On Freud's equations for exponential weights

AbstractLet {pn}n = 0∞ be the sequence of orthonormal polynomials associated with the weight exp(−f(x)), x ϵ(−∞, ∞), where f is a polynomial of even degree with positive leading coefficient. The coefficients of the three-term recurrence relation an + 1 Pn + 1(x) = (x − bn) pn(x) − anpn − 1(x), are shown to be unique “admissible” solution of the equations Fn(a, b) = 0, n = 1, 2,…,Gn(a, b) = 0, n = 0, 1 2,…, already considered by Freud for f(x) = x2m. Using these equations, we prove an important special case of Freud's Conjecture. More precisely, we establish the asymptotic behaviour of {an} and {bn} for the weight exp(−f(x)). Further, we suggest extensions of the method used here, which should lead to a proof in the general case f(x) = ¦x¦α, α > 1