More on the Bernoulli*–Taylor Formula for Extended Umbral Calculus

One presents here the *_ψ-Bernoulli-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 ψ-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 - and recent publications do confirm this historically established experience. *see below: a historical remark based on Academician N. Y. Sonin article published in Petersburg in 19-th century. We owe this information and the article to Professor O. V. Viscov from Moscow. **see: C. Graves, On the principles which regulate the interchange of symbols in certain symbolic equations, Proc. Royal Irish Academy 6 (1853-1857), 144-152.