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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References
A. Brauer: On a problem of partitions,Amer. J. Math. 64 (1942), 299–312.
S. Chowla, I. N. Herstein, andW. K. Moore: On recursions connected with symmetric groups I,Can. J. Math. 3 (1951), 328–334.
S. Chowla, I. N. Herstein, andM. R. Scott: The solutions ofx d=1 in symmetric groups,Norske Vid. Selsk. 25 (1952), 29–31.
N. G. de Bruijn:Asymptotic Methods in Analysis, Dover Publ., New York, 1981.
A. Dress andT. Müller: Decomposable functors and the exponential principle,Adv. in Math. 129 (1997), 188–221.
W. K. Hayman: A generalization of Stirling's formula,J. Reine Angew. Math. 196 (1956), 67–95.
B. Harris andL. Schoenfeld: The number of idempotent elements in symmetric semigroups.J. Comb. Theory 3 (1967), 122–135.
B. Harris andL. Schoenfeld: Asymptotic expansion for the coefficients of analytic functions,Illinois J. Math. 12 (1968), 264–277.
T. Müller: Subgroup growth of free products,Invent. Math. 126 (1996), 111–131.
T. Müller: Subgroup growth of cyclic covers, preprint.
L. Moser andM. Wyman: On solutions ofx d=1 in symmetric groups,Can. J. Math. 7 (1955), 159–168.
L. Moser andM. Wyman: Asymptotic expansions,Can. J. Math. 8 (1956), 225–233.
G. Pólya: Über die Nullstellen sukzessiver Derivierten,Math. Z. 12 (1922), 36–60.
G. Szegő:Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. XXIII, Providence, Rhode Island, 1967.
W. Van Assche: Weighted zero distribution for polynomials orthogonal on an infinite interval,Siam J. Math. Anal. 16 (1985), 1317–1334.
H. S. Wilf: The asymptotics ofe P(z) and the number of elements of each order inS n Bull. Amer. Math. Soc. 15 (1986), 228–232.
E. T. Whittaker andG. N. Watson:A Course of Modern Analysis, Cambridge, 1958.
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Research supported by Deutsche Forschungsgemeinschaft through a Heisenberg-Fellowship.