Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

Abstract

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

Keywords

• fractional exponential function
• Bernoulli numbers with fractional indices
• fractional Bernoulli polynomials
• Euler numbers with fractional indices
• fractional Euler polynomials