solving the partial diffential ODE $ y^{s} (x)=y(x) $

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how could i solve the differential equation with frational derivatives

$$ \frac{d^{s}}{dx^{s}}y(x) =y(x)$$

here 's' is a real number

my idea is to make the ansatz with the series $ y(x)= \sum_{n=0}^{\infty}a_{n}x^{n} $ and apply this series to the linear ODE.

but i have problems to get what cauchy initial conditions should i put in case of fractional derivatives..