Generalised Freud's equation and level densities with polynomial potential

We study orthogonal polynomials with weight $\exp [-NV(x)]$, where $V(x)={\sum }_{k=1}^d{a}_{2k}{x}^{2k}\mathrm{/}2k$ is a polynomial of order 2d. We derive the generalised Freud’s equations for $d=3$, 4 and 5 and using this obtain $R_{\mu }=h_{\mu }\mathrm{/}{h}_{\mu -1}$, 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 $N\to \mathrm{\infty }$ are derived.