在数学 中，以法国 数学家 埃德蒙·拉盖尔 拉盖尔多项式 定义为拉盖尔方程 的标准解。
x
y
″
+
(
1
−
x
)
y
′
+
n
y
=
0
{\displaystyle x\,y''+(1-x)\,y'+n\,y=0\,}
这是一个二阶线性微分方程 。
这个方程只有当n 非负时，才有非奇异解。拉盖尔多项式可用在高斯积分法 中，计算形如
∫
0
∞
f
(
x
)
d
x
{\displaystyle \int _{0}^{\infty }f(x)dx}
这些多项式（通常用L 0 , L 1 等表示）构成一个多项式序列 罗德里格公式 递推得到。
L
n
(
x
)
=
e
x
n
!
d
n
d
x
n
(
e
−
x
x
n
)
.
{\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right).}
在按照下式定义的内积构成的内积空间 中，拉盖尔多项式是正交多项式 。
⟨
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.}
拉盖尔多项式构成一个Sheffer序列
拉盖尔多项式在量子力学中有重要应用。氢原子薛定谔方程 的解的径向部分，就是拉盖尔多项式。
物理学家通常采用另外一种拉盖尔多项式的定义形式，即在上面的形式的基础上乘上一个n !。
前几个拉盖尔多项式 [ 编辑 ] 前几个拉盖尔多项式的表达式与函数图像如下：
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)\,}
前六个拉盖尔多项式 递归定义 [ 编辑 ] 拉盖尔多项式也可以通过递归的方式进行定义。首先，规定前两个拉盖尔多项式为：
L
0
(
x
)
=
1
{\displaystyle L_{0}(x)=1\,}
L
1
(
x
)
=
1
−
x
{\displaystyle L_{1}(x)=1-x\,}
然后运用下面的递推关系 得到更高阶的多项式。
L
k
+
1
(
x
)
=
1
k
+
1
(
(
2
k
+
1
−
x
)
L
k
(
x
)
−
k
L
k
−
1
(
x
)
)
.
{\displaystyle L_{k+1}(x)={\frac {1}{k+1}}\left((2k+1-x)L_{k}(x)-kL_{k-1}(x)\right).}
广义拉盖尔多项式 [ 编辑 ] 上面提到的拉盖尔多项式的正交性，也可以用另外一种方式表达。即：如果X 是一个服从指数分布 的随机变量 （即，概率密度函数 如下式）：
f
(
x
)
=
{
e
−
x
if
x
>
0
,
0
if
x
<
0
,
{\displaystyle f(x)=\left\{{\begin{matrix}e^{-x}&{\mbox{if}}\ x>0,\\0&{\mbox{if}}\ x<0,\end{matrix}}\right.}
那么：
E
[
L
n
(
X
)
L
m
(
X
)
]
=
0
whenever
n
≠
m
.
{\displaystyle E\left[L_{n}(X)L_{m}(X)\right]=0\ {\mbox{whenever}}\ n\neq m.}
指数分布不是唯一的伽玛分布 ，对于任意的伽玛分布（概率密度函数如下，α > −1，参见Γ函数 ）
f
(
x
)
=
{
x
α
e
−
x
/
Γ
(
1
+
α
)
if
x
>
0
,
0
if
x
<
0
,
{\displaystyle f(x)=\left\{{\begin{matrix}x^{\alpha }e^{-x}/\Gamma (1+\alpha )&{\mbox{if}}\ x>0,\\0&{\mbox{if}}\ x<0,\end{matrix}}\right.}
相应的正交多项式为形如下式的广义拉盖尔多项式 （可以通过罗德里格公式 得到）：
L
n
(
α
)
(
x
)
=
x
−
α
e
x
n
!
d
n
d
x
n
(
e
−
x
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).}
有时也将上面的多项式称为连带（联属，伴随）拉盖尔多项式 。当取α = 0时，就回到拉盖尔多项式：
L
n
(
0
)
(
x
)
=
L
n
(
x
)
.
