Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2017
Mittag–Leﬄer Functions and the Truncated
J. Vanterler da C. Sousa and E. Capelas de Oliveira
Abstract. In this paper, we introduce a new type of fractional derivative,
which we called truncated V-fractional derivative, for α-diﬀerentiable
functions, by means of the six-parameter truncated Mittag–Leﬄer func-
tion. One remarkable characteristic of this new derivative is that it
generalizes several diﬀerent fractional derivatives, recently introduced:
conformable fractional derivative, alternative fractional derivative, trun-
cated alternative fractional derivative, M-fractional derivative and trun-
cated M-fractional derivative. This new truncated V-fractional deriv-
ative satisﬁes several important properties of the classical derivatives
of integer order calculus: linearity, product rule, quotient rule, func-
tion composition and the chain rule. Also, as in the case of the Caputo
derivative, the derivative of a constant is zero. Since the six parameters
Mittag–Leﬄer function is a generalization of Mittag–Leﬄer functions of
one, two, three, four and ﬁve parameters, we were able to extend some
of the classical results of the integer-order calculus, namely: Rolle’s the-
orem, the mean value theorem and its extension. In addition, we present
a theorem on the law of exponents for derivatives and as an application
we calculate the truncated V-fractional derivative of the two-parameter
Mittag–Leﬄer function. Finally, we present the V-fractional integral
from which, as a natural consequence, new results appear as applica-
tions. Speciﬁcally, we generalize the inverse property, the fundamen-
tal theorem of calculus, a theorem associated with classical integration
by parts, and the mean value theorem for integrals. We also calculate
the V-fractional integral of the two-parameter Mittag–Leﬄer function.
Further, we were able to establish the relation between the truncated
V-fractional derivative and the truncated V-fractional integral and the
fractional derivative and fractional integral in the Riemann–Liouville
sense when the order parameter α lies between 0 and 1 (0 <α<1).
Mathematics Subject Classiﬁcation. 26A06, 26A24, 26A33, 26A39, 26A42.
Keywords. Mittag–Leﬄer functions, V-fractional derivative, V-fractional
integral, Riemann–Liouville derivative, Riemann–Liouville integral.
This work was completed with the support of our T