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The paper deals with extensions of the finite operator calculus of G.-C. Rota called $\psi$-extensions which give a framework for corresponding quantum group investigations. This also covers the instance of the well-known $q$-analogue of umbral calculus. The article also contains glossaries of the most important terms and notations used by Ward, Viskov, Markowsky and Roman on one side and the Rota-oriented notations on the other side.

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