  Let $h:\mathbb Z \to \mathbb Z_{\geq 0}$ be a nonzero function with $h(k)=0$
for $k\ll 0$. We define the Hilbert depth of $h$ by
$\operatorname{hdepth}(h)=\max\{d\;:\; \sum_{j\leq k}
(-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$. We show that
$\operatorname{hdepth}(h)$ is a natural generalization for the Hilbert depth of
a subposet $\operatorname{P}\subset 2^{[n]}$ and we prove some basic properties
of it.
  Given $h(j)=\begin{cases} aj^n+b,& j\geq 0 \\ 0, & j<0 \end{cases}$, with
$a,b,n$ positive integers, we compute $\operatorname{hdepth}(h)$ for $n=1,2$
and we give upper bounds for $\operatorname{hdepth}(h)$ for $n\geq 3$. More
generally, if $h(j)=\begin{cases} P(j),& j\geq 0 \\ 0,& j<0 \end{cases}$, where
$P(j)$ is a polynomial of degree $n$, with non-negative integer coefficients,
and $P(0)>0$, we show that $\operatorname{hdepth}(h)\leq 2^{n+1}$.
