A Unified Calculus Using the Generalized Bernoulli Polynomials

AbstractWe introduce an α-calculus with the help of the generalized Bernoulli polynomials. The parameter α is the order of a Bessel function of the first kind. The differential α-calculus can be put in a general context where the concept of supporting function is an important tool for practical purposes. Our somewhat more restrictive point of view has the advantage of permitting a consistent definition of an α-integral with several interesting properties. It results in the possibility of expressing a remainder, in the aforementioned context, in a completely new form in our case