  In this paper we extend the results of the research started by the first
author, in which Karlin-McGregor diagonalization of certain reversible Markov
chains over countably infinite general state spaces by orthogonal polynomials
was used to estimate the rate of convergence to a stationary distribution.
  We use a method of Koornwinder to generate a large and interesting family of
random walks which exhibits a lack of spectral gap, and a polynomial rate of
convergence to the stationary distribution. For the Chebyshev type subfamily of
Markov chains, we use asymptotic techniques to obtain an upper bound of order
$O({\log{t} \over \sqrt{t}})$ and a lower bound of order $O({1 \over
\sqrt{t}})$ on the distance to the stationary distribution regardless of the
initial state. Due to the lack of a spectral gap, these results lie outside the
scope of geometric ergodicity theory.
