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Is it possible to form differential equation of curves basically defined in fractional calculus

$$F(x,q)=\frac{x^(1 - q)}{\Gamma[2 - q]} + 2 \sum_{k=1}^{n-1} unitstep(x - 2 k + 1)(-1)^k\frac{(x - 2k + 1)^(1 - q)}{ \Gamma[2 - q]} ? $$

e.g.,for the following

$$ F( x,-\frac12 )? $$

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