{\displaystyle L_{n}^{(0)}(x)=L_{n}(x).}
广义拉盖尔多项式的性质与应用 [ 编辑 ] 拉盖尔函数可以由合流超几何函数 和Kummer变换得到：
L
n
(
α
)
(
x
)
:=
(
n
+
α
n
)
M
(
−
n
,
α
+
1
,
x
)
=
(
n
+
α
n
)
∑
i
=
0
(
−
1
)
i
(
n
i
)
(
α
+
i
i
)
x
i
{\displaystyle L_{n}^{(\alpha )}(x):={n+\alpha \choose n}M(-n,\alpha +1,x)={n+\alpha \choose n}\sum _{i=0}(-1)^{i}{\frac {n \choose i}{\alpha +i \choose i}}x^{i}\,}
=
e
x
⋅
(
n
+
α
n
)
M
(
α
+
n
+
1
,
α
+
1
,
−
x
)
{\displaystyle =e^{x}\cdot {n+\alpha \choose n}M(\alpha +n+1,\alpha +1,-x)}
=
e
x
sin
(
n
π
)
sin
(
(
n
+
α
)
π
)
L
−
α
−
n
−
1
(
α
)
(
−
x
)
{\displaystyle ={\frac {e^{x}\sin(n\pi )}{\sin((n+\alpha )\pi )}}L_{-\alpha -n-1}^{(\alpha )}(-x)}
=
e
x
⋅
∑
i
=
0
(
−
1
)
i
(
α
+
n
+
i
n
)
x
i
i
!
.
{\displaystyle =e^{x}\cdot \sum _{i=0}(-1)^{i}{\alpha +n+i \choose n}{\frac {x^{i}}{i!}}.}
n
{\displaystyle n}
n
{\displaystyle n}
n
{\displaystyle n}
莱布尼茨乘积求导公式
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!}}}
n 阶拉盖尔多项式的首项系数为(−1)n n !；拉盖尔多项式在x=0的取值（常数项 ）为
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)}};}
L n (α ) 有n 个实 的正根 （应该注意到
(
(
−
1
)
n
−
i
L
n
−
i
(
α
)
)
i
=
0
n
{\displaystyle \left((-1)^{n-i}L_{n-i}^{(\alpha )}\right)_{i=0}^{n}}
施图姆序列 ），且这些根全部位于区间
(
0
,
n
+
α
+
(
n
−
1
)
n
+
α
]
{\displaystyle (0,n+\alpha +(n-1){\sqrt {n+\alpha }}]}
当
n
{\displaystyle n}
α
{\displaystyle \alpha }
x
>
0
{\displaystyle x>0}
L
n
(
α
)
(
x
)
≈
n
α
2
−
1
4
π
e
x
2
x
α
2
+
1
4
cos
(
2
x
(
n
+
α
+
1
2
)
−
π
2
(
α
+
1
2
)
)
{\displaystyle L_{n}^{(\alpha )}(x)\approx {\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 {x\left(n+{\frac {\alpha +1}{2}}\right)}}-{\frac {\pi }{2}}\left(\alpha +{\frac {1}{2}}\right)\right)}
L
n
(
α
)
(
−
x
)
≈
n
α
2
−
1
4
2
π
e
−
x
2
x
α
2
+
1
4
exp
(
2
x
(
n
+
α
+
1
2
)
)
{\displaystyle L_{n}^{(\alpha )}(-x)\approx {\frac {n^{{\frac {\alpha }{2}}-{\frac {1}{4}}}}{2{\sqrt {\pi }}}}{\frac {e^{-{\frac {x}{2}}}}{x^{{\frac {\alpha }{2}}+{\frac {1}{4}}}}}\exp \left(2{\sqrt {x\left(n+{\frac {\alpha +1}{2}}\right)}}\right)}
[1]
L
0
(
α
)
(
x
)
=
1
{\displaystyle L_{0}^{(\alpha )}(x)=1}
L
1
(
α
)
(
x
)
=
−
x
+
α
+
1
{\displaystyle L_{1}^{(\alpha )}(x)=-x+\alpha +1}
L
2
(
α
)
(
x
)
=
x
2
2
−
(
α
+
2
)
x
+
(
α
+
2
)
(
α
+
1
)
2
{\displaystyle L_{2}^{(\alpha )}(x)={\frac {x^{2}}{2}}-(\alpha +2)x+{\frac {(\alpha +2)(\alpha +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 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}}}
根据拉盖尔多项式的定义，可以使用秦九韶算法 计算拉盖尔多项式，程序代码如下： function LaguerreL(n, alpha, x) {
LaguerreL:= 1; bin:= 1
for i:= n to 1 step -1 {
bin:= bin* (alpha+ i)/ (n+ 1- i)
LaguerreL:= bin- x* LaguerreL/ i
}
return LaguerreL;
}
递推关系 [ 编辑 ] 拉盖尔多项式满足以下的递推关系：
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),}
特别地，有
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)}
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)}
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);}
还有
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)\\&=(-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}}}
运用以上式子可以得到以下四条关系式：
L
n
(
α
)
(
x
)
=
L
n
(
α
+
1
)
(
x
)
−
L
n
−
1
(
α
+
1
)
(
x
)
{\displaystyle L_{n}^{(\alpha )}(x)=L_{n}^{(\alpha +1)}(x)-L_{n-1}^{(\alpha +1)}(x)}
