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An algebraic structure for one-dimensional quantum walks is introduced. Thisstructure characterizes, in some sense, one-dimensional quantum walks. Anatural computation using this algebraic structure leads us to obtain aneffective formula for the characteristic function of the transitionprobability. Then, the weak limit theorem for the transition probability ofquantum walks is deduced by using simple properties of the Chebyshevpolynomials.

An algebraic structure for one-dimensional quantum walks and a new proof of the weak limit theorem