Generalised Freud's equation and level densities with polynomial potential

Abstract

We study orthogonal polynomials with weight $\exp[-NV(x)]$, where $V(x)=\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\mu}=h_{\mu}/h_{\mu -1}$, where $h_{\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\rightarrow\infty$ are derived.