  Starting with a quaternion difference equation with boundary conditions, a
parameterized sequence which is complete in finite dimensional quaternion
Hilbert space is derived. By employing the parameterized sequence as the kernel
of discrete transform, we form a quaternion function space whose elements have
sampling expansions. Moreover, through formulating boundary-value problems, we
make a connection between a class of tridiagonal quaternion matrices and
polynomials with quaternion coefficients. We show that for a tridiagonal
symmetric quaternion matrix, one can always associate a quaternion
characteristic polynomial whose roots are eigenvalues of the matrix. Several
examples are given to illustrate the results.
