Long-story-short, I'm required to come up with an unconventional concept for a presentation, and thought of the following:
An operator $D$ such that for any recurrence relation, we get that$$D(x_n) = n x_{n-1}$$similar to the standard derivative, with an anti-derivative analog$$I(x_n) = \frac{x_{n+1}}{n+1}$$To see it in action, given a difference operator $\Delta$ we get the example:$$D(\Delta x_n) - \Delta x_{n-1} = n x_{n} - (n-1)x_{n-1}$$or, given a generating function$$G_a(x) = \sum_{n=1}^{\infty}a_n x^n$$we can see that$$D(G_a) = \sum_{n=1}^{\infty}na_{n-1} x^n \approx x\frac{d G_a}{dx}$$if $a_n$ allows, or at least looks very similar in form to $x\frac{d G_a}{dx}$ suggesting an actual potential use through some algebraic manipulations. I need to build a presentation of a study of the concept, so my question is what other things can we do with this? If it has ever been used, etc?