Closed-form solution for the first moment μ0(τ; κ) at general κ

Derive a closed-form expression for the first moment μ0(τ; κ) = ∫_{−∞}^{∞} exp(−x^6 + τ x^4 − κ τ^2 x^2) dx for general κ by solving the third‑order linear ordinary differential equation in τ given in equation (eq:mu0gen), extending beyond the special cases κ ∈ {1/4, 1/3, 0} where closed forms are provided.

Background

In Section 6, the authors consider the special reduction t = −κτ^2 and derive a third-order linear ODE in τ (equation (eq:mu0gen)) for the first moment μ0(τ; κ). They obtain explicit closed-form solutions in special cases κ = 1/4, κ = 1/3, and κ = 0, expressed in terms of modified Bessel or hypergeometric functions.

For general κ, despite presenting the governing ODE and its singularity structure, the authors do not have a closed-form solution and explicitly state this as an unresolved point.