Information on some recent applications of umbral extensions to discrete mathematics

TL;DR

This paper explores recent applications of umbral extensions in discrete mathematics, deriving new formulas and extending solutions to difference calculus equations with specific focus on q-calculus and fibonomial calculus.

Contribution

It introduces new Dobinski-like formulas from umbral extensions of Stirling numbers and extends Ward solutions to extended difference calculus, including q-calculus and fibonomial calculus.

Findings

Derived new Dobinski-like formulas from umbral extensions.

Extended Ward solutions to nonhomogeneous difference equations.

Explicitly applied to q-calculus and fibonomial calculus cases.

Abstract

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 .