Fractional Differential Equation

The solution to the differential equation

$$[D^{2v}+aD^v+bD^0]y(t)=0$$

is

$$y(t)=\cases{ e_\alpha(t)-e_\beta(t)\cr \qquad {\rm for\ }\... ...u-1}\over\Gamma(2v)}\cr \qquad {\rm for\ }\alpha=\beta=0,\cr}$$

where

$$q = {1\over v}$$

$$e_\beta(t) = \sum_{k=0}^{q-1} \beta^{q-k-1} E_t(-kv,\beta^q),$$

$E_t(a,x)$ is the Et-Function, and $\Gamma(n)$ is the Gamma Function.

References

Miller, K. S. ``Derivatives of Noninteger Order.'' Math. Mag. 68, 183-192, 1995.