Touchard Polynomials
图沙德多项式是1939年法兰西数学家Jacques Touchard提出的多项式。定义如下[1]:
Touchard Polynomials
![{\displaystyle T_{n}(x)=\sum _{k=0}^{n}S(n,k)x^{k}=\sum _{k=0}^{n}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}x^{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab46c9f53efeac337ba622dda3a41dd9359a47c2)
其中
是第二类斯特林数。
前面几个图沙德多项式是:
![{\displaystyle T_{0}(x)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9414e090394b17d0c05d9f4312baf7a82c571c66)
![{\displaystyle T_{1}(x)=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/404d7dcada1494e3807fb23f14eb118d6f42c7bd)
![{\displaystyle T_{2}(x)=x+x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1160dae3ff876d04f101b2b55c9ee2c20d2d593b)
![{\displaystyle T_{3}(x)=x+3x^{2}+x^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0957442e6e7b2ee037c05ca935376b8a438efb)
![{\displaystyle T_{4}(x)=x+7x^{2}+6x^{3}+x^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c037b7e1f0c391d1ae6b4397bb1429ea12eae3eb)
![{\displaystyle T_{5}(x)=x+15x^{2}+25x^{3}+10x^{4}+x^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c953647bf0164fcb60d4cf44b544c6980343d3)
生成函数
图沙德多项式的生成函数为
![{\displaystyle \sum _{n=0}^{\infty }{T_{n}(x) \over n!}t^{n}=e^{x\left(e^{t}-1\right)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/707e2b88461bb3b4ce4d23bbc2cb5a2e6c9e05bd)
参考文献
- ^ Touchard, Jacques, Sur les cycles des substitutions, Acta Mathematica, 1939, 70 (1): 243–297, ISSN 0001-5962, MR 1555449, doi:10.1007/BF02547349