, Volume 112, Issue 1, pp 301-325

Asymptotics of multinomial sums and identities between multi-integrals

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Abstract

We calculate the asymptotics of combinatorial sums ∑ α f(α)( α n ) β , whereα = (α 1, …,α h ) withα i =α j for certaini, j. Hereh is fixed and theα i ’s are natural numbers. This implies the asymptotics of the correspondingS n -character degrees ∑λ f(λ)d λ β . For certain sequences ofS n characters which involve Young’s rule, the latter asymptotics were obtained earlier [1] by a different method. Equating the two asymptotics, we obtain equations between multi-integrals which involve Gaussian measures. Special cases here give certain extensions of the Mehta integral [5], [6].

Supported by the Weizmann Institute of Science, Rehovot, Israel; by the Institute for Advanced Study, Princeton, New Jersey, USA; NSF grant number DMS 9304580; and by the Centre National de Recherche Scientifique, Lille, France.
This work was partially supported by an NSF grant number DMS 94-01197.