Laguerre-Type Bernoulli and Euler Numbers and Related Fractional Polynomials

Abstract

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0).

Keywords

• Bernoulli numbers and polynomials
• Euler numbers and polynomials
• Laguerre-type exponential functions
• generating functions
• generalized Laplace transform