On extended umbral calculus, oscillator-like algebras and Generalized Clifford Algebra

Some quantum algebras build from deformed oscillator algebras ala Jordan-Schwinger may be described in terms of a particular case of psi-calculus. We give here an example of a specific relation between such certain quantum algebras and generalized Clifford algebras also in the context of Levy-Leblond's azimuthal quantization of angular momentum which was interpreted afterwards as the finite dimensional quantum mechanics by Santhanam et. all. ψ-calculus used for that as a framework is that of classical operator calculus of Rota. By its nature ψ-umbral calculus supplies a simple mathematical underpinning for ψ-deformed quantum-like oscillator algebras and-at least for the ψ n (q) = [n q !] −1 case [1-3]. It provides the natural underpinning for quantum group investigation. Moreover-the other way around-one may formulate q-extended finite operator calculus with help of the " quantum q-plane " q-commuting variables. .. ψ-calculus is expected to be useful in C * algebraic [4] description of " ψ-quantum processes " with various parastatistics [5].