More on the Bernoulli and Taylor Formula for Extended Umbral Calculus

TL;DR

This paper introduces an extended Bernoulli and Taylor formula within the framework of extended umbral calculus, emphasizing the significance of such formulas and their connections to combinatorics and difference calculus.

Contribution

It presents a new form of Bernoulli and Taylor formulas with a Cauchy-type remainder term in the context of $$-difference calculus, expanding the theoretical framework.

Findings

Derived a new extended Bernoulli and Taylor formula

Established links between umbral calculus and combinatorics

Highlighted the importance of these formulas in mathematical analysis

Abstract

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.