On the moments of roots of Laguerre-polynomials and the Marchenko–Pastur law

Abstract. In this paper we compute the leading terms in the sum of the \(k^{th}\) power of the roots of \(L_{p}^{(\alpha)}\), the Laguerre-polynomial of degree \(p\) with parameter \(\alpha\). The connection between the Laguerre-polynomials and the Marchenko–Pastur distribution is expressed by the fact, among others, that the limiting distribution of the empirical distribution of the normalized roots of the Laguerre-polynomials is given by the Marchenko–Pastur distribution. We give a direct proof of this statement based on the recursion satisfied by the Laguerre-polynomials. At the same time, our main result gives that the leading term in \(p\) and \((\alpha+p)\) of the sum of the \(k^{th}\) power of the roots of \(L_{p}^{(\alpha)}\) coincides with the \(k^{th}\) moment of the Marchenko–Pastur law. We also mention the fact that the expectation of the characteristic polynomial of a \(XX^T\) type random covariance matrix, where \(X\) is a \(p\times n\) random matrix with iid elements, is \(\ell^{(n-p)}_p\), i.e. the monic version of the \(p^{th}\) Laguerre polynomial with parameter \(n-p\).