=
∑
j
=
0
k
(
k
j
)
L
n
−
j
(
α
−
k
+
j
)
(
x
)
,
{\displaystyle =\sum _{j=0}^{k}{k \choose j}L_{n-j}^{(\alpha -k+j)}(x),}
n
L
n
(
α
)
(
x
)
=
(
n
+
α
)
L
n
−
1
(
α
)
(
x
)
−
x
L
n
−
1
(
α
+
1
)
(
x
)
,
{\displaystyle nL_{n}^{(\alpha )}(x)=(n+\alpha )L_{n-1}^{(\alpha )}(x)-xL_{n-1}^{(\alpha +1)}(x),}
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
)
,
{\displaystyle {\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),}
n
L
n
(
α
+
1
)
(
x
)
=
(
n
−
x
)
L
n
−
1
(
α
+
1
)
(
x
)
+
(
n
+
α
)
L
n
−
1
(
α
)
(
x
)
{\displaystyle nL_{n}^{(\alpha +1)}(x)=(n-x)L_{n-1}^{(\alpha +1)}(x)+(n+\alpha )L_{n-1}^{(\alpha )}(x)}
x
L
n
(
α
+
1
)
=
(
n
+
α
)
L
n
−
1
α
(
x
)
−
(
n
−
x
)
L
n
(
α
)
(
x
)
;
{\displaystyle xL_{n}^{(\alpha +1)}=(n+\alpha )L_{n-1}^{\alpha }(x)-(n-x)L_{n}^{(\alpha )}(x);}
将它们组合在一起，就得到了最常用的递推关系式：
L
n
+
1
(
α
)
(
x
)
=
1
n
+
1
(
(
2
n
+
1
+
α
−
x
)
L
n
(
α
)
(
x
)
−
(
n
+
α
)
L
n
−
1
(
α
)
(
x
)
)
.
{\displaystyle L_{n+1}^{(\alpha )}(x)={\frac {1}{n+1}}\left((2n+1+\alpha -x)L_{n}^{(\alpha )}(x)-(n+\alpha )L_{n-1}^{(\alpha )}(x)\right).}
当
i
{\displaystyle i}
n
{\displaystyle n}
(
−
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);}
进一步可以得到部分分式分解 ：
L
n
(
α
)
(
x
)
(
n
+
α
n
)
=
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 {\frac {L_{n}^{(\alpha )}(x)}{n+\alpha \choose n}}=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}}.}
拉盖尔多项式的导函数 [ 编辑 ] 将拉盖尔多项式对自变量x 求导k 次，得到：
d
k
d
x
k
L
n
(
α
)
(
x
)
=
(
−
1
)
k
L
n
−
k
(
α
+
k
)
(
x
)
;
{\displaystyle {\frac {\mathrm {d} ^{k}}{\mathrm {d} x^{k}}}L_{n}^{(\alpha )}(x)=(-1)^{k}L_{n-k}^{(\alpha +k)}(x)\,;}
进一步有：
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 {\mathrm {d} ^{k}}{\mathrm {d} x^{k}}}x^{\alpha }L_{n}^{(\alpha )}(x)={n+\alpha \choose k}x^{\alpha -k}L_{n}^{(\alpha -k)}(x),}
运用柯西多重积分公式
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.}
将拉盖尔多项式对参变量
α
{\displaystyle \alpha }
d
d
α
L
n
(
α
)
(
x
)
=
∑
i
=
0
n
−
1
L
i
(
α
)
(
x
)
n
−
i
.
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \alpha }}L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n-1}{\frac {L_{i}^{(\alpha )}(x)}{n-i}}.}
广义拉盖尔多项式满足下面的微分方程：
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,\,}
可以与拉盖尔多项式的k阶导数所满足的微分方程作一比较。
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,\,}
仅在此式中，
L
n
(
k
)
(
x
)
≡
d
L
n
(
x
)
d
x
k
{\displaystyle L_{n}^{(k)}(x)\equiv {\frac {dL_{n}(x)}{dx^{k}}}}
于是，当
α
=
0
{\displaystyle \alpha =0}
L
n
(
k
)
(
x
)
=
(
−
1
)
k
d
L
n
+
k
(
x
)
d
x
k
{\displaystyle L_{n}^{(k)}(x)=(-1)^{k}{\frac {dL_{n+k}(x)}{dx^{k}}}}
正交性 [ 编辑 ] 伴随拉盖尔多项式在区间[0, ∞)上以权函数x α e −x 正交：
∫
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},}
这可由下式得到：
∫
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 ').}
伴随对称核多项式可以用拉盖尔多项式表示为：
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}}}
也有下面的递推关系：
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}}.}
进一步地，在伴L 2 [0, ∞)空间上，有：
y
α
e
−
y
K
n
(
α
)
(
⋅
,
y
)
→
δ
(
y
−
⋅
)
,
{\displaystyle y^{\alpha }e^{-y}K_{n}^{(\alpha )}(\cdot ,y)\rightarrow \delta (y-\,\cdot ),}
在氢原子的量子力学处理中用到了下面的公式：
∫
0
∞
x
α
+
1
e
−
x
[
L
n
(
α
)
]
2
d
x
=
(
n
+
α
)
!
n
!
