I think the problem is a general tensor analysis one rather semiconductor physics (GaAs), since no particular values of any d-symbol is of any importance? Anyway, here is my effort to produce an answer.
The piezoelectric coefficient $d_{ijk}=d^{[110]}_{ijk}$ is a third rank Cartesian tensor which transforms from the reference frame $[x_1, x_2, x_3]$ to $[\bar {x}_1,\bar {x}_2,\bar {x}_3]$ giving $\bar {d}_{pqr}= d^{[\bar110]}_{pqr}$ according to the general rule:
${d}^{110}_{ijk} =\Sigma \bar {d}^{\bar {1}10}_{pqr}a_{pi}a_{qj}a_{prk} $
where $a_{\lambda\mu}$ are the cosines between coordinates in the two frames, and the summation is over repeated indices.
In your particular coordinate transformation $[- x,y,z]$ to $[x,y,z]$ which transforms $\bar {d}_{pqr}$ to $d_{ijk}$ you need to bear in mind that $a_{pi}=\delta_{pi}$ etc so you get the general relationship with the $\delta$ symbols multiplying each other in each term of the summation . They are not exponents (powers) to base (-1).
The expression you have written doesn't seem to give the correct relationship between $d^{[110]}_{ijk}$ and $d_{pqr}=d^{[\bar {1}10]}_{pqr}$.
I hope this helps.
