There's a generalization of the mean value theorem that can be applied here, namely that if $f$ is smooth enough then for each $x$ there is $y$ such that $$\Delta^kf(x)=f^{(k)}(y)$$ and $x < y < x+k$.

Added see Is it true that all the "irrational power" functions are almost polynomial ? .

There's a generalization of the mean value theorem that can be applied here, namely that if $f$ is smooth enough then for each $x$ there is $y$ such that $$\Delta^kf(x)=f^{(k)}(y)$$ and $x < y < x+k$.

Added see Is it true that all the "irrational power" functions are almost polynomial ? .

There's a generalization of the mean value theorem that can be applied here, namely that if $f$ is smooth enough then for each $x$ there is $y$ such that $$\Delta^kf(x)=f^{(k)}(y)$$ and $x < y < x+k$.

There's a generalization of the mean value theorem that can be applied here, namely that if $f$ is smooth enough then for each $x$ there is $y$ such that $$\Delta^kf(x)=f^{(k)}(y)$$ and $x < y < x+k$.

Added see Is it true that all the "irrational power" functions are almost polynomial ? .