Average position in quantum walks with a U(2) coin

We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position $<x>=\max (<x>\sin (\alpha +\gamma )$, while the initial state is $1\mathrm{/}\sqrt{2}(\mid 0L>+i\mid 0R>)$. We prove the result and get some symmetry properties of quantum walks with a U(2) coin with $\mid 0L>$ and $\mid 0R>$ as the initial state.