In this paper the integrated three-valued telegraph process X = X(t), t ≧ 0 is examined. In particular, the third-order equations governing the distributions Pr{X(t) ≦ x | N(t) = k}, k ≧ 0, (where N(t) denotes the number of changes of the telegraph process up to time t) are derived and recurrence relationships for them are obtained by solving suitable initial-value problems. These recurrence formulas are related to the Fourier transform of the conditional distributions and are used to obtain explicit results for small values of k. The conditional mean values E{X(t) | N(t) = k, V(0) = j}, k ≧ 0, j = 1, 0, –1 (where V(0) denotes the initial velocity of motions) are obtained and discussed.
