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