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

Let ${p_n}(x) = {\gamma _n}{x^n} + \cdots$ denote the $n$th polynomial orthonormal with respect to the weight $\exp ( - {x^\beta }/\beta )$ where $\beta > 0$ is an even integer. G. Freud conjectured and Al. Magnus proved that, writing ${a_n} = {\gamma _{n - 1}}/{\gamma _n}$, the expression ${a_n}{n^{ - 1/\beta }}$ has a limit as $n \to \infty$. 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.