Recently, the orthogonal polynomial techniques of Karlin and McGregor were extended in order to determine the rates at which stationary reversible Markov processes converge to their stationary distribution. In the current work, we investigate the other direction by considering well-known sets of orthogonal polynomials and adjusting them by attaching a point mass at unity using the techniques of Koornwinder. These new orthogonal polynomials will represent a class of recurrent nearest-neighbor Markov chains over non-negative integers. The spectral properties of these Markov chains are studied. In the absence of spectral gap, we obtain polynomial rates of convergence to stationarity. This presentation is based on joint work with Y. Kovchegov.