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arXiv · 2303.00796

Fractional Finite Sums Theory: a motivation and some new results

Leonardo F. Bielinski·Giuliano G. La Guardia·Jocemar Q. Chagas
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Abstract

This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a function). Examples of the essences of functions are presented, including the Bernoulli numbers, which are obtained as essences of the functions $f(x)=x^a$, $a \in \mathbb{N}^*$. Based on the new results, we can expand the comprehension of the Euler-Maclaurin summation formula for a real number instead of only considering natural numbers. Moreover, we utilize the concept of the essence of a function to propose a new regularization method for divergent series.

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Leonardo F. Bielinski, Giuliano G. La Guardia, Jocemar Q. Chagas. 2023-03-01. Fractional Finite Sums Theory: a motivation and some new results. https://arxiv.org/abs/2303.00796

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