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Bell Polynomial


Two different Generating Functions for the Bell polynomials for $n>0$ are given by

\phi_n(x) \equiv e^{-x} \sum_{k=0}^\infty {k^n x^k\over k!}


\phi_n(x)=x\sum_{k=1}^n {n-1\choose k-1}\phi_{k-1}(x),

where ${n\choose k}$ is a Binomial Coefficient.

The Bell polynomials are defined such that $\phi_n(1)=B_n$, where $B_n$ is a Bell Number. The first few Bell polynomials are

$\displaystyle \phi_0(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle \phi_1(x)$ $\textstyle =$ $\displaystyle x$  
$\displaystyle \phi_2(x)$ $\textstyle =$ $\displaystyle x+x^2$  
$\displaystyle \phi_3(x)$ $\textstyle =$ $\displaystyle x+3x^2+x^3$  
$\displaystyle \phi_4(x)$ $\textstyle =$ $\displaystyle x+7x^2+6x^3+x^4$  
$\displaystyle \phi_5(x)$ $\textstyle =$ $\displaystyle x+15x^2+25x^3+10x^4+x^5$  
$\displaystyle \phi_6(x)$ $\textstyle =$ $\displaystyle x+31x^2+90x^3+65x^4+15x^5+x^6.$  

See also Bell Number


Bell, E. T. ``Exponential Polynomials.'' Ann. Math. 35, 258-277, 1934.

© 1996-9 Eric W. Weisstein