1. Introduction Let w be a weight function supported on L (a subset of ℝ) that has finite moments of all orders

With w(x) we associate the Hankel matrix, where i, j = 0 , … , n—1. The purpose of this paper is the determination of

for large n with suitable conditions on w. If L is a single interval, say [—1,1], then the asymptotic form of such determinants was computed by Szego [1918] and later by Hirschmann [1966] for quite general w.

Our main result is as follows. Suppose we replace w(x) by a function given in the form wo(x)U(x) where w0(x) is the weight e-xxv. Then for appropriate functions w, the determinants are given asymptotically as n → ∞ by

and G is the Barnes function (see Section 2).

In Section 2 we establish an identity relating Dn[w], Dn[wo], and a certain Predholm determinant and a description of the Predholm determinant from a “linear statistics” point of view. A computation of Dn[wo] is also included. Then in Section 3 the Coulomb fluid approach is used to compute the asymptotics of the Predholm determinant. This, along with the computation of Dn[wo] allows us to give a heuristic, Coulomb fluid derivation of the formula.