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Summary:
This paper deals with $\varphi(q)$ 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 $\varphi_n(q)= [n_{q^1}]^{-1}$ is known to be relevant for quantum group investigations. It is shown here that such $\varphi(q)$ 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 $\varphi(q)$ calculus in which new families of $\varphi(q)$ extensions of Poisson processes and $q$-Bernoulli-Taylor formula with the rest $q$-term of Cauchy type are derived besides other results.
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