  We study an infinite one-dimensional Ising spin chain where each particle
interacts only with its nearest neighbors and is in contact with a heat bath
with temperature decaying hyperbolically along the chain. The time evolution of
the magnetization (spin expectation value) is governed by a semi-infinite
Jacobi matrix. The matrix belongs to a three-parameter family of Jacobi
matrices whose spectral problem turns out to be solvable in terms of the basic
hypergeometric series. As a consequence, we deduce the essential properties of
the corresponding orthogonal polynomials, which seem to be new. Finally, we
return to the Ising model and study the time evolution of magnetization and
two-spin correlations.
