  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.
