The Grünwald–Letnikov definition of the fractional derivative of order

of

is given by the expression

,

,

where

represents the gamma function and

the increment. We verify that

,

.

The expression

can be viewed as the expected value of the discrete random variable that for

takes the value

with probability

.

The Grünwald–Letnikov definition gets the slope of a triangle with upper corners

and

. The factor

in the denominator expression means that, for large values of

, we have a slow variation, while for small values of

we have a fast variation.

The implementation of the Grünwald–Letnikov definition of the fractional derivative corresponds to an

-term truncated series given by

.

[1] K. B. Oldham and J. Spanier,

*The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order*, New York: Academic Press, 1974.

[2] I. Podlubny,

*Fractional Differential Equations*, San Diego: Academic Press, 1999.

[3] J. A. Tenreiro Machado, "Discrete-Time Fractional-Order Controllers,"

*Journal of Fractional Calculus & Applied Analysis*,

**4**(1), 2001 pp. 47–66.

[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,

*Theory and Applications of Fractional Differential Equations*, Vol. 204, Amsterdam: Elsevier, 2006.

[5] J. A. Tenreiro Machado, "Fractional Derivatives: Probability Interpretation and Frequency Response of Rational Approximations,"

*Communications in Nonlinear Science and Numerical Simulations*,

**14**(9–10), 2009 pp. 3492–3497.