Ali-Bulent Ekin

Affiliation: Ankara University

Email: ekin@science.ankara.edu.tr

Title Of Talk: Some relations among the components of the partition generating function

Abstract: Partition Generating function is $$ \sum_{n=0}^\infty p(n)q^n=\prod_{n=1}^\infty (1-q^n)^{-1}, $$ where $p(n)$ denotes the number of partitions of nonnegative integer $n$. Components are defined $$ {\mathbf P}^{(r)}:=q^r\sum_{n=0}^\infty p(mn+r)q^{mn}, $$ where $m\geq 2$ is positive integer and $r=0,1,2,\ldots, m-1$. Kolberg gave interesting relations among ${\mathbf P}^{(r)}$ when $m=2,3,5$ and $7$. Following Kolberg here we study the components when $m=11$ and $13$.

Atkin and Swinnerton-Dyer's method is based on expressing a power series as a polynomial whose coefficients are also power series. We follow the same way to get the components of the generating function of partitions in the cases mod 11 and 13. But, these are to be simplified to get a simple form and the congruence properties of the components. We explain how a certain subgroup of $\mbox{SL}_2(\mathbb{Z})$ acts on the components.

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