本德爾·鄧恩多項式(Bender-Dunne polynomials)是一個正交多項式，定義如下：[1]. 本德爾-鄧恩多項式 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)} Remove ads例子 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}}} Remove ads參考文獻Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads
![{\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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c8eeae0caa31a7fc12b98e7fb6c955cfb619e62f)