Alexander Berkovich

Affiliation: University of Florida

Email: alexb@ufl.edu

Title Of Talk: Partition inequalities

Abstract: It is a well known corollary of the celebrated Rogers-Ramanujan identities that the coefficients in $q$-series expansion of the difference of two infinite products $$ \frac{1}{(q,q^4;q^5)_\infty}-\frac{1}{(q^2,q^3;q^5)_\infty} $$ are all non-negative. Surprisingly, it is also true for finite products $$ \frac{1}{(q,q^4;q^5)_L}-\frac{1}{(q^2,q^3;q^5)_L}, $$ where $L$ is a positive integer.
In my talk I will discuss a simple injective argument due to Frank Garvan and myself that proves it.
I will also discuss a new theorem by George Andrews:
The $q$-series expansion of $$ \frac{1}{(q,q^5,q^6;q^8)_L}-\frac{1}{(q^2,q^3,q^7;q^8)_L} $$ for any positive integer $L$, has non-negative coefficients.
Finally, I will discuss my recent work with Keith Grizzell, where we proved that for any $L>0$ and any odd $y>1$, the $q$-series expansion of $$ \frac{1}{(q,q^{y+2},q^{2y};q^{2y+2})_L}-\frac{1}{(q^2,q^{y},q^{2y+1};q^{2y+2})_L} $$ has non-negative coefficients.

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Last update made Tue Oct 23 16:45:27 EDT 2012.
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