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(α+γ), while the initial state is 1/√2(|0L〉+ i|0R〉). We verify the result, and obtain some symmetry properties of quantum walks with a U(2) coin with |0L〉 and |0R〉 as the initial state.