Máté, Attila; Nevai, Paul; Zaslavsky, Thomas Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights. (English) Zbl 0536.42023 Trans. Am. Math. Soc. 287, 495-505 (1985). Let \(p_ n(x)=\gamma_ nx^ n+..\). 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 that, writing \(a_ n=\gamma_{n-1}/\gamma_ n\), the expression \(a_ nn^{-1/\beta}\) has a limit as \(n\to \infty\). Assuming this conjecture, 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. Cited in 24 Documents MSC: 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:asymptotic expansion; combinatorial enumerations; Freud’s conjecture; Jensen’s identity; orthogonal polynomials PDFBibTeX XMLCite \textit{A. Máté} et al., Trans. Am. Math. Soc. 287, 495--505 (1985; Zbl 0536.42023) Full Text: DOI