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Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights. (English) Zbl 0536.42023

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.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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