{i}^{(\alpha )}(y)}{\alpha +i \choose i}}\\[4pt]&={\frac {1}{\Gamma (\alpha +1)}}{\frac {L_{n}^{(\alpha )}(x)L_{n+1}^{(\alpha )}(y)-L_{n+1}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)}{{\frac {x-y}{n+1}}{n+\alpha \choose n}}}\\[4pt]&={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {x^{i}}{i!}}{\frac {L_{n-i}^{(\alpha +i)}(x)L_{n-i}^{(\alpha +i+1)}(y)}{{\alpha +n \choose n}{n \choose i}}};\end{aligned}}}
recursively
K
n
(
α
)
(
x
,
y
)
=
y
α
+
1
K
n
−
1
(
α
+
1
)
(
x
,
y
)
+
1
Γ
(
α
+
1
)
L
n
(
α
+
1
)
(
x
)
L
n
(
α
)
(
y
)
(
α
+
n
n
)
.
{\displaystyle K_{n}^{(\alpha )}(x,y)={\frac {y}{\alpha +1}}K_{n-1}^{(\alpha +1)}(x,y)+{\frac {1}{\Gamma (\alpha +1)}}{\frac {L_{n}^{(\alpha +1)}(x)L_{n}^{(\alpha )}(y)}{\alpha +n \choose n}}.}
Moreover,[clarification needed
y
α
e
−
y
K
n
(
α
)
(
⋅
,
y
)
→
δ
(
y
−
⋅
)
.
{\displaystyle y^{\alpha }e^{-y}K_{n}^{(\alpha )}(\cdot ,y)\to \delta (y-\cdot ).}
Turán's inequalities can be derived here, which is
L
n
(
α
)
(
x
)
2
−
L
n
−
1
(
α
)
(
x
)
L
n
+
1
(
α
)
(
x
)
=
∑
k
=
0
n
−
1
(
α
+
n
−
1
n
−
k
)
n
(
n
k
)
L
k
(
α
−
1
)
(
x
)
2
>
0.
{\displaystyle L_{n}^{(\alpha )}(x)^{2}-L_{n-1}^{(\alpha )}(x)L_{n+1}^{(\alpha )}(x)=\sum _{k=0}^{n-1}{\frac {\alpha +n-1 \choose n-k}{n{n \choose k}}}L_{k}^{(\alpha -1)}(x)^{2}>0.}
The following integral is needed in the quantum mechanical treatment of the hydrogen atom ,
∫
0
∞
x
α
+
1
e
−
x
[
L
n
(
α
)
(
x
)
]
2
d
x
=
(
n
+
α
)
!
n
!
(
2
n
+
α
+
1
)
.
{\displaystyle \int _{0}^{\infty }x^{\alpha +1}e^{-x}\left[L_{n}^{(\alpha )}(x)\right]^{2}dx={\frac {(n+\alpha )!}{n!}}(2n+\alpha +1).}
Series expansions Let a function have the (formal) series expansion
f
(
x
)
=
∑
i
=
0
∞
f
i
(
α
)
L
i
(
α
)
(
x
)
.
{\displaystyle f(x)=\sum _{i=0}^{\infty }f_{i}^{(\alpha )}L_{i}^{(\alpha )}(x).}
Then
f
i
(
α
)
=
∫
0
∞
L
i
(
α
)
(
x
)
(
i
+
α
i
)
⋅
x
α
e
−
x
Γ
(
α
+
1
)
⋅
f
(
x
)
d
x
.
{\displaystyle f_{i}^{(\alpha )}=\int _{0}^{\infty }{\frac {L_{i}^{(\alpha )}(x)}{i+\alpha \choose i}}\cdot {\frac {x^{\alpha }e^{-x}}{\Gamma (\alpha +1)}}\cdot f(x)\,dx.}
The series converges in the associated Hilbert space L 2 [0, ∞)if and only if
‖
f
‖
L
2
2
:=
∫
0
∞
x
α
e
−
x
Γ
(
α
+
1
)
|
f
(
x
)
|
2
d
x
=
∑
i
=
0
∞
(
i
+
α
i
)
|
f
i
(
α
)
|
2
<
∞
.
{\displaystyle \|f\|_{L^{2}}^{2}:=\int _{0}^{\infty }{\frac {x^{\alpha }e^{-x}}{\Gamma (\alpha +1)}}|f(x)|^{2}\,dx=\sum _{i=0}^{\infty }{i+\alpha \choose i}|f_{i}^{(\alpha )}|^{2}<\infty .}
Further examples of expansions Monomials are represented as
x
n
n
!
