The Newton's series, i.e. the discrete analogue of the continuum Taylor expansion, involves classical iterated difference operators $\Delta$ defined by $\Delta f(k) = f(k+1) - f(k)$. Indeed, Newton's series writes $$f(x) = \sum_{k=0}^{\infty}\frac{\Delta^{k}f(a)}{k!}(x-a)_{k},$$

where $(x)_{k} = x(x-1)(x-2)...(x-k+1)$.

There also exist a generalized difference operator defined by

$$\Delta^{\mu}f(x) = \sum_{k=0}^{\infty}\mu_{k}f(x+k),$$

where $\mu = (\mu_{1}, \mu_{2}, ...)$ is a sequence of real numbers such that $\sum_{k=0}^{\infty}\mu_{k} < \infty$.

My question: is there a "discrete Taylor expansion" like the one presented above involving $\Delta^{\mu}$ instead of the classical $\Delta$ ?

Thank you for your answers !