  At the first part of the paper we show how specific umbral extensions of the
Stirling numbers of the second kind result in new type of Dobinski-like
formulas. In the second part among others one recovers how and why Ward
solution of uncountable family of extended difference calculus nonhomogeneous
equations extends to Ward-Appell polynomials case . Illustrative specifications
to q-calculus case and fibonomial calculus case are made explicit due to the
usage of the so called upside down notation for objects of extended finite
operator calculus .
