Abstract
We establish an asymptotic expansion for the number |Hom (G,S n )| of actions of a finite groupG on ann-set in terms of the order |G|=m and the numbers G (d) of subgroups of indexd inG ford|m. This expansion and related results on the enumeration of finite group actions follow from more general results concerning the asymptotic behaviour of the coefficients of entire functions of finite genus with finitely many zeros. As another application of these analytic considerations we establish an asymptotic property of the Hermite polynomials, leading to the explicit determination of the coefficientsC ν(α;z) in Perron's asymptotic expansion for Laguerre polynomials in the cases α=±1/2.
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