I have recently been working on a problem using fractional calculus and have come across something in Maple's fracdiff  command that makes no sense to me.

Consider the function y:=a+b*(x-q)+c*(x-q)^2

z:=subs(x=q,fracdiff(y,x,1)) gives the correct answer of z:=b, z:=subs(x=q,fracdiff(y,x,2)) gives the correct answer of z:=2c, z:=subs(x=q,fracdiff(y,x,3/2)) gives the answer of z:=4*sqrt(q)*c/sqrt(pi)

Surely the differential of a function has to be invariant under translation in x? That is if you translate a function by distance q along the x-axis f(x-q), then surely the derivative functions must also translate by q along the x-axis. If this is so, the last example should give z as a function of a,b,c only, but not q.

The first and second differentials in the example above are invariant under this translation, so why is the 3/2 differential different in this respect?   (Maple reckons it scales in amplitude by sqrt(q))

I guess it has something to do with the fact that the fractional differential operator is non-commutative and non-additive but my argument is why should fractional derviatives not be invariant under translation in x-axis when all the integer derivatives are?

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