(
2
n
+
α
+
1
)
.
{\displaystyle \int _{0}^{\infty }x^{\alpha +1}e^{-x}\left[L_{n}^{(\alpha )}\right]^{2}dx={\frac {(n+\alpha )!}{n!}}(2n+\alpha +1).}
级数展开 [ 编辑 ] 设一个函数具有以下的级数展开形式：
f
(
x
)
=
∑
i
=
0
f
i
(
α
)
L
i
(
α
)
(
x
)
.
{\displaystyle f(x)=\sum _{i=0}f_{i}^{(\alpha )}L_{i}^{(\alpha )}(x).}
则展开式的系数由下式给出
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.}
这个级数在Lp空间
L
2
[
0
,
∞
)
{\displaystyle L^{2}[0,\infty )}
当且仅当
‖
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}{i+\alpha \choose i}|f_{i}^{(\alpha )}|^{2}<\infty .}
一个相关的展开式为：
f
(
x
)
=
e
γ
1
+
γ
x
⋅
∑
i
=
0
L
i
(
α
)
(
x
1
+
γ
)
(
1
+
γ
)
i
+
α
+
1
∑
n
=
0
i
γ
i
−
n
(
i
n
)
f
n
(
α
)
;
{\displaystyle f(x)=e^{{\frac {\gamma }{1+\gamma }}x}\cdot \sum _{i=0}{\frac {L_{i}^{(\alpha )}\left({\frac {x}{1+\gamma }}\right)}{(1+\gamma )^{i+\alpha +1}}}\sum _{n=0}^{i}\gamma ^{i-n}{i \choose n}f_{n}^{(\alpha )};}
特别地
e
−
γ
x
⋅
L
n
(
α
)
(
x
(
1
+
γ
)
)
=
∑
i
=
n
L
i
(
α
)
(
x
)
(
1
+
γ
)
i
+
α
+
1
γ
i
−
n
(
i
n
)
,
{\displaystyle e^{-\gamma x}\cdot L_{n}^{(\alpha )}(x(1+\gamma ))=\sum _{i=n}{\frac {L_{i}^{(\alpha )}(x)}{(1+\gamma )^{i+\alpha +1}}}\gamma ^{i-n}{i \choose n},}
这可由下式得到：
L
n
(
α
)
(
x
1
+
γ
)
=
1
(
1
+
γ
)
n
∑
i
=
0
n
γ
n
−
i
(
n
+
α
n
−
i
)
L
i
(
α
)
(
x
)
.
{\displaystyle L_{n}^{(\alpha )}\left({\frac {x}{1+\gamma }}\right)={\frac {1}{(1+\gamma )^{n}}}\sum _{i=0}^{n}\gamma ^{n-i}{n+\alpha \choose n-i}L_{i}^{(\alpha )}(x).}
还有，当
Re
(
2
α
−
β
)
>
−
1
{\displaystyle \operatorname {Re} {(2\alpha -\beta )}>-1}
x
α
−
β
f
(
x
)
Γ
(
α
−
β
+
1
)
=
(
α
β
)
∑
i
=
0
L
i
(
β
)
(
x
)
(
β
+
i
i
)
∑
n
=
0
i
(
−
1
)
i
−
n
(
α
−
β
i
−
n
)
(
α
+
n
n
)
f
n
(
α
)
,
{\displaystyle {\frac {x^{\alpha -\beta }f(x)}{\Gamma (\alpha -\beta +1)}}={\alpha \choose \beta }\sum _{i=0}{\frac {L_{i}^{(\beta )}(x)}{\beta +i \choose i}}\sum _{n=0}^{i}(-1)^{i-n}{\alpha -\beta \choose i-n}{\alpha +n \choose n}f_{n}^{(\alpha )},}
这个结果可以由下式导出，
x
α
−
β
L
n
(
α
)
(
x
)
Γ
(
α
−
β
+
1
)
=
(
α
β
)
(
α
+
n
n
)
∑
i
=
n
(
−
1
)
i
−
n
(
α
−
β
i
−
n
)
L
i
(
β
)
(
x
)
(
β
+
i
i
)
{\displaystyle {\frac {x^{\alpha -\beta }L_{n}^{(\alpha )}(x)}{\Gamma (\alpha -\beta +1)}}={\alpha \choose \beta }{\alpha +n \choose n}\sum _{i=n}(-1)^{i-n}{\alpha -\beta \choose i-n}{\frac {L_{i}^{(\beta )}(x)}{\beta +i \choose i}}}
更多的例子 [ 编辑 ] 幂函数 可以展开为：
x
n
n
!
