  We use the "tridiagonal representation approach" to solve the
time-independent Schr\"odinger equation for bound states in a basis set of
finite size. We obtain two classes of solutions written as finite series of
square integrable functions that support a tridiagonal matrix representation of
the wave operator. The differential wave equation becomes an algebraic
three-term recursion relation for the expansion coefficients of the series,
which is solved in terms of finite polynomials in the energy and/or potential
parameters. These orthogonal polynomials contain all physical information about
the system. The basis elements in configuration space are written in terms of
either the Romanovski-Bessel polynomial or the Romanovski-Jacobi polynomial.
The maximum degree of both polynomials is limited by the polynomial
parameter(s). This makes the size of the basis set finite but sufficient to
give a very good approximation of the bound states wavefunctions that improves
with an increase in the basis size.
