# 2011-10 Multivariable polynomial

Let $$t_1,t_2,\ldots,t_n$$ be positive integers. Let $$p(x_1,x_2,\dots,x_n)$$ be a polynomial with n variables such that $$\deg(p)\le t_1+t_2+\cdots+t_n$$. Prove that $$\left(\frac{\partial}{\partial x_1}\right)^{t_1} \left(\frac{\partial}{\partial x_2}\right)^{t_2}\cdots \left(\frac{\partial}{\partial x_n}\right)^{t_n} p$$ is equal to $\sum_{a_1=0}^{t_1} \sum_{a_2=0}^{t_2}\cdots \sum_{a_n=0}^{t_n} (-1)^{t_1+t_2+\cdots+t_n+a_1+a_2+\cdots+a_n}\left( \prod_{i=1}^n \binom{t_i}{a_i} \right)p(a_1,a_2,\ldots,a_n).$

GD Star Rating