=
∑
i
=
0
n
(
−
1
)
i
(
n
+
α
n
−
i
)
L
i
(
α
)
(
x
)
=
(
−
1
)
n
∑
i
=
0
n
L
i
(
α
−
i
)
(
x
)
(
−
α
n
−
i
)
,
{\displaystyle {\frac {x^{n}}{n!}}=\sum _{i=0}^{n}(-1)^{i}{n+\alpha \choose n-i}L_{i}^{(\alpha )}(x)=(-1)^{n}\sum _{i=0}^{n}L_{i}^{(\alpha -i)}(x){-\alpha \choose n-i},}
二项式 可以展开为：
(
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 ).}
进一步可以得到：
e
−
γ
x
=
∑
i
=
0
γ
i
(
1
+
γ
)
i
+
α
+
1
L
i
(
α
)
(
x
)
{\displaystyle e^{-\gamma x}=\sum _{i=0}{\frac {\gamma ^{i}}{(1+\gamma )^{i+\alpha +1}}}L_{i}^{(\alpha )}(x)}
Re
(
γ
)
>
−
1
2
{\displaystyle \operatorname {Re} {(\gamma )}>-{\frac {1}{2}}}
更一般地
x
β
e
−
γ
x
Γ
(
β
+
1
)
=
(
α
+
β
α
)
∑
i
=
0
L
i
(
α
)
(
x
)
(
α
+
i
i
)
∑
j
=
0
i
(
−
1
)
j
(
1
+
γ
)
α
+
β
+
j
+
1
(
α
+
β
+
j
j
)
(
α
+
i
i
−
j
)
.
{\displaystyle {\frac {x^{\beta }e^{-\gamma x}}{\Gamma (\beta +1)}}={\alpha +\beta \choose \alpha }\sum _{i=0}{\frac {L_{i}^{(\alpha )}(x)}{\alpha +i \choose i}}\sum _{j=0}^{i}{\frac {(-1)^{j}}{(1+\gamma )^{\alpha +\beta +j+1}}}{\alpha +\beta +j \choose j}{\alpha +i \choose i-j}.}
对于非负的整数
β
{\displaystyle \beta }
x
n
e
−
γ
x
n
!
=
∑
i
=
0
γ
i
L
i
(
α
)
(
x
)
(
1
+
γ
)
i
+
n
+
α
+
1
∑
j
=
0
n
(
−
1
)
n
−
j
γ
j
(
n
+
α
j
)
(
i
n
−
j
)
,
{\displaystyle {\frac {x^{n}e^{-\gamma x}}{n!}}=\sum _{i=0}{\frac {\gamma ^{i}L_{i}^{(\alpha )}(x)}{(1+\gamma )^{i+n+\alpha +1}}}\sum _{j=0}^{n}(-1)^{n-j}\gamma ^{j}{n+\alpha \choose j}{i \choose n-j},}
当
γ
=
0
{\displaystyle \gamma =0}
x
β
Γ
(
β
+
1
)
=
(
α
+
β
α
)
∑
i
=
0
(
−
1
)
i
(
β
i
)
L
i
(
α
)
(
x
)
(
α
+
i
i
)
,
{\displaystyle {\frac {x^{\beta }}{\Gamma (\beta +1)}}={\alpha +\beta \choose \alpha }\sum _{i=0}(-1)^{i}{\beta \choose i}{\frac {L_{i}^{(\alpha )}(x)}{\alpha +i \choose i}},}
x
β
L
n
(
γ
)
(
x
)
Γ
(
β
+
1
)
=
(
α
+
β
α
)
∑
i
=
0
L
i
(
α
)
(
x
)
(
α
+
i
i
)
∑
j
=
0
n
(
−
1
)
i
−
j
(
n
+
γ
n
−
j
)
(
β
+
j
i
)
(
α
+
β
+
j
j
)
.
