# Prove that $a_{n-k}$ can be expressed in the terms of $∇a_n, ∇a_n, ∇^2a_n,…,∇^ka_n$

Prove that $a_{n-k}$ can be expressed in the terms of $$∇a_n, ∇a_n, ∇^2a_n,...,∇^ka_n$$

-I'm brand new to the del operator, and unsure of how to utilize it/ work with it in this proof, any help is appreciated.

Note that if we denote by $B$ the backshift $(Ba)_n = a_{n-1}$ on sequences, we have that $$\def\d{\nabla}\d a_n = a_n - a_{n-1} = a_n - Ba_n = (I - B)a_n$$ that is $\d = I - B$ or $B = I - \d$. As $\d$ and $I$ commute, this implies $$B^k = (I - \d)^k = \sum_{i=0}^k \binom ki (-1)^i \d^i$$ hence, as $B^ka_n = a_{n-k}$, we have $$a_{n-k} = (I-\d)^k = \sum_{i=0}^k \binom ki (-1)^i \d^i a_n$$