In mathematics, there is a finite difference operator $\Delta$ defined by $\Delta_n f(n) = f(n + 1) - f(n)$. This operator shares many properties with the continuous derivative $\mathcal{D}$. However, it lacks a simple composition relation. This operator finds use in calculating sums, demonstrated by the relation: $$ \sum_{n = a}^{b - 1} \Delta_n f(n) = f(b) - f(a). $$
My question is why isn't there an operator $_2\Delta$ defined as $_2 \Delta_n f(n) = \frac{f(n + 1)}{f(n)}$? This operator would be called the finite ratio operator. It is related to the variadic product operator $\prod$ given this relation: $$ \prod_{n = a}^{b - 1} {_2\Delta_n} f(n) = \frac{f(b)}{f(a)}. $$
This operator seems at the very least fairly useful for computing products, but I think it would also allow for an extension of the Umbral Calculus. I'm not quite sure how this extension would work (mainly because I don't quite know what an analogous ratio continuous derivative ($_2\mathcal{D}$) would look like, but I have some thoughts).
With the normal Umbral Calculus (at least according to my current understanding of this video), one is better able to transition between the discrete world of mathematics and the continuous world. This is done by introducing an operator $\phi$ so that $\Delta\phi = \mathcal{D}\phi$. I was thinking about introducing another operator $_2\phi$ where $_2\Delta_2\phi = {_2\phi}{_2\mathcal{D}}$, an operator $\alpha$ that relates $\Delta$ and $_2\Delta$, and an operator $\alpha'$ that relates $\mathcal{D}$ and $_2\mathcal{D}$.
The $_2\Delta$ could be generalized as $_n\Delta$, representing the "hyper-ratios" of higher hyperoperations.
Does such an operator like $_2\Delta$ already exist? If so, what has it been used for?