{\displaystyle {\frac {x^{\beta }L_{n}^{(\gamma )}(x)}{\Gamma (\beta +1)}}={\alpha +\beta \choose \alpha }\sum _{i=0}{\frac {L_{i}^{(\alpha )}(x)}{\alpha +i \choose i}}\sum _{j=0}^{n}(-1)^{i-j}{n+\gamma \choose n-j}{\beta +j \choose i}{\alpha +\beta +j \choose j}.}
雅可比Theta 函数 有下面的表示：
∑
k
∈
Z
e
−
k
2
π
x
=
∑
i
=
0
L
i
(
α
)
(
x
t
)
∑
k
∈
Z
(
k
2
π
t
)
i
(
1
+
k
2
π
t
)
i
+
α
+
1
;
{\displaystyle \sum _{k\in \mathbb {Z} }e^{-k^{2}\pi x}=\sum _{i=0}L_{i}^{(\alpha )}\left({\frac {x}{t}}\right)\sum _{k\in \mathbb {Z} }{\frac {(k^{2}\pi t)^{i}}{(1+k^{2}\pi t)^{i+\alpha +1}}};}
随意选定参量t，贝塞尔函数 可以表示为：
J
α
(
x
)
(
x
2
)
α
=
e
−
t
Γ
(
α
+
1
)
∑
i
=
0
L
i
(
α
)
(
x
2
4
t
)
(
i
+
α
i
)
t
i
i
!
;
{\displaystyle {\frac {J_{\alpha }(x)}{\left({\frac {x}{2}}\right)^{\alpha }}}={\frac {e^{-t}}{\Gamma (\alpha +1)}}\sum _{i=0}{\frac {L_{i}^{(\alpha )}\left({\frac {x^{2}}{4t}}\right)}{i+\alpha \choose i}}{\frac {t^{i}}{i!}};}
Γ函数 可以展开为：
Γ
(
α
)
=
x
α
∑
i
=
0
L
i
(
α
)
(
x
)
α
+
i
(
ℜ
(
α
)
<
1
2
)
;
{\displaystyle \Gamma (\alpha )=x^{\alpha }\sum _{i=0}{\frac {L_{i}^{(\alpha )}(x)}{\alpha +i}}\qquad \left(\Re (\alpha )<{\frac {1}{2}}\right);}
低阶不完全伽玛函数 可展开为：
γ
(
s
;
z
)
t
s
Γ
(
s
)
=
(
z
t
)
α
Γ
(
α
+
1
)
∑
i
=
0
L
i
(
α
)
(
z
t
)
(
α
+
i
i
)
∑
j
=
0
i
(
−
1
)
j
(
1
+
t
)
s
+
j
(
s
−
1
+
j
j
)
(
α
−
1
+
i
i
−
j
)
,
{\displaystyle {\frac {\gamma (s;z)}{t^{s}\Gamma (s)}}={\frac {\left({\frac {z}{t}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\sum _{i=0}{\frac {L_{i}^{(\alpha )}\left({\frac {z}{t}}\right)}{\alpha +i \choose i}}\sum _{j=0}^{i}{\frac {(-1)^{j}}{(1+t)^{s+j}}}{s-1+j \choose j}{\alpha -1+i \choose i-j},}
γ
(
s
;
z
)
t
s
Γ
(
s
)
=
(
α
+
s
α
+
1
)
∑
i
=
0
(
α
+
i
+
1
i
+
1
)
−
L
i
+
1
(
α
)
(
z
t
)
(
α
+
i
+
1
i
)
∑
j
=
0
i
(
−
1
)
j
(
1
+
t
)
α
+
1
+
s
+
j
(
α
+
s
+
j
j
)
(
α
+
i
+
1
i
−
j
)
.
