Bounds for certain Freud-type orthogonal polynomials

AbstractLet wQ(x) = exp(−Q(x)) be a weight function and {Pn} the system of polynomials orthonormal with respect to wQ2 on R. We show that if Q satisfies certain technical conditions, then ¦WQ(x)pn(x)¦ ⩽ c1qn−12, for ¦x¦ ⩽ c2qn, n = 1, 2, 3, …, where c1, c2 are constants depending upon Q alone and qnQ′(qn) = n, n = 1, 2, …. The weights considered include exp(−¦x¦α) when α ⩾ 4. The proof involves the use of certain “infinite-finite range inequalities” to estimate the coefficients in a differential equation satisfied by pn. These estimates, in turn, enables us to use a concavity argument