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

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