Generalized Hunt-Dyson formula and Bohnenblust-Spitzer theorem

Abstract

Let m be a natural number, and assume that a1,…,am are real. Let n be an arbitrary natural power. Consider the maximum of zero and the accumulating sums a1, a1+a2, and up to a1+a2+…+am. We prove a combinatorial identity for this maximum raised to the power n and summed over the set of all permutations of the variables a1,…,am. We call this identity the generalized Hunt-Dyson formula (gHD). In the case of the power n=1 it reduces to the usual Hunt-Dyson formula (HD). In this paper, we derive the gHD from the renowned Bohnenblust-Spitzer combinatorial theorem (BSt). We give also a sketch of an independent proof of the gHD based on a generalization of Dyson's proof of the HD. After that we derive the BSt from the gHD providing the former with a new proof. The formula gHD is related to two problems in analysis. The first one is a computation of lower order asymptotic terms in a generalization of the strong Szegö limit theorem for a pseudodifferential operator on a Zoll manifold of any dimension, in particular for the standard sphere of any dimension. The second application of the gHD is a formula for an arbitrary order moment of the maximum of a random walk with independent identically distributed steps.