  In this work we consider open quantum random walks on the non-negative
integers. By considering orthogonal matrix polynomials we are able to describe
transition probability expressions for classes of walks via a matrix version of
the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for
simpler classes of examples, we consider path counting and the corresponding
combinatorial tools. A non-commutative version of the gambler's ruin is studied
by obtaining the probability of reaching a certain fortune and the mean time to
reach a fortune or ruin in terms of generating functions. In the case of the
Hadamard coin, a counting technique for boundary restricted paths in a lattice
is also presented. We discuss an open quantum version of Foster's Theorem for
the expected return time together with applications.
