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K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products

Alexander Berkovich·Frank G. Garvan
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Abstract

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [17] and Garvan, Kim and Stanton [10]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.

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Alexander Berkovich, Frank G. Garvan. 2007-02-01. K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products. https://arxiv.org/abs/math/0702027

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