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Symmetric Functions & Probability Theory Seminar

A power series with positive coefficients

Siddhartha Sahi (Rutgers)

Location:  Hill 705
Date & time: Wednesday, 10 September 2025 at 10:45AM - 11:45PM

Date: 09/03/2025
Speaker: Siddhartha Sahi (Rutgers)

Title: A power series with positive coefficients 

Abstract: Consider the $n$-variable polynomial $P(x)=P(x_1,..,x_n)= (1-x_1-...-x_n) prod_{i=1}^n (1-x_i)$. We show that the power series expansion of $1-P(x)^a$ has positive coefficients for $0le ale 1/4$.

A similar positivity result was conjectured by H. Lewy and K. Friedrichs in the course of their work on the discretized wave equation, and proved by G. Szego. This result has been extended considerably over the years by Askey-Gasper, Gillis-Resnick-Zeilberger, Scott-Sokal and others, but these techniques do not seem to apply to our problem. 

We will explain the genesis of this problem, which has to do with the speaker's conjectural generalization of the FKG inequality; and give a self-contained proof, which involves a couple of new ideas.

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