In mathematics , the Laguerre polynomials , named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation:
x
y
″
+
(
1
−
x
)
y
′
+
n
y
=
0
{\displaystyle xy''+(1-x)y'+ny=0}
which is a second-order linear differential equation . This equation has 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 Gaussian 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 , ..., are a polynomial sequence which may be defined by the 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 orthogonal polynomials with respect to an 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 sequence of Laguerre polynomials n ! Ln is a Sheffer sequence ,
d
d
x
L
n
=
(
d
d
x
−
1
)
L
n
−
1
.
{\displaystyle {\frac {d}{dx}}L_{n}=\left({\frac {d}{dx}}-1\right)L_{n-1}.}
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 Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space . They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator .
Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of n ! than the definition used here. Likewise in quantum mechanics the definition for the associated Laguerre polynomials differs.
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 {\scriptstyle {\frac {1}{2}}}(x^{2}-4x+2)\,}
3
1
6
(
−
x
3
+
9
x
2
−
18
x
+
6
)
{\displaystyle {\scriptstyle {\frac {1}{6}}}(-x^{3}+9x^{2}-18x+6)\,}
4
1
24
(
x
4
−
16
x
3
+
72
x
2
−
96
x
+
24
)
{\displaystyle {\scriptstyle {\frac {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 {\scriptstyle {\frac {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 {\scriptstyle {\frac {1}{720}}}(x^{6}-36x^{5}+450x^{4}-2400x^{3}+5400x^{2}-4320x+720)\,}
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}}.}
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
∞
t
n
L
n
(
x
)
=
1
1
−
t
e
−
t
x
1
−
t
.
{\displaystyle \sum _{n}^{\infty }t^{n}L_{n}(x)={\frac {1}{1-t}}e^{-{\frac {tx}{1-t}}}.}
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''+(\alpha +1-x)\,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 of the generalized Laguerre polynomials:
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
(
d
d
x
−
1
)
n
n
!
x
n
+
α
.
{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)&={x^{-\alpha }e^{x} \over n!}{d^{n} \over dx^{n}}\left(e^{-x}x^{n+\alpha }\right)\\&=x^{-\alpha }{\frac {({\frac {d}{dx}}-1)^{n}}{n!}}x^{n+\alpha }.\end{aligned}}}
The generating function for them is
∑
n
∞
t
n
L
n
(
α
)
(
x
)
=
1
(
1
−
t
)
α
+
1
e
−
t
x
1
−
t
.
{\displaystyle \sum _{n}^{\infty }t^{n}L_{n}^{(\alpha )}(x)={\frac {1}{(1-t)^{\alpha +1}}}e^{-{\frac {tx}{1-t}}}.}
The first few generalized Laguerre polynomials, Ln (k ) (x )
Explicit examples and properties of the generalized Laguerre polynomials
L
n
(
α
)
(
x
)
:=
(
n
+
α
n
)
M
(
−
n
,
α
+
1
,
x
)
.
{\displaystyle L_{n}^{(\alpha )}(x):={n+\alpha \choose n}M(-n,\alpha +1,x).}
(
n
+
α
n
)
{\displaystyle {n+\alpha \choose n}}
is a generalized binomial coefficient . When n is an integer the function reduces to a polynomial of degree n . It has the alternative expression[4]
L
n
(
α
)
(
x
)
=
(
−
1
)
n
n
!
U
(
−
n
,
α
+
1
,
x
)
{\displaystyle L_{n}^{(\alpha )}(x)={\frac {(-1)^{n}}{n!}}U(-n,\alpha +1,x)}
in terms of Kummer's function of the second kind .
The closed form for these generalized Laguerre polynomials of degree n is[5]
L
n
(
α
)
(
x
)
=
∑
i
=
0
n
(
−
1
)
i
(
n
+
α
n
−
i
)
x
i
i
!
{\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}(-1)^{i}{n+\alpha \choose n-i}{\frac {x^{i}}{i!}}}
derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
The first few generalized Laguerre polynomials are:
L
0
(
α
)
(
x
)
=
1
L
1
(
α
)
(
x
)
=
−
x
+
α
+
1
L
2
(
α
)
(
x
)
=
x
2
2
−
(
α
+
2
)
x
+
(
α
+
2
)
(
α
+
1
)
2
L
3
(
α
)
(
x
)
=
−
x
3
6
+
(
α
+
3
)
x
2
2
−
(
α
+
2
)
(
α
+
3
)
x
2
+
(
α
+
1
)
(
α
+
2
)
(
α
+
3
)
6
{\displaystyle {\begin{aligned}L_{0}^{(\alpha )}(x)&=1\\L_{1}^{(\alpha )}(x)&=-x+\alpha +1\\L_{2}^{(\alpha )}(x)&={\frac {x^{2}}{2}}-(\alpha +2)x+{\frac {(\alpha +2)(\alpha +1)}{2}}\\L_{3}^{(\alpha )}(x)&={\frac {-x^{3}}{6}}+{\frac {(\alpha +3)x^{2}}{2}}-{\frac {(\alpha +2)(\alpha +3)x}{2}}+{\frac {(\alpha +1)(\alpha +2)(\alpha +3)}{6}}\end{aligned}}}
L
n
(
α
)
(
0
)
=
(
n
+
α
n
)
≈
n
α
Γ
(
α
+
1
)
;
{\displaystyle L_{n}^{(\alpha )}(0)={n+\alpha \choose n}\approx {\frac {n^{\alpha }}{\Gamma (\alpha +1)}};}
If α is non-negative, then L n (α ) has n real , strictly positive roots (notice that
(
(
−
1
)
n
−
i
L
n
−
i
(
α
)
)
i
=
0
n
{\displaystyle \left((-1)^{n-i}L_{n-i}^{(\alpha )}\right)_{i=0}^{n}}
is a Sturm chain ), which are all in the interval
(
0
,
n
+
α
+
(
n
−
1
)
n
+
α
]
.