{\displaystyle {\frac {\gamma (s;z)}{t^{s}\Gamma (s)}}={\alpha +s \choose \alpha +1}\sum _{i=0}{\frac {{\alpha +i+1 \choose i+1}-L_{i+1}^{(\alpha )}\left({\frac {z}{t}}\right)}{\alpha +i+1 \choose i}}\sum _{j=0}^{i}{\frac {(-1)^{j}}{(1+t)^{\alpha +1+s+j}}}{\alpha +s+j \choose j}{\alpha +i+1 \choose i-j}.}
还有：
γ
(
s
,
z
)
=
γ
s
Γ
(
1
−
s
)
∑
i
=
0
L
i
+
1
(
−
s
)
(
0
)
−
L
i
+
1
(
−
s
)
(
z
γ
)
(
1
+
γ
)
i
+
1
∑
n
=
0
i
γ
i
−
n
(
i
n
)
n
+
1
−
s
;
{\displaystyle \gamma (s,z)={\frac {\gamma ^{s}}{\Gamma (1-s)}}\sum _{i=0}{\frac {L_{i+1}^{(-s)}(0)-L_{i+1}^{(-s)}\left({\frac {z}{\gamma }}\right)}{(1+\gamma )^{i+1}}}\sum _{n=0}^{i}\gamma ^{i-n}{\frac {i \choose n}{n+1-s}};}
于是，高阶不完全伽玛函数就是：
Γ
(
s
,
z
)
z
s
e
−
z
=
∑
k
=
0
L
k
(
α
)
(
z
)
(
k
+
1
)
(
k
+
1
+
α
−
s
k
+
1
)
(
ℜ
(
s
−
α
2
)
<
1
4
)
=
∑
k
=
0
L
k
(
α
)
(
z
t
)
⋅
2
F
1
(
1
+
α
+
k
,
1
+
k
;
2
+
α
+
k
−
s
;
t
−
1
t
)
t
k
(
k
+
1
)
(
1
+
α
+
k
−
s
1
+
k
)
=
t
s
∑
k
=
0
L
k
(
α
)
(
z
t
)
⋅
2
F
1
(
1
−
s
,
1
+
α
−
s
;
2
+
α
+
k
−
s
;
t
−
1
t
)
(
k
+
1
)
(
1
+
α
+
k
−
s
1
+
k
)
=
t
1
+
α
∑
k
=
0
L
k
(
α
)
(
z
t
)
⋅
2
F
1
(
1
+
α
+
k
,
1
+
α
−
s
;
2
+
α
+
k
−
s
;
1
−
t
)
(
k
+
1
)
(
1
+
α
+
k
−
s
1
+
k
)
,
{\displaystyle {\begin{aligned}{\frac {\Gamma (s,z)}{z^{s}e^{-z}}}&=\sum _{k=0}{\frac {L_{k}^{(\alpha )}(z)}{(k+1){k+1+\alpha -s \choose k+1}}}\qquad \left(\Re \left(s-{\frac {\alpha }{2}}\right)<{\frac {1}{4}}\right)\\&=\sum _{k=0}L_{k}^{(\alpha )}(z\,t)\cdot {\frac {_{2}F_{1}\left(1+\alpha +k,1+k;2+\alpha +k-s;{\frac {t-1}{t}}\right)}{t^{k}(k+1){1+\alpha +k-s \choose 1+k}}}\\&=t^{s}\sum _{k=0}L_{k}^{(\alpha )}(z\,t)\cdot {\frac {_{2}F_{1}\left(1-s,1+\alpha -s;2+\alpha +k-s;{\frac {t-1}{t}}\right)}{(k+1){1+\alpha +k-s \choose 1+k}}}\\&=t^{1+\alpha }\sum _{k=0}L_{k}^{(\alpha )}(z\,t)\cdot {\frac {_{2}F_{1}\left(1+\alpha +k,1+\alpha -s;2+\alpha +k-s;1-t\right)}{(k+1){1+\alpha +k-s \choose 1+k}}},\end{aligned}}}
2
F
1
{\displaystyle _{2}F_{1}}
超几何函数 。
围道积分表示 [ 编辑 ] 拉盖尔多项式可以用围道积分 表示，如下式所示：
L
n
(
α
)
(
x
)
=
1
2
π
i
∮
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 {\frac {e^{-{\frac {xt}{1-t}}}}{(1-t)^{\alpha +1}\,t^{n+1}}}\;dt}
积分方向逆时针绕原点一周。
与埃爾米特多項式的关系 [ 编辑 ] 广义拉盖尔多项式与埃爾米特多項式有下列关系：
H
2
n
(
x
)
=
(
−
1
)
n
2
2
n
n
!
L
n
(
−
1
/
2
)
(
x
2
)
{\displaystyle H_{2n}(x)=(-1)^{n}\ 2^{2n}\ n!\ L_{n}^{(-1/2)}(x^{2})}
以及
H
2
n
+
1
(
x
)
=
(
−
1
)
n
2
2
n
+
1
n
!
x
L
n
(
1
/
2
)
(
x
2
)
{\displaystyle H_{2n+1}(x)=(-1)^{n}\ 2^{2n+1}\ n!\ x\ L_{n}^{(1/2)}(x^{2})}
这里的H n x 2 )的埃爾米特多項式 （所谓的“物理学家形式”）。
正因为这样，广义拉盖尔多项式也在量子谐振子 的量子力学处理中出现。
与超几何函数的关系 [ 编辑 ] 拉盖尔多项式可以用超几何函数 来定义，具体地说，是用合流超几何函数 定义：
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)}
(
a
)
n
{\displaystyle (a)_{n}}
阶乘幂 ，这里表示升阶乘 。
与贝塞尔函数的关系 [ 编辑 ] 拉盖尔多项式与变形贝塞尔函数 之间有以下关系：
L
n
(
α
)
(
x
)
=
e
x
2
(
x
4
)
n
+
1
2
2
π
(
n
+
1
)
!
