Further remarks on extended umbral calculus

Kwaśniewski, A. K.; Grądzka, E.

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More on deformed oscillator algebras and extended umbral calculus

Kwaśniewski, A. K., Grądzka, E.

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This paper deals with $ϕ (q)$ calculus which is an extension of finite operator calculus due to Rota, and leading results of Rota’s calculus are easily $ϕ$ -extendable. The particular case $ϕ_{n} (q) = {[n_{q^{1}}]}^{- 1}$ is known to be relevant for quantum group investigations. It is shown here that such $ϕ (q)$ umbral calculus leads to infinitely many new $ϕ$ -deformed quantum like oscillator algebra representations. The authors point to several references dealing with new applications of $q$ umbral and $ϕ (q)$ calculus in which new families...

Extended finite operator calculus-an example of algebraization of analysis

Andrzej Kwaśniewski, Ewa Borak (2004)

Open Mathematics

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“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota-Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form-a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota-Mullin or equivalently-of umbral calculus of Roman and Rota. At the...