  Biconfluent Heun equation (BHE) is a confluent case of the general Heun
equation which has one more regular singular points than the Gauss
hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by
a Nevanlinna theory (complex oscillation theory) approach, we have established
a theory of \textit{periodic} BHE (PBHE) in parallel with the Lam\'e equation
verses the Heun equation, and the Mathieu equation verses the confluent Heun
equation. We have established condition that lead to explicit construction of
eigen-solutions of PBHE, and their single and double orthogonality, and a
related first-order Fredholm-type integral equation for which the corresponding
eigen-solutions must satisfy. We have also established a Bessel polynomials
analogue at the BHE level which is based on the observation that both the
Bessel equation and the BHE have a regular singular point at the origin and an
irregular singular point at infinity on the Riemann sphere $\hat{\mathbb{C}}$,
and that the former equation has orthogonal polynomial solutions with respect
to a complex weight. Finally, we relate our results to an equation considered
by Turbiner, Bender and Dunne, etc concerning a quasi-exact solvable
Schr\"odinger equation generated by first order operators such that the second
order operators possess a finite-dimensional invariant subspace in a Lie
algebra of $SL_2(\mathbb{C})$
