  We obtain a class of exact solutions of a Bessel-type differential equation,
which is a six-parameter linear ordinary differential equation of the second
order with irregular (essential) singularity at the origin. The solutions are
obtained using the Tridiagonal Representation Approach (TRA) as bounded series
of square integrable functions written in terms of the Bessel polynomial on the
real line. The expansion coefficients of the series are orthogonal polynomials
in the equation parameters space. We use our findings to obtain solutions of
the Schr\"odinger equation for some novel potential functions.
