Hypergeometric bernoulli polynomials and numbers of fractional index

In this paper, we shall study what we referred to as Bernoulli numbers and polynomials of fractional index and prove some of their properties. While these numbers and polynomials are similar to the classical Bernoulli numbers and polynomials, they differ in many aspects. Among the properties we shall study are the recurrence formulas satisfied by these numbers and polynomials. We shall also introduce the higher order Bernoulli polynomials of fractional order and establish some of their properties. Finally, we will show their connections with Hermite polynomials.