Write the expression of Bell polynomials $B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right)$ for $g(x)=4-x^2$.

According to Faà di Bruno's formula, $${d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right)$$ where $B_{n,k}(x_{1},...,x_{n−k+1})$ are the Bell polynomials. In my case $g(x)=4-x^2$. How can I write the expression of Bell polynomials if $g'(x)=-2x$, $g''(x)=-2$ and $g^{(k)}(x)=0$ for $k\geq3$?

Solution 1:

In the sum defining $B_{n,k}$,

suppose that for $p\geq 3, x_p=0$

for $p\geq 3$, if $j_p>0$ over $x_{j_p}=0$ then the term is zero.

It remains the terms such that for $p\geq 3$, $j_p=0$ (and we have $j_p!=1$)

The sum becomes

$B_{n, k}(x_1,x_2, ...,x_{n-k+1})=\displaystyle\sum_{j_1+j_2=k, j_1+2j_2=n}\dfrac{k!}{j_1!j_2!}x_1^{j_1} x_2^{j_2}$

For n=1, k=1, the conditions are only verified for $(j_1,j_2)=(1,0)$

$B_{1,1}(g'(x),g''(x), ...)=g'(x)$

For n=2,

k=1, $(j_1,j_2)=(0,1)$

k=2, $(j_1,j_2)=(2,0)$

$B_{2,1}(g'(x),g''(x), ...)=g''(x)$

$B_{2,2}(g'(x),g''(x), ...)=[g'(x)]^2$