q-quantum plane, ψ(q)-umbral calculus, and all that ^*)

It is known that one can formulate q-extended finite operator calculus with help of “quantum q-plane” q-commuting variables A, B: AB - qBA \equiv [A, B]_q = 0.

We shall recall this simple fact in its natural entourage which is the so-called ψ(q)-extension of Rota's finite operator calculus. We aim to convince the audience that this is a natural and elementary method for formulation and treatment of q-extended and possibly R-extended or ψ(q)-extended models for quantum-like ψ(q)-deformed oscillators.