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 $X{X}^T$ type random covariance matrix, where $X$ is a $p\times n$ random matrix with iid elements, is ${\ell }_p^{(n-p)}$, i.e. the monic version of the $p^{th}$ Laguerre polynomial with parameter $n-p$.