This is a polynomial that I found in my research, and I'd like to know a simplified form.
I have a very simple degree-$n$ multivariate polynomial $p(x,z_1,\cdots,z_n) = (x + z_1 + \cdots +z_n)^n$.
I'd like to apply a set of simple operations to convert $p(x,z_1,\cdots,z_n)$ into a univariate polynomial $q(x)$.
\begin{equation*} q(x) = \left[ \left( \prod_{i=1}^n \left(1 - \frac{\partial}{\partial z_i}\right) \right) p(x,z_1,\cdots,z_n) \right] \Bigg|_{z_1=\cdots=z_n=0}. \end{equation*}
One can easily see that the conversion is composed of two kinds of operations:
- Restriction: Setting a certain variable $z_i = 0$ in a multivariate polynomial; that is, $r(x,z_1,\cdots,z_{n-1},z_n)|_{z_n=0} = r(x,z_1,\cdots,z_{n-1},0)$.
- Differentiation: $(1 - \frac{\partial}{\partial z_i})r(x,z_1,\cdots,z_n) = r(x,z_1,\cdots,z_n) - \frac{\partial}{\partial z_i}r(x,z_1,\cdots,z_n)$.
I'd like to know how $q(x)$ looks for any $n$. Is there any closed form for this univariate polynomial $q(x)$? Can I actually expand the definition of $q(x)$ and find a simplified form?