=
∑
i
=
0
n
(
−
1
)
i
(
n
+
α
n
−
i
)
L
i
(
α
)
(
x
)
,
{\displaystyle {\frac {x^{n}}{n!}}=\sum _{i=0}^{n}(-1)^{i}{n+\alpha \choose n-i}L_{i}^{(\alpha )}(x),}
while
binomials have the parametrization
(
n
+
x
n
)
=
∑
i
=
0
n
α
i
i
!
L
n
−
i
(
x
+
i
)
(
α
)
.
{\displaystyle {n+x \choose n}=\sum _{i=0}^{n}{\frac {\alpha ^{i}}{i!}}L_{n-i}^{(x+i)}(\alpha ).}
This leads directly to
e
−
γ
x
=
∑
i
=
0
∞
γ
i
(
1
+
γ
)
i
+
α
+
1
L
i
(
α
)
(
x
)
convergent iff
ℜ
(
γ
)
>
−
1
2
{\displaystyle e^{-\gamma x}=\sum _{i=0}^{\infty }{\frac {\gamma ^{i}}{(1+\gamma )^{i+\alpha +1}}}L_{i}^{(\alpha )}(x)\qquad {\text{convergent iff }}\Re (\gamma )>-{\tfrac {1}{2}}}
for the exponential function. The
incomplete gamma function has the representation
Γ
(
α
,
x
)
=
x
α
e
−
x
∑
i
=
0
∞
L
i
(
α
)
(
x
)
1
+
i
(
ℜ
(
α
)
>
−
1
,
x
>
0
)
.
{\displaystyle \Gamma (\alpha ,x)=x^{\alpha }e^{-x}\sum _{i=0}^{\infty }{\frac {L_{i}^{(\alpha )}(x)}{1+i}}\qquad \left(\Re (\alpha )>-1,x>0\right).}
In quantum mechanics In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial.[11]
Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.[12]
Multiplication theorems Erdélyi gives the following two multiplication theorems [13]
t
n
+
1
+
α
e
(
1
−
t
)
z
L
n
(
α
)
(
z
t
)
=
∑
k
=
n
∞
(
k
n
)
(
1
−
1
t
)
k
−
n
L
k
(
α
)
(
z
)
,
e
(
1
−
t
)
z
L
n
(
α
)
(
z
t
)
=
∑
k
=
0
∞
(
1
−
t
)
k
z
k
k
!
L
n
(
α
+
k
)
(
z
)
.
{\displaystyle {\begin{aligned}&t^{n+1+\alpha }e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=n}^{\infty }{k \choose n}\left(1-{\frac {1}{t}}\right)^{k-n}L_{k}^{(\alpha )}(z),\\[6pt]&e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=0}^{\infty }{\frac {(1-t)^{k}z^{k}}{k!}}L_{n}^{(\alpha +k)}(z).\end{aligned}}}
Relation to Hermite polynomials The generalized Laguerre polynomials are related to the
Hermite polynomials
:
H
2
n
(
x
)
=
(
−
1
)
n
2
2
n
n
!
L
n
(
−
1
/
2
)
(
x
2
)
H
2
n
+
1
(
x
)
=
(
−
1
)
n
2
2
n
+
1
n
!
x
L
n
(
1
/
2
)
(
x
2
)
{\displaystyle {\begin{aligned}H_{2n}(x)&=(-1)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})\\[4pt]H_{2n+1}(x)&=(-1)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})\end{aligned}}}
where the
H n x ) are the
Hermite polynomials
based on the weighting function
exp(−x 2 ) , the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator .
Relation to hypergeometric functions The Laguerre polynomials may be defined in terms of hypergeometric functions , specifically the confluent hypergeometric functions , as
L
n
(
α
)
(
x
)
=
(
n
+
α
n
)
M
(
−
n
,
α
+
1
,
x
)
=
(
α
+
1
)
n
n
!
1
F
1
(
−
n
,
α
+
1
,
x
)
{\displaystyle L_{n}^{(\alpha )}(x)={n+\alpha \choose n}M(-n,\alpha +1,x)={\frac {(\alpha +1)_{n}}{n!}}\,_{1}F_{1}(-n,\alpha +1,x)}
where
(
a
)
n
{\displaystyle (a)_{n}}
is the
Pochhammer symbol
(which in this case represents the rising factorial).