(
−
1
2
n
+
1
)
⋅
⋅
∑
k
=
0
n
(
−
1
)
k
+
1
(
2
n
+
1
n
−
k
)
(
n
+
α
n
)
(
α
+
2
n
+
1
n
−
k
)
(
n
−
k
+
α
n
−
k
)
(
k
+
1
2
)
K
k
+
1
2
(
x
2
)
=
e
x
2
(
4
x
)
n
+
α
+
1
2
Γ
(
α
+
1
2
)
(
−
α
−
1
n
)
(
−
α
−
1
2
n
)
⋅
⋅
n
!
∑
k
=
0
n
(
−
2
n
−
1
−
2
α
k
−
n
)
(
−
2
n
−
1
−
α
k
−
n
)
(
−
α
−
1
k
−
n
)
(
α
+
1
2
+
k
)
I
α
+
1
2
+
k
(
x
2
)
,
{\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)=&e^{\frac {x}{2}}\left({\frac {x}{4}}\right)^{n+{\frac {1}{2}}}{\frac {2}{{\sqrt {\pi }}(n+1)!{-{\frac {1}{2}} \choose n+1}}}\cdot \\&\cdot \sum _{k=0}^{n}(-1)^{k+1}{2n+1 \choose n-k}{\frac {{n+\alpha \choose n}{\alpha +2n+1 \choose n-k}}{n-k+\alpha \choose n-k}}\left(k+{\frac {1}{2}}\right)K_{k+{\frac {1}{2}}}\left({\frac {x}{2}}\right)\\=&e^{\frac {x}{2}}\left({\frac {4}{x}}\right)^{n+\alpha +{\frac {1}{2}}}\Gamma \left(\alpha +{\frac {1}{2}}\right){-\alpha -1 \choose n}{-\alpha -{\frac {1}{2}} \choose n}\cdot \\&\cdot n!\sum _{k=0}^{n}{\frac {{-2n-1-2\alpha \choose k-n}{-2n-1-\alpha \choose k-n}}{-\alpha -1 \choose k-n}}\left(\alpha +{\frac {1}{2}}+k\right)I_{\alpha +{\frac {1}{2}}+k}\left({\frac {x}{2}}\right)\end{aligned}},}
进一步有：
L
n
(
α
)
(
x
)
=
2
4
n
(
2
n
+
1
)
(
−
1
2
n
)
∑
k
=
0
n
(
k
+
1
2
)
(
2
n
+
1
n
−
k
)
(
n
k
)
2
(
n
+
α
k
)
(
2
n
+
α
+
1
n
−
k
)
x
n
−
k
(
n
−
k
)
!
L
k
−
2
k
−
1
(
x
)
.
{\displaystyle L_{n}^{(\alpha )}(x)={\frac {2}{4^{n}(2n+1){-{\frac {1}{2}} \choose n}}}\sum _{k=0}^{n}\left(k+{\frac {1}{2}}\right){\frac {2n+1 \choose n-k}{{n \choose k}^{2}}}{n+\alpha \choose k}{2n+\alpha +1 \choose n-k}{\frac {x^{n-k}}{(n-k)!}}L_{k}^{-2k-1}(x).}
外部链接 [ 编辑 ]
^ Abramowitz, p. 506, 13.3.8 （页面存档备份 ，存于互联网档案馆 ）
参考文献 [ 编辑 ] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22 （页面存档备份 ，存于互联网档案馆 ）", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
B Spain, M G Smith, Functions of mathematical physics , Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
Eric W. Weisstein, "Laguerre Polynomial （页面存档备份 ，存于互联网档案馆 ）", From MathWorld—A Wolfram Web Resource.
George Arfken and Hans Weber. Mathematical Methods for Physicists. Academic Press. 2000. ISBN 0-12-059825-6 S. S. Bayin (2006), Mathematical Methods in Science and Engineering , Wiley, Chapter 3. 
![{\displaystyle E\left[L_{n}(X)L_{m}(X)\right]=0\ {\mbox{whenever}}\ n\neq m.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfce23a3e74e527e067dcbe0fb05ad0cf768a63)
![{\displaystyle (0,n+\alpha +(n-1){\sqrt {n+\alpha }}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f94def16cdcc2ced729d4e45d7152000ed5453)
![{\displaystyle \int _{0}^{\infty }x^{\alpha +1}e^{-x}\left[L_{n}^{(\alpha )}\right]^{2}dx={\frac {(n+\alpha )!}{n!}}(2n+\alpha +1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb41ec3fea5cdbbd653107cdb8067455a1cbccb9)
