I need a formula for $\Delta^k \frac 1 u$, where $u(x)$ is a strictly positive function, $\Delta^k$ is the difference operator defined recursively as $\Delta^k=\Delta^1 \Delta^{k-1}$ and $\Delta^1 u(x)=u(x+h)-u(x)$, with $h$ fixed. It is trivial to get a recursive formula by expanding the identity $\Delta^k(u\cdot\frac 1 u)=0$ via the discrete Leibnitz' formula. Thus it should be easy to prove the formula by induction. But it is not easy to guess the correct expression. 

I suspect the formula is known, although possibly not widely known (like the formula for the $k$-th derivative of $1/u$). Any pointers to existing literature would be appreciated.