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6. Grunwald-Letnikov Derivative

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Grunwald-Letnikov derivative or also named Grunwald-Letnikov differintegral, is a generalization of the derivative analogous to our generalization by the binomial formula (4.6), but it is based on the direct generalization of the equation (1.4). The idea behind is that h should approach 0 as n approaches infinity,

 (6.1)

being a negative infinity. However, the generalization is done so that any wished a less than x can be chosen. The above equation can be generalized now to get a formula equivalent to (4.6)

 (6.2)

or equivalently,

 (6.3)

These generalizations have the inconvenient that if a is a negative integer they are not valid for the gamma function is infinity in negative integers and zero. However, for negative a values the following can be used

 (6.4)

so that for negative values equations (6.2) and (6.3) can be changed to

 (6.5)

and

 (6.6)

The reason why x-a is considered in the above equations instead of a single positive number can be seen considering the case of a=-1, which must give the same result that integrating the function,

 (6.7)

Yes, in fractional derivatives limits must be considered as they must be in integrals. Limits only vanish with integer order derivatives. This also means that fractional derivatives are nonlocal, which may be the reason that makes this kind of derivatives less useful in describing Nature. We shall see later more about the integration -or better say differentiation- limits.

In the case of the derivative generalized by the binomial formula (4.6), since n goes to infinity regardless of h, x-a must be infinity, so that the derivative defined in (4.4) and (4.6) is equivalent to the Grunwald-Letnikov derivative with a lower limit of negative infinity.