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Summary:
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.
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