Finite Group Actions and Asymptotic Expansion of eP(z)

Footnote for page 8: For the form of Lagrange's theorem used here see for example 4, Abstract. We establish an asymptotic expansion for the number jHom(G; S n)j of actions of a nite group G on an n-set in terms of the order jGj = m and the number s G (d) of subgroups of index d in G for djm: This expansion and related results on the enumeration of nite group actions follow from more general results concerning the asymptotic behaviour of the coeecients of entire functions of nite genus with nitely many zeros. As another application of these analytic considerations we establish an as-ymptotic property of the Hermite polynomials, leading to the explicit determination of the coeecients C (; z) in Perron's asymptotic expansion for Laguerre polynomials in the cases = 1=2: Contents 1. Introduction