  In this paper we present closed-form expressions for the wave function that
governs the evolution of the discrete-time quantum walk on a line when the coin
operator is arbitrary. The formulas were derived assuming that the walker can
either remain put in the place or proceed in a fixed direction but never move
backward, although they can be easily modified to describe the case in which
the particle can travel in both directions. We use these expressions to explore
the properties of magnitudes associated to the process, as the probability mass
function or the probability current, even though we also consider the
asymptotic behavior of the exact solution. Within this approximation, we will
estimate upper and lower bounds, consider the origins of an emerging
approximate symmetry, and deduce the general form of the stationary probability
density of the relative location of the walker.
