Bernoulli-Taylor formula in the case of Q-umbral Calculus

TL;DR

This paper derives a Q-difference Bernoulli-Taylor formula with a Cauchy form remainder using Viskov's method, extending Q-umbral calculus techniques with a Q-extended Kwasniewski's *-product.

Contribution

It introduces a new derivation of the Bernoulli-Taylor formula in Q-umbral calculus using an extended Kwasniewski's *-product and Viskov's method.

Findings

Derived Q-difference Bernoulli-Taylor formula with Cauchy form rest term

Extended Q-umbral calculus techniques with Q-extended Kwasniewski's *-product

Provided theoretical foundation for further applications in Q-calculus

Abstract

In this note we derive the Q-difference Bernoulli-Taylor formula with the rest term of the Cauchy form by the Viskov's method. This is an extension of technique by the use of Q-extented Kwasniewski's *-product . The main theorems of Q-umbral calculus were given by G. Markowsky in 1968 and extented by A.K.Kwasniewski.