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Article

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

by
Diego Caratelli
1,2,
Pierpaolo Natalini
3 and
Paolo Emilio Ricci
4,*
1
Department of Research and Development, The Antenna Company, High Tech Campus 29, 5656 AE Eindhoven, The Netherlands
2
Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
3
Department of Mathematics and Physics, Roma Tre University, Largo San Leonardo Murialdo 1, 00146 Rome, Italy
4
Department of Mathematics, International Telematic University UniNettuno, Corso Vittorio Emanuele II 39, 00186 Rome, Italy
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(10), 1900; https://doi.org/10.3390/sym15101900
Submission received: 1 September 2023 / Revised: 21 September 2023 / Accepted: 7 October 2023 / Published: 10 October 2023
(This article belongs to the Special Issue Mathematical Theories, Models and Methods in Applied Sciences: Deterministic, Stochastic and Symmetry Perspectives)
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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 fractional exponential function; Bernoulli numbers with fractional indices; fractional Bernoulli polynomials; Euler numbers with fractional indices; fractional Euler polynomials

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MDPI and ACS Style

Caratelli, D.; Natalini, P.; Ricci, P.E. Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory. Symmetry 2023, 15, 1900. https://doi.org/10.3390/sym15101900

AMA Style

Caratelli D, Natalini P, Ricci PE. Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory. Symmetry. 2023; 15(10):1900. https://doi.org/10.3390/sym15101900

Chicago/Turabian Style

Caratelli, Diego, Pierpaolo Natalini, and Paolo Emilio Ricci. 2023. "Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory" Symmetry 15, no. 10: 1900. https://doi.org/10.3390/sym15101900

APA Style

Caratelli, D., Natalini, P., & Ricci, P. E. (2023). Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory. Symmetry, 15(10), 1900. https://doi.org/10.3390/sym15101900

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