Asymptotic Expansions of Ratios of Coefficients of Orthogonal Polynomials with Exponential Weights

Let p,(x) = ynxn+ . . . denote the nth polynomial orthonormal with respect to the weight exp(-x8//3) where /3 > 0 is an even integer. G. Freud conjectured and Al. Magnus proved that, writing a, = y,,/y,, the expression ~ , n ' / ~ has a limit as n + w. 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 combinatonal identity of J. L. W. V. Jensen is explored. Consider the polynomials p, that are orthonormal with respect to the weight function exp(-lxlP/P) on the real line, where P is a positive real number. Denoting by yn the leading coefficient of p, (n 2 0) and writing a, = yn-,/y, for n > 1 and a, = 0 for n < 0, G. Freud conjectured that P 1 -'/P lim an/nl/P = n + m holds for every positive even P (see [3, Conjecture, p. 51; his conjecture has a slightly different form, as he considered the weight function lxlPexp(-lxlP) rather than the one above). He also entertained the possibility that thls conjecture is valid for all positive real p. In case /3 > 0 is even, he proved that if the limit on the left exists then it must have the value on the right-hand side (see [3, Theorem 1on p. 4]), and he established the conjecture for P = 2,4, and 6 (see [3, pp. 5-61). He accomplished these by extracting information from the formula Received by the editors October 30, 1983. 1980 Mathematics Subject Classification. Primary 42C05; Secondary 05A15, 05A19, 41A60.