Asymptotic expansions of the recurrence coefficients for truncated Freud polynomials

Determine the full asymptotic expansions, as n→∞, of the recurrence coefficients a_n and b_n in the three-term recurrence relation x P_n(x) = P_{n+1}(x) + b_n P_n(x) + a_n P_{n-1}(x) for the monic orthogonal polynomials associated with the truncated Freud linear functional u_z defined by ⟨u_z, p⟩ = ∫_0^∞ p(x) e^{-z x^4} dx with z > 0.

Background

In Section 4 the authors derive Laguerre–Freud equations governing the recurrence coefficients a_n and b_n of the three-term recurrence for the monic orthogonal polynomials with respect to the linear functional u_z induced by the weight e^{-z x^4} on (0, ∞). Using an ansatz a_n ∼ A n^{r} z^{-1/2} and b_n ∼ B n^{s} z^{-1/4}, they identify the leading exponents r = 1/2, s = 1/4 and constants A = (140)^{-1/2}, B = 2⋅(140)^{-1/4}, yielding the leading-order behavior a_n ∼ √(n/(140 z)) and b_n ∼ 2 (n/(140 z))^{1/4}.

Following these leading-order results, the authors explicitly state that the problem of obtaining the full asymptotic expansions in n for a_n and b_n remains open, motivating further analysis beyond the leading term for these recurrence parameters in the truncated Freud setting.