Closed-form solutions for higher moments μ2n(τ; κ) at general κ

Derive closed‑form expressions for the even moments μ2n(τ; κ) = ∫_{−∞}^{∞} x^{2n} exp(−x^6 + τ x^4 − κ τ^2 x^2) dx for general κ and n ≥ 0 by solving the third‑order linear ordinary differential equation in τ given in equation (eq:mungen), since closed forms are currently available only for κ ∈ {1/4, 1/3, 0}.

Background

Beyond μ0, the authors derive a third-order linear ODE in τ (equation (eq:mungen)) satisfied by the moments μ2n(τ; κ). They present exact closed-form solutions for the three special cases κ = 1/4, κ = 1/3, and κ = 0, but do not provide general κ solutions.

They explicitly note the absence of closed forms for general κ and highlight singularity structures of the ODE, indicating that obtaining solutions is nontrivial.