Asymptotics of the coefficients of polynomials, asymptotic zero distribution and free probability

The asymptotic behavior of the coefficients of a sequence of polynomials is given under the conditions that all the zeros are real and positive and the zeros have an asymptotic zero distribution on $(0,\mathrm{\infty })$. We show how the limit for the coefficients is related to the Stieltjes transform of the asymptotic zero distribution and make a connection with the $S$-transform in free probability. We illustrate with some examples involving orthogonal and multiple orthogonal polynomials.