Eigenfunction of LTI causal system Z-transform
Hint: $y[n] = H(e^{j (\pi / 2)}) x[n]$. 

Since $x[n]$ is a complex exponential, it is an eigenfunction of a LTI system with eigenvalue given by the frequency response at the frequency of $x[n]$ (which in this case is $H(e^{j (\pi/2)})$. In less fancy words, if you put a complex exponential into a LTI system, you get the same complex exponential out, except scaled by the transfer function evaluated at the frequency of the complex exponential.