Hardy–Hille formula The generalized Laguerre polynomials satisfy the Hardy–Hille formula[14] [15]
∑
n
=
0
∞
n
!
Γ
(
α
+
1
)
Γ
(
n
+
α
+
1
)
L
n
(
α
)
(
x
)
L
n
(
α
)
(
y
)
t
n
=
1
(
1
−
t
)
α
+
1
e
−
(
x
+
y
)
t
/
(
1
−
t
)
0
F
1
(
;
α
+
1
;
x
y
t
(
1
−
t
)
2
)
,
{\displaystyle \sum _{n=0}^{\infty }{\frac {n!\,\Gamma \left(\alpha +1\right)}{\Gamma \left(n+\alpha +1\right)}}L_{n}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)t^{n}={\frac {1}{(1-t)^{\alpha +1}}}e^{-(x+y)t/(1-t)}\,_{0}F_{1}\left(;\alpha +1;{\frac {xyt}{(1-t)^{2}}}\right),}
where the series on the left converges for
α
>
−
1
{\displaystyle \alpha >-1}
and
|
t
|
<
1
{\displaystyle |t|<1}
. Using the identity
0
F
1
(
;
α
+
1
;
z
)
=
Γ
(
α
+
1
)
z
−
α
/
2
I
α
(
2
z
)
,
{\displaystyle \,_{0}F_{1}(;\alpha +1;z)=\,\Gamma (\alpha +1)z^{-\alpha /2}I_{\alpha }\left(2{\sqrt {z}}\right),}
(see
generalized hypergeometric function ), this can also be written as
∑
n
=
0
∞
n
!
Γ
(
1
+
α
+
n
)
L
n
(
α
)
(
x
)
L
n
(
α
)
(
y
)
t
n
=
1
(
x
y
t
)
α
/
2
(
1
−
t
)
e
−
(
x
+
y
)
t
/
(
1
−
t
)
I
α
(
2
x
y
t
1
−
t
)
.
{\displaystyle \sum _{n=0}^{\infty }{\frac {n!}{\Gamma (1+\alpha +n)}}L_{n}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)t^{n}={\frac {1}{(xyt)^{\alpha /2}(1-t)}}e^{-(x+y)t/(1-t)}I_{\alpha }\left({\frac {2{\sqrt {xyt}}}{1-t}}\right).}
This formula is a generalization of the
Hermite polynomials
, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.
Physics Convention The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals.[16] [17] [18] [19]
L
n
(
α
)
(
x
)
=
Γ
(
α
+
n
+
1
)
Γ
(
α
+
1
)
n
!
1
F
1
(
−
n
;
α
+
1
;
x
)
,
{\displaystyle L_{n}^{(\alpha )}(x)={\frac {\Gamma (\alpha +n+1)}{\Gamma (\alpha +1)n!}}\,_{1}F_{1}(-n;\alpha +1;x),}
where
1
F
1
(
a
;
b
;
x
)
{\displaystyle \,_{1}F_{1}(a;b;x)}
confluent hypergeometric function .
In the physics literature,[18]
L
¯
n
(
α
)
(
x
)
=
[
Γ
(
α
+
n
+
1
)
]
2
Γ
(
α
+
1
)
n
!
1
F
1
(
−
n
;
α
+
1
;
x
)
.
{\displaystyle {\bar {L}}_{n}^{(\alpha )}(x)={\frac {\left[\Gamma (\alpha +n+1)\right]^{2}}{\Gamma (\alpha +1)n!}}\,_{1}F_{1}(-n;\alpha +1;x).}
The physics version is related to the standard version by
L
¯
n
(
α
)
(
x
)
=
(
n
+
α
)
!
L
n
(
α
)
(
x
)
.
{\displaystyle {\bar {L}}_{n}^{(\alpha )}(x)=(n+\alpha )!L_{n}^{(\alpha )}(x).}
There is yet another, albeit less frequently used, convention in the physics literature [20] [21] [22]
L
~
n
(
α
)
(
x
)
=
(
−
1
)
α
L
¯
n
−
α
(
α
)
.
{\displaystyle {\tilde {L}}_{n}^{(\alpha )}(x)=(-1)^{\alpha }{\bar {L}}_{n-\alpha }^{(\alpha )}.}
Umbral Calculus Convention Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for
D
/
(
D
−
I
)
{\displaystyle D/(D-I)}
n
!
{\displaystyle n!}
[23]
L
n
(
x
)
=
n
!
L
n
(
−
1
)
(
x
)
=
∑
k
=
0
n
L
(
n
,
k
)
(
−
x
)
k
{\displaystyle {\mathcal {L}}_{n}(x)=n!L_{n}^{(-1)}(x)=\sum _{k=0}^{n}L(n,k)(-x)^{k}}
where
L
(
n
,
k
)
=
(
n
−
1
k
−
1
)
n
!
k
!
{\textstyle L(n,k)={\binom {n-1}{k-1}}{\frac {n!}{k!}}}
are the signless
Lah numbers .
(
L
n
(
x
)
)
n
∈
N
{\textstyle ({\mathcal {L}}_{n}(x))_{n\in \mathbb {N} }}
is a sequence of polynomials of
binomial type ,
ie they satisfy
L
n
(
x
+
y
)
=
∑
k
=
0
n
(
n
k
)
L
k
(
x
)
L
n
−
k
(
y
)
{\displaystyle {\mathcal {L}}_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}{\mathcal {L}}_{k}(x){\mathcal {L}}_{n-k}(y)}
See also Notes
. ^ A&S p. 781
^ A&S p. 509
^ A&S p. 510
^ A&S p. 775
^ Szegő, p. 198.
^ A&S equation (22.12.6), p. 785
. ^ "Associated Laguerre Polynomial" .^ Ratner, Schatz, Mark A., George C. (2001). Quantum Mechanics in Chemistry . 0-13-895491-7: Prentice Hall. pp. 90–91. {{cite book }}: CS1 maint: location (link ) CS1 maint: multiple names: authors list (link ). ^ C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions ", Proceedings of the National Academy of Sciences, Mathematics , (1950) pp. 752–757.
^ Szegő, p. 102.
^ W. A. Al-Salam (1964), "Operational representations for Laguerre and other polynomials" , Duke Math J. 31 (1): 127–142.
. . ^ .
. . . . .
References . G. Szegő, Orthogonal polynomials , 4th edition, Amer. Math. Soc. Colloq. Publ. , vol. 23, Amer. Math. Soc., Providence, RI, 1975.
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials" , in B. Spain, M.G. Smith, Functions of mathematical physics , Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
"Laguerre polynomials" , Encyclopedia of Mathematics EMS Press , 2001 [1994]Eric W. Weisstein , "Laguerre Polynomial ", From MathWorld—A Wolfram Web Resource.. External links
International National Other
_with_n_as_-1_divided_by_9_and_x_as_z_to_the_power_of_4_from_-2-2i_to_2%2B2i.svg)
![{\displaystyle \left(0,n+\alpha +(n-1){\sqrt {n+\alpha }}\,\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/268387cf6ed314caa46b2a11e87acbaf9695875d)
![{\displaystyle {\begin{aligned}&L_{n}^{(\alpha )}(x)={\frac {n^{{\frac {\alpha }{2}}-{\frac {1}{4}}}}{\sqrt {\pi }}}{\frac {e^{\frac {x}{2}}}{x^{{\frac {\alpha }{2}}+{\frac {1}{4}}}}}\sin \left(2{\sqrt {nx}}-{\frac {\pi }{2}}\left(\alpha -{\frac {1}{2}}\right)\right)+O\left(n^{{\frac {\alpha }{2}}-{\frac {3}{4}}}\right),\\[6pt]&L_{n}^{(\alpha )}(-x)={\frac {(n+1)^{{\frac {\alpha }{2}}-{\frac {1}{4}}}}{2{\sqrt {\pi }}}}{\frac {e^{-x/2}}{x^{{\frac {\alpha }{2}}+{\frac {1}{4}}}}}e^{2{\sqrt {x(n+1)}}}\cdot \left(1+O\left({\frac {1}{\sqrt {n+1}}}\right)\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65232999dd61f4298266965f24f79d6d0923b05)
![{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)-\sum _{j=0}^{\Delta -1}{n+\alpha \choose n-j}(-1)^{j}{\frac {x^{j}}{j!}}&=(-1)^{\Delta }{\frac {x^{\Delta }}{(\Delta -1)!}}\sum _{i=0}^{n-\Delta }{\frac {n+\alpha \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(\alpha +\Delta )}(x)\\[6pt]&=(-1)^{\Delta }{\frac {x^{\Delta }}{(\Delta -1)!}}\sum _{i=0}^{n-\Delta }{\frac {n+\alpha -i-1 \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(n+\alpha +\Delta -i)}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e449381aff6f2b593def172ab03c17cb1bb57dfb)
