Towards ψ-extension of Rotaʹs finite operator calculus

ψ-extension of Rotaʹs finite operator calculus is further developed. The extension relies on the notion of ∂ψ-shift invariance and ∂ψ-delta operators. Main statements of Rotaʹs finite operator calculus are given by their ψ-counterparts. This includes among others the properties of Sheffer ψ-polynomials and Rodrigues formula. Such ψ-extended calculus delivers an elementary umbral underpinning for a model of q-deformed quantum oscillator and its possible generalisations. ta 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 ψn(q) = [R(qn)!]−1 and R(x) = 1−x1−q, we arrive at the well-known q-deformed oscillator. Because the reduced incidence algebra R(L(S)) is isomorphic to the algebra Φψ of ψ-exponential formal power series, the ψ-extensions of finite operator calculus provide a vast family of representations of R(L(S)).