{\displaystyle \left(0,n+\alpha +(n-1){\sqrt {n+\alpha }}\right].}
[citation needed ]
The polynomials' asymptotic behaviour for large n , but fixed α and x > 0 , is given by[6] [7]
L
n
(
α
)
(
x
)
=
n
α
2
−
1
4
π
e
x
2
x
α
2
+
1
4
cos
(
2
n
x
−
π
2
(
α
+
1
2
)
)
+
O
(
n
α
2
−
3
4
)
,
{\displaystyle 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}}}}}\cos \left(2{\sqrt {nx}}-{\frac {\pi }{2}}\left(\alpha +{\frac {1}{2}}\right)\right)+O\left(n^{{\frac {\alpha }{2}}-{\frac {3}{4}}}\right),}
L
n
(
α
)
(
−
x
)
=
(
n
+
1
)
α
2
−
1
4
2
π
e
−
x
2
x
α
2
+
1
4
e
2
x
(
n
+
1
)
⋅
(
1
+
O
(
1
n
+
1
)
)
,
{\displaystyle L_{n}^{(\alpha )}(-x)={\frac {(n+1)^{{\frac {\alpha }{2}}-{\frac {1}{4}}}}{2{\sqrt {\pi }}}}{\frac {e^{-{\frac {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),}
and summarizing by
L
n
(
α
)
(
x
n
)
n
α
≈
e
x
2
n
⋅
J
α
(
2
x
)
x
α
,
{\displaystyle {\frac {L_{n}^{(\alpha )}\left({\frac {x}{n}}\right)}{n^{\alpha }}}\approx e^{\frac {x}{2n}}\cdot {\frac {J_{\alpha }\left(2{\sqrt {x}}\right)}{{\sqrt {x}}^{\alpha }}},}
where
J
α
{\displaystyle J_{\alpha }}
is the Bessel function .
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^{-{\frac {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
+
j
)
(
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+j)}(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)}
is a monic polynomial of degree
n
{\displaystyle n}
in
α
{\displaystyle \alpha }
,
there is the 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 = 0 ) of the formula above: for integer α = k the generalized polynomial may be written
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)^{\prime }=n\cdot x^{\alpha }e^{-x}\cdot L_{n}^{(\alpha )}(x),}
which shows that L (α) n is an eigenvector for the eigenvalue n .
Orthogonality
The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with 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)}
denotes the Gamma distribution then the orthogonality relation can be written as
∫
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}}\\&{=}{\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}}}\\&{=}{\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 Limit as n goes to infinity? ]
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).}
Associated Laguerre Polynomial (Quantum Mechanics)
In quantum mechanics the associated Laguerre polynomial is a component of the radial part of the schrodinger equation for the hydrogen atom.[11] The associated Laguerre polynomial is defined as
L
n
m
(
x
)
=
d
m
d
x
m
L
n
(
x
)
{\displaystyle L_{n}^{m}(x)={\frac {d^{m}}{dx^{m}}}L_{n}(x)}
[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
)
,
{\displaystyle t^{n+1+\alpha }e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=n}{k \choose n}\left(1-{\frac {1}{t}}\right)^{k-n}L_{k}^{(\alpha )}(z),}
e
(
1
−
t
)
z
L
n
(
α
)
(
z
t
)
=
∑
k
=
0
(
1
−
t
)
k
z
k
k
!
L
n
(
α
+
k
)
(
z
)
.
{\displaystyle e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\sum _{k=0}{\frac {(1-t)^{k}z^{k}}{k!}}L_{n}^{(\alpha +k)}(z).}
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})\\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^{-{\frac {(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 \left(\alpha +1\right)z^{-{\frac {\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 \left(1+\alpha +n\right)}}L_{n}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)t^{n}={\frac {1}{\left(xyt\right)^{\frac {\alpha }{2}}\left(1-t\right)}}e^{-{\frac {\left(x+y\right)t}{1-t}}}I_{\alpha }\left({\frac {2{\sqrt {xyt}}}{1-t}}\right).}
This formula is a generalization of the Mehler kernel for Hermite polynomials , which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.
See also
Transverse mode , an important application of Laguerre polynomials to describe the field intensity within a waveguide or laser beam profile.
Notes
^ N. Sonine (1880). "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries" . Math. Ann. 16 (1): 1–80. doi :10.1007/BF01459227 .
^ A&S p. 781
^ A&S p.509
^ A&S p.510
^ A&S p. 775
^ Szegő, p. 198.
^ D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", SIAM J. Numer. Anal. , vol. 46 (2008), no. 6, pp. 3285-3312 doi :10.1137/07068031X
^ A&S equation (22.12.6), p. 785
^ W. Koepf, "Identities for families of orthogonal polynomials and special functions. ", Integral Transforms and Special Functions 5 , (1997) pp.69-102. (Theorem 10)
^ http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
^ 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 )
^ 1937-, Steiner, Erich, (2008). The chemistry maths book (2nd ed.). Oxford: Oxford University Press. ISBN 9781613441145 . OCLC 743217755 . CS1 maint: extra punctuation (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
Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. "Chapter 22" . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0 . LCCN 64-60036 . MR 0167642 . LCCN 65-12253 .
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 Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 .
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.
George Arfken and Hans Weber (2000). Mathematical Methods for Physicists . Academic Press. ISBN 0-12-059825-6 .
S. S. Bayin (2006), Mathematical Methods in Science and Engineering , Wiley, Chapter 3.
External links