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The zeros of the random Laurent series $1/\mu - \sum_{j=1}^\infty c_j/z^j$,where each $c_j$ is an independent standard complex Gaussian, is known tocorrespond to the scaled eigenvalues of a particular additive rank 1perturbation of a standard complex Gaussian matrix. For the correspondingrandom Maclaurin series obtained by the replacement $z \mapsto 1/z$, we showthat these same zeros correspond to the scaled eigenvalues of a particularmultiplicative rank 1 perturbation of a random unitary matrix. Since thecorrelation functions of the latter are known, by taking an appropriate limitthe correlation functions for the random Maclaurin series can be determined.Only for $|\mu| \to \infty$ is a determinantal point process obtained. For theone and two point correlations, by regarding the Maclaurin series as the limitof a random polynomial, a direct calculation can also be given.

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices