  The subject of this paper is quantum walks, which are expected to simulate
several kinds of quantum dynamical systems. In this paper, we define
analyticity for quantum walks on Z. Almost all the quantum walks on
$\mathbb{Z}$ which have been already studied are analytic. In the framework of
analytic quantum walks, we can enlarge the theory of quantum walks. We obtain
not only several generalizations of known results, but also new types of
theorems. It is proved that every analytic space-homogeneous quantum walk on Z
is essentially a composite of shift operators and continuous-time analytic
space-homogeneous quantum walks. We also prove existence of the weak limit
distribution for analytic space-homogeneous quantum walks on Z.
