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We construct an expansion of the solutions of the bi-confluent Heun equationin terms of the Hermite functions. The series is governed by a three-termrecurrence relation between successive coefficients of the expansion. Weexamine the restrictions that are imposed on the involved parameters in orderthat the series terminates thus resulting in closed-form finite-sum solutionsof the bi-confluent Heun equation. A physical application of the closed-formsolutions is discussed. We present the five six-parametric potentials for whichthe general solution of the one-dimensional Schr\"odinger equation is writtenin terms of the bi-confluent Heun functions and further identify a particularconditionally integrable potential for which the involved bi-confluent Heunfunction admits a four-term finite-sum expansion in terms of the Hermitefunctions. This is an infinite well defined on a half-axis. We present theexplicit solution of the one-dimensional Schr\"odinger equation for thispotential and discuss the bound states supported by the potential. We derivethe exact equation for the energy spectrum and construct an accurateapproximation for the bound-state energy levels.

Solutions of the bi-confluent Heun equation in terms of the Hermite functions