本德爾-鄧恩多項式 本德爾·鄧恩多項式 (Bender-Dunne polynomials)是一個正交多項式 ，定義如下：[ 1]
P
0
(
x
)
=
1
{\displaystyle P_{0}(x)=1}
P
1
(
x
)
=
x
{\displaystyle P_{1}(x)=x}
當
n
>
1
{\displaystyle n>1}
P
n
(
x
)
=
x
P
n
−
1
(
x
)
+
16
(
n
−
1
)
(
n
−
J
−
1
)
(
n
+
2
s
−
2
)
P
n
−
2
(
x
)
{\displaystyle P_{n}(x)=xP_{n-1}(x)+16(n-1)(n-J-1)(n+2s-2)P_{n-2}(x)}
P
[
2
]
=
x
2
+
32
∗
s
−
32
∗
s
∗
J
P
[
3
]
=
x
3
+
160
∗
x
∗
s
−
96
∗
x
∗
s
∗
J
+
64
∗
x
−
32
∗
x
∗
J
P
[
4
]
=
x
4
+
448
∗
x
2
∗
s
−
192
∗
x
2
∗
s
∗
J
+
352
∗
x
2
−
128
∗
x
2
∗
J
+
9216
∗
s
−
12288
∗
s
∗
J
+
9216
∗
s
2
−
12288
∗
s
2
∗
J
+
3072
∗
s
∗
J
2
+
3072
∗
s
2
∗
J
2
.
{\displaystyle {\begin{aligned}P[2]&=x^{2}+32*s-32*s*J\\P[3]&=x^{3}+160*x*s-96*x*s*J+64*x-32*x*J\\P[4]&=x^{4}+448*x^{2}*s-192*x^{2}*s*J+352*x^{2}-128*x^{2}*J+9216*s-12288*s*J+9216*s^{2}-12288*s^{2}*J+3072*s*J^{2}+3072*s^{2}*J^{2}.\end{aligned}}}
^ Bender, Carl M.; Dunne, Gerald V. (1988), "Polynomials and operator orderings", Journal of Mathematical Physics 29 (8): 1727–1731, doi:10.1063/1.527869, ISSN 0022-2488, MR 955168
![{\displaystyle {\begin{aligned}P[2]&=x^{2}+32*s-32*s*J\\P[3]&=x^{3}+160*x*s-96*x*s*J+64*x-32*x*J\\P[4]&=x^{4}+448*x^{2}*s-192*x^{2}*s*J+352*x^{2}-128*x^{2}*J+9216*s-12288*s*J+9216*s^{2}-12288*s^{2}*J+3072*s*J^{2}+3072*s^{2}*J^{2}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8eeae0caa31a7fc12b98e7fb6c955cfb619e62f)