Appell expansions and general-to-confluent Heun degeneration in a bounded recovery diffusion

Corporate recovery under state-dependent uncertainty can be represented as first passage of a bounded latent financial-health diffusion between distress and recovery thresholds, with an additional killing channel for abrupt failure. When volatility is asymmetric across financial states and failure intensity is state dependent, the recovery functional satisfies a constrained general Heun equation. We specialize an origin-centered Appell F1 expansion to this family and derive a compact four-term recurrence in four dimensionless parameters. Probabilistic admissibility ensures convergence on every compact subset of the state interval in the principal parameter region. We then track the limit ξ → 0 (|a| → ∞) simultaneously through the differential equation, normalized expansion basis, and coefficient recurrence. Each Appell basis element converges to an incomplete-beta function, while the recurrence converges coefficientwise to the four-term relation of a Whittaker-Ince confluent-Heun equation. Only the additional removal of state dependence from killing gives the Gauss hypergeometric equation. Thus the hierarchy corresponds economically to removing, successively, volatility asymmetry and state dependence in abrupt-failure risk. Terminating and lower-term recurrences remain exceptional. Numerical calculations verify the specialized series and illustrate an approximately first-order approach of the bounded recovery solution to its confluent limit for the selected parameter set.