More on the Bernoulli and Taylor Formula for Extended Umbral Calculus

One delivers here the extended Bernoulli and Taylor formula of a new sort with the rest term of the Cauchy type recently derived by the author in the case of the so called $\psi$-difference calculus which constitutes the representative for the purpose case of extended umbral calculus. The central importance of such a type formulas is beyond any doubt. Recent publications do confirm this historically established experience. Its links via umbrality to combinatorics are known at least since Rota and Mullin source papers then up to recently extended by many authors to be indicated in the sequel.