![{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)&=L_{n}^{(\alpha +1)}(x)-L_{n-1}^{(\alpha +1)}(x)=\sum _{j=0}^{k}{k \choose j}L_{n-j}^{(\alpha +k)}(x),\\[10pt]nL_{n}^{(\alpha )}(x)&=(n+\alpha )L_{n-1}^{(\alpha )}(x)-xL_{n-1}^{(\alpha +1)}(x),\\[10pt]&{\text{or }}\\{\frac {x^{k}}{k!}}L_{n}^{(\alpha )}(x)&=\sum _{i=0}^{k}(-1)^{i}{n+i \choose i}{n+\alpha \choose k-i}L_{n+i}^{(\alpha -k)}(x),\\[10pt]nL_{n}^{(\alpha +1)}(x)&=(n-x)L_{n-1}^{(\alpha +1)}(x)+(n+\alpha )L_{n-1}^{(\alpha )}(x)\\[10pt]xL_{n}^{(\alpha +1)}(x)&=(n+\alpha )L_{n-1}^{(\alpha )}(x)-(n-x)L_{n}^{(\alpha )}(x);\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c41078870f1a019f4dd86d6716be8fa960adbf8)
![{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)&=\left(2+{\frac {\alpha -1-x}{n}}\right)L_{n-1}^{(\alpha )}(x)-\left(1+{\frac {\alpha -1}{n}}\right)L_{n-2}^{(\alpha )}(x)\\[10pt]&={\frac {\alpha +1-x}{n}}L_{n-1}^{(\alpha +1)}(x)-{\frac {x}{n}}L_{n-2}^{(\alpha +2)}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc32e1d35979f9126c29a1b4fe1d3ae723d4641d)
![{\displaystyle xL_{n}^{[k]\prime \prime }(x)+(k+1-x)L_{n}^{[k]\prime }(x)+(n-k)L_{n}^{[k]}(x)=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18d6805dbdf16f74423e00fc23ca5ea675eabd49)
![{\displaystyle L_{n}^{[k]}(x)\equiv {\frac {d^{k}L_{n}(x)}{dx^{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94adaf2e2b6fca61a722420bcb1a66576e10593a)
![{\displaystyle {\begin{aligned}K_{n}^{(\alpha )}(x,y)&:={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {L_{i}^{(\alpha )}(x)L_{i}^{(\alpha )}(y)}{\alpha +i \choose i}}\\[4pt]&={\frac {1}{\Gamma (\alpha +1)}}{\frac {L_{n}^{(\alpha )}(x)L_{n+1}^{(\alpha )}(y)-L_{n+1}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)}{{\frac {x-y}{n+1}}{n+\alpha \choose n}}}\\[4pt]&={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {x^{i}}{i!}}{\frac {L_{n-i}^{(\alpha +i)}(x)L_{n-i}^{(\alpha +i+1)}(y)}{{\alpha +n \choose n}{n \choose i}}};\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbc06de0ef9a87d70e951a09cacec0b7da13396)
![{\displaystyle \int _{0}^{\infty }x^{\alpha +1}e^{-x}\left[L_{n}^{(\alpha )}(x)\right]^{2}dx={\frac {(n+\alpha )!}{n!}}(2n+\alpha +1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb8646b87b220d879aec2c64ff9e7e15228b32a)
![{\displaystyle {\begin{aligned}&t^{n+1+\alpha }e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=n}^{\infty }{k \choose n}\left(1-{\frac {1}{t}}\right)^{k-n}L_{k}^{(\alpha )}(z),\\[6pt]&e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=0}^{\infty }{\frac {(1-t)^{k}z^{k}}{k!}}L_{n}^{(\alpha +k)}(z).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/419e3fd18bf2b0ef484d747b9670be1f38888684)
![{\displaystyle {\begin{aligned}H_{2n}(x)&=(-1)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})\\[4pt]H_{2n+1}(x)&=(-1)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/015273f34e288fc1847a90dd793e433e9dbb23d0)
![{\displaystyle {\bar {L}}_{n}^{(\alpha )}(x)={\frac {\left[\Gamma (\alpha +n+1)\right]^{2}}{\Gamma (\alpha +1)n!}}\,_{1}F_{1}(-n;\alpha +1;x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12bfdcde9315fd8f61b7556fb8b93652a9ba59f2)
