  A unit evolution step of discrete-time quantum walks is determined by both a
coin-flip operator and a position-shift operator. The behavior of quantum
walkers after many steps delicately depends on the coin-flip operator and an
initial condition of the walk. To get the behavior, a lot of long-time limit
distributions for the quantum walks starting with a localized initial state
have been derived. In the present paper, we compute limit distributions of a
2-state quantum walk with a delocalized initial state, not a localized initial
state, and discuss how the walker depends on the coin-flip operator. The
initial state induced from the Fourier series expansion, which is called the
$(\alpha,\beta)$ delocalized initial state in this paper, provides different
limit density functions from the ones of the quantum walk with a localized
initial state.
