Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights

Let pₙ(x) = γ ₙxⁿ + ⋯ denote the nth polynomial orthonormal with respect to the weight exp ( - x^β /β ) where β > 0 is an even integer. G. Freud conjectured and Al. Magnus proved that, writing aₙ = γ n - 1/γ ₙ, the expression aₙn- 1/β has a limit as n → ∞. It is shown that this expression has an asymptotic expansion in terms of negative even powers of n. In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.