  The sextic oscillator is discussed as a potential obtained from the
bi-confluent Heun equation after a suitable variable transformation. Following
earlier results, the solutions of this differential equation are expressed as a
series expansion of Hermite functions with shifted and scaled arguments. The
expansion coefficients are obtained from a three-term recurrence relation. It
is shown that this construction leads to the known quasi-exactly solvable form
of the sextic oscillator when some parameters are chosen in a specific way. By
forcing the termination of the recurrence relation, the Hermite functions turn
into Hermite polynomials with shifted arguments, and, at the same time, a
polynomial expression is obtained for one of the parameters, the roots of which
supply the energy eigenvalues. With the $\delta=0$ choice the quartic potential
term is cancelled, leading to the {\it reduced} sextic oscillator. It was found
that the expressions for the energy eigenvalues and the corresponding wave
functions of this potential agree with those obtained from the quasi-exactly
solvable formalism. Possible generalizations of the method are also presented.
