Towards psi-extension of Rota`s Finite Operator Calculus

ψ-extension of Gian-Carlo Rota's finite operator calculus due to Viskov [1, 2] is further developed. The extension relies on the notion of ∂ ψ-shift invariance and ∂ ψ-delta operators. Main statements of Rota's finite operator calculus are given their ψ-counterparts. This includes Sheffer ψ-polynomials properties and Rodrigues formula-among others. Such ψ-extended calculus delivers an elementary um-bral underpinning for q-deformed quantum oscillator model and its possible generalisations. ∂ q-delta operators and their duals and similarly ∂ ψ-delta operators with their duals are pairs of generators of ψ(q)-extended quantum oscillator algebras. With the choice ψ n (q) = [R(q n)!] −1 and R(x) = 1−x 1−q we arrive at the well known q-deformed oscillator. Because the reduced incidence algebra R(L(S)) is isomorphic to the algebra Φ ψ of ψ-exponential formal power series-the ψ-extensions of finite operator calculus provide a vast family of representations of R(L(S)).