On recurrence coefficients for rapidly decreasing exponential weights

Let, for example,W fenced(x) = exp fenced(- exp_k fenced(1 - x²)^{- α}), x ∈ fenced(- 1, 1),where α > 0, k ≥ 1, and exp_k = exp fenced(exp fenced(... exp fenced())) denotes the kth iterated exponential. Let {} fenced(A_n) denote the recurrence coefficients in the recurrence relationxp_n fenced(x) = A_n p_{n + 1} fenced(x) + A_{n - 1} p_{n - 1} fenced(x)for the orthonormal polynomials {} fenced(p_n) associated with W². We prove that as n → ∞,frac(1, 2) - A_n = frac(1, 4) fenced(log_k n)^{- 1 / α} fenced(1 + o fenced(1)),where log_k = log fenced(log fenced(... log fenced())) denotes the kth iterated logarithm. This illustrates the relationship between the rate of convergence to frac(1, 2) of the recurrence coefficients, and the rate of decay of the exponential weight at ± 1. More general non-even exponential weights on a non-symmetric interval fenced(a, b) are also considered.