Hankel determinant and orthogonal polynomials for a perturbed Gaussian weight: From finite n to large n asymptotics

Abstract

We study the monic polynomials Pn(x; t), orthogonal with respect to a symmetric perturbed Gaussian weight function w(x)=w(x;t)≔e−x21+tx2λ,x∈R, with t>0,λ∈R. This problem is related to single-user multiple-input multiple-output systems in information theory. It is shown that the recurrence coefficient βn(t) is related to a particular Painlevé V transcendent, and the sub-leading coefficient p(n, t) of Pn(x; t) (Pn(x; t) = xn + p(n, t)xn−2 + ⋯) satisfies the Jimbo–Miwa–Okamoto σ-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by βn(t) and p(n, t), respectively. This enables us to obtain the large n full asymptotic expansions for βn(t) and p(n, t) with the aid of Dyson’s Coulomb fluid approach in the one-cut case [i.e., λt ≤ 1 (t > 0)]. We also consider the Hankel determinant Dn(t), generated by the perturbed Gaussian weight. It is found that Φn(t), a quantity allied to the logarithmic derivative of Dn(t) via Φn(t)=2t2ddtlnDn(t)−2nλt, can be expressed in terms of βn(t) and p(n, t). Based on this result, we obtain the large n asymptotic expansion of Φn(t) and then that of the Hankel determinant Dn(t) in the one-cut case.