On Freud's equations for exponential weights (Q1089561)

Let \(\{p_ n\}^{\infty}_{n=0}\) be the sequence of orthonormal polynomials associated with the weight exp(-f(x)), \(x\in (- \infty,\infty)\), and let \(a_{n+1}p_{n+1}(x)=(x-b_ n)p_ n(x)-a_ np_{n-1}(x)\) be the corresponding three-term recurrence relation. For the case \(f(x)=| x|^{\alpha},\alpha >1\), Freud formulated a conjecture concerning the asymptotic behavior, as \(n\to \infty\), of the recursion coefficients \(a_ n\) and \(b_ n\). In this important paper the author proves Freud's conjecture when f(x) is a polynomial of even degree with positive leading coefficient. Extensions of the method used for the proof, which should lead to a proof in the general case, are suggested.